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SanDisk Sansa MP3 Players/m200

Driver for Linux
Most modern distros support this product right out of the box, as a mass storage device (formatted as vfat). If you are going to use the device from both Linux and Windows, be sure to set your player to USB Mass Storage mode (see Changing USB mode below), or otherwise you may experience that music uploaded from one system will not be accessible from the other.

Music files (mp3 or wav format) may be uploaded to it (in the root folder and in user created folders) and will be found automatically by the player.

It is also possible to upload playlists, as follows:

 * 1) Upload the music files which will be referenced by the playlist to the player.  The advice below suggests placing the music in a "Music"folder, but this does not seem to be necessary.
 * 2) Create the playlist in your favorite music management program and save is as a M3U file (EXTM3U format).
 * 3) Edit the M3U file (it is just a text file) to eliminate paths from filenames.  That is, a line of the form "/path/to/my_file.mp3" should be replaced by just "my_file.mp3".
 * 4) Convert the newline format of the M3U file from Unix-style newlines to DOS-style newlines.  (There are scripts available to do this, such as 'unix2dos'.  Your favorite text-editor may also be able to do this.)
 * 5) Upload the M3U file to your player.  (Again location should not matter.)

Driver for windows 98 and windows ME
SanDisk does not support windows 98 for the m200 series of players, so they will not make a driver available. The e100 series driver does not work.

Refer to a discussion on CNET about the need for a win98 driver, where a working driver can be found. Someone has made a generic driver for USB mass storage device class that works with the m200 series. Both links lead to the same patch.

If you've got a windows 98 language version other than English, or have Windows ME, you may need to read the whole thread. It should be possible to use the driver, but it requires special care. Some sub versions have been made, so read the whole thread.

Changing USB mode
If you want to set your player to USB Mass Storage mode, try this: Menu button - Settings - USB - MSC There might be a difference in firmware versions (Europe and USA have different firmware), so check if yours is the same.

The American version (USA) has a higher volume setting and Fm reception while the E(Europe version) Has no Fm reception and has a volume Cap On It.

Playlist creation
There's a post on the Post No Bills blog with a response from Sandisk about how you can create playlists.

One approach is to put all the MP3 files in one directory (Music), and create playlists that refer to those files.

The problem is that one directory is hard to maintain when you want to find files and delete them. The m200 firmware is able to read mp3 ID3 tags versions 1 and 2, including the track number field. For best results, you should use mp3 ID3 tags version 2 and use the track number field, so the original track order is preserved.

Solution 1
In the Sansa drive in Windows Explorer select the songs that you want in the playlist. Right click and create playlist. You can do minimal playlist editing through the playlist file properties. Drag files to the references tab to add files.

There is a free playlist creator which creates playlists for each folder on the sansa player with one click. it is written in .NET and does not alter the music files (http://www.infosys.tuwien.ac.at/staff/cp/playlist_creator.zip).

There is also one written in Python that should work on both linux and Windows (http://people.ifm.liu.se/yohell/make_playlists.py).

Solution 2
Try Musicmatch Jukebox or MediaMonkey. Both of those should be able to create a music library using only files from the device. Then sort them by artist and album, and delete them from the library, deleting the files as well. Transfer new files to the device as files (MSC USB mode on the player), not by syncing. You can create a new music library on the fly if you've made extensive changes. It doesn't take much time. Playlists should not contain drive letters or folder names, so edit them in Notepad, unless you have an app that makes them on the fly (suggestions?).

You can keep your main database in MediaMonkey, and open the Sansa device (see the plus and click on it). You'll be able to delete files from there. But if you double click on a song on the Sansa device, the wrong song will play. Dunno why, but that's what happens on my computer. But deleting files works.

Solution 3
Rename all your tracks so artist (look for batch renaming feature in your software. Add example of freeware editor?), album and song name is in the file name. That won't be visible in the player, but can aid administration if you use the player in MSC mode.

Solution 4
A how-to for using Windows Media Player and syncing to make playlists can be found here.

Solution 5
The following solution appears to allow you to create an extended m3u (EXTM3U) playlist file using Windows Media 9 and allows you to keep album directories. It is unclear what the maximum file and folder count are on this device.

 * 1) SanDisk m200 is set to auto-detect USB mode (MTP).  Also tested and verified on m250.
 * 2) Create the "MUSIC" directory on the SanDisk m200, if it's not already there.
 * 3) Copy all music files and their respective album directories into the "MUSIC" directory of the device.
 * 4) Run Windows Media 9.  Click on Media Library on the left navigation.  Click on the album node in the left window pane.  Click on the Add button near the top of the application window and then click "add from directory" from the resulting  menu.  You are going to add albums to your computer's Media Library from your m200.  Do not load files from your computer.  Add each directory you want to the Media Library.  They will appear as albums in the left window pane.  Windows Media 9 will find all the files within that directory.
 * 5) Click on the Playlist node in left window pane.  Click the Playlists button near the top of the application window and create a new playlist.
 * 6) Now click on the album you want in the left window pane.  You will see the files belonging to that album in the right window pane.  Drag and drop the file from the right window pane over to the playlist.  That will add the file to the playlist.  Keep doing this until you have the files you want.
 * 7) When you are done, click on the playlist to highlight it.  Then, click File menu and click the Save Playlist As option. Save the playlist to the m200 in the "MUSIC" directory as a m3u playlist (you will need to change the filetype dropdown in the Save File Dialog window).  Windows Media Player 9 will save as an EXTM3U file.
 * 8) This m3u playlist is a text file that you can view in a text editor if you want to check if the files are listed correctly.
 * 9) Disconnect the m200 from the computer.  The m200 will rebuild its internal database.
 * 10) Go through the m200 menus and select the Play Music menu.  Then, pick the Playlists option.  You should see your playlist.

You can add more playlists to the m200, and they will show up individually. This will allow you to make multiple playlists to suit the occasion.

Note: The m200 does not seem to recognize the regular M3U format. The music library software must be able to save playlists as EXTM3U format. It may be possible to use another program besides Windows Media Player to create playlists. One popular program is Winamp, which creates correct playlists based on the saving directory (saving the playlist directly onto the Sansa device will give correct track paths).

Solution 6
Use Winamp. Select SanDisk m200 on left pane of Media Library under Portables. Then select the songs in the media library which you want on the playlist. Right click to save the playlist.

 * 1) Connect Sansa in mtp mode
 * 2) Copy all the mp3 files in any subfolder of 'music' folder of sansa say 'rock' subfolder created by you.
 * 3) After all mp3 files are copied, select all these by pressing shift key and highlighting the icons.
 * 4) Once all selected,finally right click- you will get option of create playlist—that's it! You can create and rename playlist as you like. But you have to have winamp 5.5 installed on your PC.

Solution 7
An MSC format playlist editing trick: The m3u format that works for me is very simple:

 * #EXTM3U
 * 1st_song.mp3
 * 2nd_song.mp3
 * 3rd_song.mp3

etc.

Note that there must be a blank line after each song. Note that there must be two blank lines after the EXTM3U line. Also note that the folder/path names must not be included in the file name. The sansa will automatically search through all the folders and find the file names.

Copy the playlist to the top folder/directory of the player (MSC format) (or into the album's folder).

A big benefit of playlists for me is that songs will play in the order specified in the playlist (unless shuffle is on)!

Use a standard text editor like Notepad. If you use custom text editors, make sure they save in LF/CR (DOS) format.

Note 1: If the filename has only numbers in it (eg. 40.mp3), the playlist will cut off at that song. Rename the file to begin with a letter (eg. forty.mp3) and update the playlist.

Note 2: If the Sansa has a problem with the playlist, it will stop reading it in, but will remember all the songs right up to where it found a problem (got confused). Use this knowledge to view the playlist on the Sansa and quickly track down the problem song and fix it in the playlist (this is how I discovered Note 1 above).

Solution 8
A playlist editor that lets you actually create the list and name it. Written in Win32 C++. Only been tested in MSC mode. NOTE: You need to have the folder 'Playlists' in the same folder as your music folder.

Solution 9
This one is the most easy and most reliable. A Program that creates Playlists from the directory structure of your mp3 player, without fiddling around wit all that stupid media player stuff...

Download it at http://web.archive.org/web/20070304144931/http://www.infosys.tuwien.ac.at/Staff/cp/playlist_creator.zip if you want, its free ;-)

Solution 10
I wrote a little .net program to make it easy to create playlists for your sansa. Creates playlists in the same way as playlist_creator above, but allows you to choose what songs go in a playlist, their order, etc. Currently only works in MSC mode, but I'm working on adding MTP support. The program is here http://www.randomphotos.org/SansaPlaylists.zip, and the source is here http://www.randomphotos.org/SansaPlaylists_source.zip.

Solution 11
Place the following script into the root directory of the Sansa player, and run it (i.e. double-click it). Bash is included in OS X and nearly every flavor of Linux. For Windows, you'll need bash, find, sort, and a few others. It would be best to install Cygwin or use one of the other methods above (or switch to a Unix-like OS).

 * 1) !/bin/bash

 * 1) Written by Jason Gyorog
 * 2) Place this script on the device, at the root level.
 * 3) Any directories with mp3 files will have playlists created.

cd "`dirname "$0"`" # Set working directory to script directory.

find * -type d -depth 0 | while read DIR; do       find "$DIR" -iname [^.]*.mp3 | sort | while read TRACK; do                basename "$TRACK" done > playlist.temp

if [ -s playlist.temp ]; then echo -e "#EXTM3U\r\n\r" > "$DIR".m3u while read TRACK; do                       echo -e "\r\n$TRACK\r" >> "$DIR".m3u done < playlist.temp fi done

rm playlist.temp

Alternative to playlists
Some prefer to use custom genres in the ID3 tag to create mixes for workout, relaxation or other moods. It's more flexible than playlists and easier to add to. This should work with the e100 series as well, but it's reportedly buggy in that respect.

Playing tracks in order
The m200 series rely on ID3 (Version 1) tags for sorting songs. Most people hit Play All when they've chosen an album. If you do that, the songs will play in alphabetical order. But if you want to play the songs in track order, scroll down to the first song and hit play instead. Make sure shuffle is off. The m200 series was previously believed to not be track tag aware. Thanks to Chip_Leader for the solution.

The m240 (latest firmware) is track tag aware and should play album tracks in track order. However it will ignore both v2 and v1 if any of the MP3s in the Album list have invalid characters in the track tag. The tag must be numeric. For example, if you choose the "Track X of X" option when tagging in ID3TagIt, the software will enter "1/12" for the first track of twelve into the MP3 track tag. The Sansa will ignore this tag because of the "/" and will sort by track name instead, even if the v1 ID3 tag has a valid value in it and even if only one MP3 in the list has an invalid character in the track tag.

Alternatively, use a playlist. If shuffle is off, playlists will play in order.

Audio Books
Extra care should be given when using Audio Books, since chapters could easily play out of sequence. Batch rename the ID3 tags for song title to reflect the numbers from the file name, or create playlists. Some use several folders (directories) for easier administration of audio books, and batch rename. I'd actually consider using something like Mp3splt to split into smaller files, if you go back and forth between music and audio books. Audio book files are often way too long.

File limits
The FAT files system used has some file limits. One report appears to say that the limit is 130-140 files in the root directory.

Unknown Album and or Unknown Artist
There is a known issue regarding these players’ tendency to display “unknown artist” or “unknown album” even when Windows Explorer, or Media Player clearly reflect the correct artist and or album of a particular song when the unit is connected via the USB port. This issue appears to occur randomly and even the most recent beta firmware (version 2.2.5) has not resolved it.

See these threads. CNET MP3 players Forums Ecoustics.com/bbs/messages

Workaround in this thread: http://forum.ecoustics.com/bbs/messages/15118/209222.html

This issue is actually related to the number of files stored on the device. When using MSC mode every 100'th file uploaded to the device will appear as unknown artist. This appears to be some sort of off by one error in either the ID3 parsing routine or the database creation routine. Breaking large groups of files across multiple directories does not appear to solve this problem.

There is also an issue with the order in which WMP11 (and other converters?) stores the tags when converting from MP3. The version one M200 (and maybe others?) can't read the artist or title tags unless they come before all others as they are in a different format from all other tags except the comment tag. See this thread in the SanDisk user forum.

Comparing carefully files with a tag editor (I use Mp3tag) that came up as "unknown" and ones that came up correctly I found that all the files in the "Unknown" category had data in the "Album Artist" field and the ones that displayed correctly did not. I deleted the data in the "Album Artist" field and reloaded the files onto my player an they all appeared correctly after that.

Hidden files
A directory with the name "radio" seems to have some special significance. Files placed in a directory with such a name are not indexed by (or hidden from) the player.

title: SanDisk Sansa MP3 Players/m200


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User:Vojtech Huser~enwikibooks

Medical Informatics

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Vojtech Huser~enwikibooks


text: Template:Col-begin

title: Template:Col-begin


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French/Grammar/Moods

List of Moods

 * L'Indicatif (The Indicative Mood)
 * L'Impératif (The Imperative Mood)
 * Le Conditionnel (The Conditional Mood)
 * Le Subjonctif (The Subjunctive Mood)

Francés/Gramática/El verbo/Modos

title: French/Grammar/Moods


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MATLAB Programming/Boolean and Rational

Introduction
A large number of MATLAB's functions are operations on two types of numbers: rational numbers and boolean numbers.

Rational numbers are what we usually think of when we think of what a number is. 1, 3, and -4.5 are all rational numbers. MATLAB stores rational numbers as doubles by default, which is a measure of the number of decimal places that are stored in each variable and thus of how accurate the values are. Note that MATLAB represents irrational numbers such as pi with rational approximations, except when using the symbolic math toolbox. See that section for details.

Boolean numbers are either "TRUE" or "FALSE", represented in MATLAB by a 1 and a 0 respectively. Boolean variables in MATLAB are actually interchangable with doubles, in that boolean operators can be performed with arrays of doubles and vice versa. Any non-zero number in this case is considered "TRUE".

Most of the rational operators also work with complex numbers. Complex numbers; however, cannot be interchanged with boolean values like the real rationals can.

Note: MATLAB refers to Booleans as "logicals" and does not use the word "Boolean" in code or documentation.

Rational Operators on Single Values
MATLAB has all the standard rational operators. It is important to note, however, that Unless told otherwise, all rational operations are done on entire arrays, and use the matrix definitions. Thus, even though for now we're only talking about operations on a single value, when we get into arrays, it will be important to distinguish between matrix and component-wise multiplication, for example: Add, Subtract, Multiply, Divide, Exponent operators.

The modulo function returns the remainder when the arguments are divided together, so a modulo b means the remainder when a is divided by b. All of these functions except for the modulus work for complex numbers as well.

Relational Operators
Equality '==' returns the value "TRUE" (1) if both arguments are equal. This must not be confused with the assignment operator '=' which assigns a value to a variable. Note that in the first case, a value of 1 (true) is returned, however for the second case a gets assigned the value of b.

Greater than, less than and greater than or equal to, less than or equal to are given by >, <, >=, <= respectively. All of them return a value of true or false. Example:

Boolean Operators on Single Values
The boolean operators are & (boolean AND) | (boolean OR) and ~ (boolean NOT /negation). A value of zero means false, any non-zero value (usually 1) is considered true.

Here's what they do: The negation operation in MATLAB is given by the symbol ~, which turns any FALSE values into TRUE and vice versa: This is necessary because conditionals (IF/SWITCH/TRY) and loops (FOR/WHILE) always look for statements that are TRUE, so if you want it to do something only when the statement is FALSE you need to use the negation to change it into a true statement.

The NOT operator has precedence over the AND and OR operators in MATLAB unless the AND or OR statements are in parenthesis:

title: MATLAB Programming/Boolean and Rational


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Template:Recipe summary/doc

Usage

 * All parameters to the template are optional.
 * The "Category" tries to resolve to an actual category, but does not include the recipe in that category.
 * Difficulty is between 1 and 5 inclusive
 * The difficulty includes the recipe in one of the following categories:
 * Category:Very Easy recipes
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Named parameters, All values

Positional parameters, All values
Note that some parameters should not have any newlines or spaces around them when using positional parameters as it will break the template.

Named parameters, Partial values

Positional parameters, Partial values, Subsequent positional parameters

Positional parameters, Partial values, No subsequent parameters
If you do not need later positional parameters, you can omit the "|"s.

Positional parameters, Partial values, Named subsequent parameters
As above, if you do not need later positional parameters, you can omit the "|"s. You may wish to include these via named parameters for clarity or brevity:

title: Template:Recipe summary/doc


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Template:Helpful hint

This template can be used to display a hint to the reader of a book. The title and the body of the hint can be customized.

title: Template:Helpful hint


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Scouting/BSA/Moviemaking Merit Badge

'''BSA announced in October, 2013, that the "Cinematography" merit badge has been renamed "Moviemaking" to better describe the purpose and intent of the badge. BSA changed the ID number from 126 to 156. even though the requirements are so similar to those for Cinematography, scouts should still earn this badge.''

Requirement 1
Discuss and demonstrate the proper elements of a good motion picture. In your discussion, include visual storytelling, rhythm, the 180-axis rule, camera movement, framing and composition of camera shots, and lens selection.

Requirement 2
Do the following:

 * A. In a three- or four-paragraph treatment, tell the story you plan to produce, making sure that the treatment conveys a visual picture

 * B. Prepare a storyboard for your motion picture (This can be done with rough sketches and stick figures.)

 * C. Demonstrate the following motion picture shooting techniques.
 * 1. Using a tripod
 * 2. Panning a camera
 * 3. Framing a shot
 * 4. Selecting an angle
 * 5. Selecting proper lighting
 * 6. Hand-held shooting

 * D. Using motion picture techniques, plan ONE of the following programs. Start with a treatment, and complete the requirement by presenting this program to a pack or your troop, patrol, or class.
 * 1. Film or videotape a court of honor and show it to an audience.
 * 2. Create a short feature of your own design using the techniques you learned.
 * 3. Shoot a vignette that could be used to train a new Scout in a Scouting skill.

Requirement 3
Do ONE of the following:

 * A. With your parent's permission and your counselor's approval, visit a film set or a television production studio and watch how production work is done.

 * B. Explain to your counselor the elements of the zoom lens and three important parts.

Requirement 4
Find out about three career opportunities in moviemaking. Pick one and find out about the education, training, and experience required for this profession. Discuss this career with your counselor. Explain why this profession might interest you. I picked director, editor, and producer.

title: Scouting/BSA/Moviemaking Merit Badge


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Straw Bale Construction/All-straw vaulted building

[All-straw vaulted studio in Joshua Tree, California http://www.skillful-means.com/projects/vault/The%20Vault.html]


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Acoustics/Fundamentals of Room Acoustics

Introduction
Three theories are used to understand room acoustics :

 * 1) The modal theory
 * 2) The geometric theory
 * 3) The theory of Sabine

The modal theory
This theory comes from the homogeneous Helmoltz equation $$\nabla ^2 \hat \Phi + k^2 \hat \Phi  = 0$$. Considering a simple geometry of a parallelepiped (L1,L2,L3), the solution of this problem is with separated variables :

$$P(x,y,z)=X(x)Y(y)Z(z)$$

Hence each function X, Y and Z has this form :

$$X(x) = Ae^{ - ikx} + Be^{ikx}$$ With the boundary condition $$\frac = 0 $$, for $$x=0$$ and $$x=L1$$ (idem in the other directions), the expression of pressure is :

$$P\left( {x,y,z} \right) = C\cos \left( {\frac } \right)\cos \left( {\frac } \right)\cos \left( {\frac } \right) $$

$$k^2 = \left( {\frac} \right)^2  + \left( {\frac} \right)^2  + \left( {\frac} \right)^2$$

where $$m$$,$$n$$,$$p$$ are whole numbers

It is a three-dimensional stationary wave. Acoustic modes appear with their modal frequencies and their modal forms. With a non-homogeneous problem, a problem with an acoustic source $$Q$$ in $$r_0$$, the final pressure in $$r$$ is the sum of the contribution of all the modes described above.

The modal density $$\frac$$ is the number of modal frequencies contained in a range of 1 Hz. It depends on the frequency $$f$$, the volume of the room $$V$$ and the speed of sound $$c_0$$ :

$$\frac \simeq \fracf^2$$

The modal density depends on the square frequency, so it increase rapidly with the frequency. At a certain level of frequency, the modes are not distinguished and the modal theory is no longer relevant.

The geometry theory
For rooms of high volume or with a complex geometry, the theory of acoustical geometry is critical and can be applied. The waves are modelised with rays carrying acoustical energy. This energy decrease with the reflection of the rays on the walls of the room. The reason of this phenomenon is the absorption of the walls.

The problem is this theory needs a very high power of calculation and that is why the theory of Sabine is often chosen because it is easier.

Description of the theory
This theory uses the hypothesis of the diffuse field, the acoustical field is homogeneous and isotropic. In order to obtain this field, the room has to be sufficiently reverberant and the frequencies have to be high enough to avoid the effects of predominating modes.

The variation of the acoustical energy E in the room can be written as :

$$\frac = W_s - W_{abs}$$

Where $$W_s$$ and $$W_{abs}$$ are respectively the power generated by the acoustical source and the power absorbed by the walls.

The power absorbed is related to the voluminal energy in the room e :

$$W_{abs} = \frac{4}a$$

Where a is the equivalent absorption area defined by the sum of the product of the absorption coefficient and the area of each material in the room :

$$a = \sum\limits_i {\alpha _i S_i }$$

The final equation is : $$V\frac = W_s - \frac{4}a$$

The level of stationary energy is : $$e_{sat} = 4\frac$$

Reverberation time
With this theory described, the reverberation time can be defined. It is the time for the level of energy to decrease of 60 dB. It depends on the volume of the room V and the equivalent absorption area a :

$$T_{60} = \frac{a} $$ Sabine formula

This reverberation time is the fundamental parameter in room acoustics and depends trough the equivalent absorption area and the absorption coefficients on the frequency. It is used for several measurement :

 * Measurement of an absorption coefficient of a material
 * Measurement of the power of a source
 * Measurement of the transmission of a wall


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Statistics/Numerical Methods/Random Number Generation

Definition and Usage
Random Numbers  →    ”A sequence of integers or group of numbers which show absolutely no relationship to each other anywhere in the sequence. At any point, all integers have an equal chance of occurring, and they occur in an unpredictable fashion”

Many statistical methods rely on random numbers. The use of random numbers, however, has expanded beyond random sampling or random assignment of treatments to experimental units. More common uses now are in simulation studies of physical processes, of analytically intractable mathematical expressions, or of a population resampling from a given sample from that population. These three general areas of application are sometimes called simulation, Monte Carlo, and resampling.

Uniform Random Number Generation
Given an infinite sequence $$R_0, R_1,. . ., R_n,. . .$$ most people’s notion of random would include that the numbers be uniformly distributed in the interval (0, 1). We denote this distribution by U(0, 1). I will present in this paper the two ways of generating Uniform Random Numbers : Linear Congruential Generators and Tausworthe Generators.

Linear Congruential Generators
D. H. Lehmer in 1948 proposed the linear congruential generator as a source of random numbers. In this generator, each single number determines its successor. The form of the generator is                       $$ X_{i+1} $$ = (a$$X_i$$ + c) mod m  , with  0 ≤ $$X_i$$ ≤ m.

m is called modulus. $$X_0$$, a, and c are known as the seed, multiplier, and the increment respectively.

For example, consider m = 31, a = 7, c = 0 and begin with $$X_0$$ = 19. The next integers in the sequence are 9, 1, 7, 18, 2, 14, 5, 4, 28, 10, 8, 25, 20, 16, 19,

so, of course, at this point the sequence begins to repeat.

If we would have taken a = 3 instead of a = 7, we would have got:

26, 16, 17, 20, 29, 25, 13, 8, 24, 10, 30, 28, 22, 4, 12, 5, 15,              14, 11, 2, 6, 18, 23, 7, 21, 1, 3, 9, 27, 19

From the simple example above we can guess that modulus, multiplier, and increment play a role in the period length of the linear congruential generator.

The Period  → The period is the smallest positive integer λ for which $$X_{\lambda}=X_{0}$$. The period can be no greater than m. Therefore, m is chosen to be equal or nearly equal to the largest representable integer in the computer to get a long period.

A full period generator is one in which the period is m, and it is obtained iff:

1. c is relatively prime to m; 2. (a-1) is a multiple of q, for each prime factor q of m; 3. (a-1) is a multiple of 4, if m is.

The Increment →

If c > 0, we can achieve a full period by such :

1- m = $$2^b$$ → faster computer arithmetic,

2- Set (a-1) as a multiple of 4,

3- c should be odd-valued,

4- Set b as high as possible. For example b=31 in a 32-bit computer.

If c = 0, a full period is not possible. A maximum number of random variables, then, can be achieved by such :

1-A maximum period generator, with λ = m − 1, is one in which a is a primitive element modulo m, if m is prime. i. a mod m 6= 0 ii. a(m−1)/q modm 6= 1, for each prime q of (m-1)

2-Given preceding comments, m is often set to the largest prime number less than 2b. The most commonly used modulus is the Mersenne prime $$2^{31}-1$$.

maximum period → m = prime, a = primitive element modulo m

Structure in the Generated Numbers
The idea behind the generated random number is that the numbers should be really random. That means that the numbers should appear to be distributionally independent of each other; that is, the serial correlations should be small. How bad the structure of a sequence is (that is, how much this situation causes the output to appear nonrandom) depends on the structure of the lattice.

Consider the output of the generator with m =31 and a =3 that begins with $$x_0$$=19. Plot the successive pairs (27, 19), (19, 26), (26, 16)...

As it can be seen easily from the Figure 1.2, the successive pairs of random numbers lie only on three lines. The generated numbers does not seem to be random. They rather seem to be correlated. From a visual perspective we can conclude that a generator with small number of lines does not cover the space well and has a bad lattice structure.

MacLaren and Marsaglia (1965) suggest that the output stream of a linear congruential random number generator be shuffled by using another generator to permuate subsequences from the original generator.

By this way the period can be increased and the shuffling of the output can also break up the bad lattice structure.

Bays-Durham Shuffling of Uniform Deviates
Bays and Durham suggest using a single generator to fill a table of length k and then using a single stream to select a number from the table and replenish the table. After initializing a table T to contain $$x_1,x_2,...,x_k,$$ set i = k+1 and generate $$x_i$$ to use as an index to the table. Then update the table with $$x_{i+1}$$.

For example, with the generator used in Figure 1.3, which yielded the sequence

27, 19, 26, 16, 17, 20, 29, 25, 13, 8, 24, 10, 30, 28, 22, 4, 12, 5, 15,                           14, 11, 2, 6, 18, 23, 7, 21, 1, 3, 9,

we select k = 8, and initialize the table as                                 27, 19, 26, 16, 17, 20, 29, 25.

We then use the next number, 13, as the first value in the output stream, and also to form a random index into the table. If we form the index as 13 mod8 + 1, we get the sixth tabular value, 20, as the second number in the output stream. We generate the next number in the original stream, 8, and put it in the table, so we now have the table 27, 19, 26, 16, 17, 8, 29, 25.

Now we use 20 as the index to the table and get the fifth tabular value, 17, as the third number in the output stream. By continuing in this manner to yield 10,000 deviates, and plotting the successive pairs, we get Figure 1.6.

Tausworthe Generators
Tausworthe (1965) introduced a generator based on a sequence of 0`s and 1`s generated by a recurrence of the form

$$b_{i}\equiv (a_{p}b_{i-p}+a_{p-1}b_{i-p+1}+...+a_{1}b_{i-1})\textit{mod} 2$$

where all variables take on values of either 0 or 1.

For computational efficiency, most of the a`s in the equation set to be zero. Then we get,

$$b_{i}\equiv (b_{i-p}+b_{i-p+q})\textit{mod} 2$$

After this recurrence has been evaluated a sufficient number of times, say l, the l-tuple of a`s is interpreted as a number in base 2. This is referred to as an l-wise decimation of the sequence of a`s.

As an example take a primitive trinomial modulo 2,

x4 + x + 1,

and begin with the bit sequence 1, 0, 1, 0.

For this polynomial, p = 4 and q = 3 in the recurrence. Operating the generator, we obtain

1, 1, 0, 0,1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0,

at which point the sequence repeats. A 4-wise decimation yields the numbers

12, 8, 15, 5, …

As with the linear congruential generators, different values of the c`s and even the starting values of the a`s can yield either good or bad generators.

Goodness-of-Fit Tests

In this part, I will introduce you the basic Goodness-of-Fit Tests, with which we can evaluate the sequence of random number we created using the above mentioned methods.

Chi-Squared Goodness-of-Fit Tests
$$\chi^{2}=\sum^{k}_{i=1}\frac{(o_{i}-e_{i})^{2}}{e_{i}}$$

→ The null hypothesis: A random variable has a uniform (0,1) distribution

→ Count the number of observation in each of the ten intervals

(0, 0.1], (0.1, 0.2], ..., (0.9, 1.0)

→ Compare those counts with the expected numbers.

→ If the observed numbers are significantly different from the expected numbers, we have reason to reject the null hypothesis.

The Kolmogorov-Smirnov Test
This test compares the empirical cumulative distribution function (c.d.f) $$\hat{F}(.)$$ with a theoretical c.d.f F(.).

→ H0: F(x) = x, 0 _ x < 1

→ Rank the observation so that $$R^{(1)}\leq R^{(2)} ... \leq R^{n}$$

$$\hat{F}(x)=\frac{i}{n}$$     ,and           $$R^{(n+1)}\equiv1$$

→ where $$R^{(0)}\equiv0$$     and            $$R^{(n+1)}\equiv1$$.

→ The test statistic measures the size of the largest difference between these two:

$$D_{n}= max_{0\leq x<1}$$ $$\left|\hat{F}(x)- F(x)\right|$$

Linear Dependence Test
→One way to test for dependencies between numbers in a sequence is to restrict such examination to linear dependence between observations which are separated by k numbers.

→ Given a realization of n random numbers $$R_{0},R_{1},\ldots,R_{n}$$ the sample covariance of lag k is

$$C_{k}=(n-k)^{-1}\sum^{n-k}_{i=1}(R_{i}-\frac{1}{2})(R_{i+k}-\frac{1}{2}).$$

→ Under randomness $$E[C_{k}]=0$$.


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False Friends of the Slavist/Map babka

babka

title: False Friends of the Slavist/Map babka


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Windows Programming/Driver Structure

Drivers, unlike user-mode programs, have a number of different entry points, and each entry point is specifically designed to handle a single type of interface. In this chapter, we will talk about the required interfaces, and we will also talk about adding additional interfaces.

DRIVER_OBJECT
Driver routines all receive a pointer to a DRIVER_OBJECT structure. The driver must fill in the appropriate fields of this structure, so that Windows can learn the whereabouts of the different access routines, and any other information about the driver.

title: Windows Programming/Driver Structure


text:
False Friends of the Slavist/Map bьrati

bьrati

title: False Friends of the Slavist/Map bьrati


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Windows Programming/Driver API

It should come as no surprise that kernel mode drivers cannot use the same functions as user-mode applications. Device Drivers, for this reason, must rely on the available kernel mode functions. This page will talk about some of the available libraries and functions that can be utilized in a driver.

Function Prefixes
class
 * 1) include
 * 2) include
 * 3) include

title: Windows Programming/Driver API


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Talk:Trigonometry/Law of Cosines

Convention backwards
In the diagram, the sides are have upercase labels (A, B, C), and the angles have lowercase labels (a, b, c)... whereas the equations seem to use the opposite naming convention. I think it would be easier to change the image rather than the text. Can whoever uploaded the image upload a new one?132.206.205.83 20:40, 7 March 2006 (UTC)

naming convention
that is absolutely right. The convention I've seen in my experience in trigonometry is capital letters for angles and lowercase for (opposite) sides. I think this convention is used throughout other pages of wikipedia too, soo the picture should change :)

Is there a need to specify the proofs of sine and cosine?
I am studying math on an ordinary level, and I'm revising trigonometry atm. I found these proofs very helpful, even though I used some time to figure out what was exactly happening between the equations. Therefore I think it would be very helpful for me and others to have some text leading the reader on and telling which rules are used, for instance not just "using the triangle", but "using sine in a right-angled triangle" and so on. Hope someone will make these changes, as I dont feel qualified enough.

Interwiki
vlos:Hình học 10/Chương II/§3. Các hệ thức lượng trong tam giác và giải tam giác --Nguyenthephuc 07:48, 7 June 2007 (UTC)

title: Talk:Trigonometry/Law of Cosines


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False Friends of the Slavist/Map čelověkъ

celovekъ


text:
False Friends of the Slavist/Map čitati

citati

title: False Friends of the Slavist/Map čitati


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False Friends of the Slavist/Map kъnęzь

kъnezь

title: False Friends of the Slavist/Map kъnęzь


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Acoustics/How an Acoustic Guitar Works

What are sound vibrations that contribute to sound production? First of all, there are the strings. Any string that is under tension will vibrate at a certain frequency. The weight and length of the string, the tension in the string, and the compliance of the string determine the frequency at which it vibrates. The guitar controls the length and tension of six differently weighted strings to cover a very wide range of frequencies. Second, there is the body of the guitar. The guitar body is very important for the lower frequencies of the guitar. The air mass just inside the sound hole oscillates, compressing and decompressing the compliant air inside the body. In practice this concept is called a Helmholtz resonator. Without this, it would be difficult to produce the wonderful timbre of the guitar.

The strings
The strings of the guitar vary in linear density, length, and tension. This gives the guitar a wide range of attainable frequencies. The larger the linear density is, the slower the string vibrates. The same goes for the length; the longer the string is the slower it vibrates. This causes a low frequency. Inversely, if the strings are less dense and/or shorter they create a higher frequency. The resonance frequencies of the strings can be calculated by

 * $$f_1 = \frac{\sqrt{\frac{T}{\rho_l}}}{2 L}\quad \text{with}\quad T = \text{string tension},\ \rho_l = \text{linear density of string},\ L = \text{string length}.$$

The string length, $$L$$, in the equation is what changes when a player presses on a string at a certain fret. This will shorten the string which in turn increases the frequency it produces when plucked. The spacing of these frets is important. The length from the nut to bridge determines how much space goes between each fret. If the length is 25 inches, then the position of the first fret should be located (25/17.817) inches from the nut. Then the second fret should be located $(25−(25/17.817))/17.817$ inches from the first fret. This results in the equation

 * $$d = \frac{L}{17.817} \quad \text{with}\quad d = \text{spacing between frets}, L = \text{length from previous fret to bridge}.$$

When a string is plucked, a disturbance is formed and travels in both directions away from point where the string was plucked. These "waves" travel at a speed that is related to the tension and linear density and can be calculated by

 * $$c = \sqrt{\frac{T}{\rho_l}} \quad \text{with}\quad c = \text{wave speed},\ T = \text{string tension},\ \rho_l = \text{linear density}.$$

The waves travel until they reach the boundaries on each end where they are reflected back. The link below displays how the waves propagate in a string.

Plucked String @ www.phys.unsw.edu The strings themselves do not produce very much sound because they are so thin. They can't "push" the air that surrounds them very effectively. This is why they are connected to the top plate of the guitar body. They need to transfer the frequencies they are producing to a large surface area which can create more intense pressure disturbances.

The body
The body of the guitar transfers the vibrations of the bridge to the air that surrounds it. The top plate contributes to most of the pressure disturbances, because the player dampens the back plate and the sides are relatively stiff. This is why it is important to make the top plate out of a light springy wood, like spruce. The more the top plate can vibrate, the louder the sound it produces will be. It is also important to keep the top plate flat, so a series of braces are located on the inside to strengthen it. Without these braces the top plate would bend and crack under the large stress created by the tension in the strings. This would also affect the magnitude of the sound being transmitted. The warped plate would not be able to "push" air very efficiently. A good experiment to try, in order to see how important this part of the guitar is in the amplification process, is as follows:

 * 1) Start with an ordinary rubber band, a large bowl, adhesive tape, and plastic wrap.
 * 2) Stretch the rubber band and pluck it a few times to get a good sense for how loud it is.
 * 3) Stretch the plastic wrap over the bowl to form a sort of drum.
 * 4) Tape down one end of the rubber band to the plastic wrap.
 * 5) Stretch the rubber band and pluck it a few times.
 * 6) The sound should be much louder than before.

The air
The final part of the guitar is the air inside the body. This is very important for the lower range of the instrument. The air just inside the soundhole oscillates compressing and expanding the air inside the body. This is just like blowing across the top of a bottle and listening to the tone it produces. This forms what is called a Helmholtz resonator. For more information on Helmholtz resonators go to Helmholtz Resonance. This link also shows the correlation to acoustic guitars in great detail. The acoustic guitar makers often tune these resonators to have a resonance frequency between F#2 and A2 (92.5 to 110.0 Hz). Having such a low resonance frequency is what aids the amplification of the lower frequency strings. To demonstrate the importance of the air in the cavity, simply play an open A on the guitar (the fifth string - second lowest note). Now, as the string is vibrating, place a piece of cardboard over the soundhole. The sound level is reduced dramatically. This is because you've stopped the vibration of the air mass just inside the soundhole, causing only the top plate to vibrate. Although the top plate still vibrates a transmits sound, it isn't as effective at transmitting lower frequency waves, thus the need for the Helmholtz resonator.

title: Acoustics/How an Acoustic Guitar Works


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False Friends of the Slavist/Map krasьnъ

krasьnъ


text:
D Programming/Statements

To be completed.

In the D programming language there are following statements:
 * Labeled
 * Block
 * Expression
 * Declaration
 * If
 * if-else
 * if-elseif-else
 * Debug
 * Version
 * While
 * Do-While
 * For
 * Foreach
 * Switch
 * Case
 * Default
 * Continue
 * Break
 * Return
 * Goto
 * With
 * Synchronized
 * Try
 * try-catch
 * try-catch-finally
 * try-finally
 * OnScope
 * Throw
 * Volatile
 * Asm
 * Pragma

title: D Programming/Statements


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Acoustics/Bass-Reflex Enclosure Design

Bass-reflex enclosures improve the low-frequency response of loudspeaker systems. Bass-reflex enclosures are also called "vented-box design" or "ported-cabinet design". A bass-reflex enclosure includes a vent or port between the cabinet and the ambient environment. This type of design, as one may observe by looking at contemporary loudspeaker products, is still widely used today. Although the construction of bass-reflex enclosures is fairly simple, their design is not simple, and requires proper tuning. This reference focuses on the technical details of bass-reflex design. General loudspeaker information can be found here.

Effects of the Port on the Enclosure Response
Before discussing the bass-reflex enclosure, it is important to be familiar with the simpler sealed enclosure system performance. As the name suggests, the sealed enclosure system attaches the loudspeaker to a sealed enclosure (except for a small air leak included to equalize the ambient pressure inside). Ideally, the enclosure would act as an acoustical compliance element, as the air inside the enclosure is compressed and rarified. Often, however, an acoustic material is added inside the box to reduce standing waves, dissipate heat, and other reasons. This adds a resistive element to the acoustical lumped-element model. A non-ideal model of the effect of the enclosure actually adds an acoustical mass element to complete a series lumped-element circuit given in Figure 1. For more on sealed enclosure design, see the Sealed Box Subwoofer Design page.

Figure 1. Sealed enclosure acoustic circuit.

In the case of a bass-reflex enclosure, a port is added to the construction. Typically, the port is cylindrical and is flanged on the end pointing outside the enclosure. In a bass-reflex enclosure, the amount of acoustic material used is usually much less than in the sealed enclosure case, often none at all. This allows air to flow freely through the port. Instead, the larger losses come from the air leakage in the enclosure. With this setup, a lumped-element acoustical circuit has the form shown in the diagram below.

Figure 2. Bass-reflex enclosure acoustic circuit.

In this figure, $$Z_{RAD}$$ represents the radiation impedance of the outside environment on the loudspeaker diaphragm. The loading on the rear of the diaphragm has changed when compared to the sealed enclosure case. If one visualizes the movement of air within the enclosure, some of the air is compressed and rarified by the compliance of the enclosure, some leaks out of the enclosure, and some flows out of the port. This explains the parallel combination of $$M_{AP}$$, $$C_{AB}$$, and $$R_{AL}$$. A truly realistic model would incorporate a radiation impedance of the port in series with $$M_{AP}$$, but for now it is ignored. Finally, $$M_{AB}$$, the acoustical mass of the enclosure, is included as discussed in the sealed enclosure case. The formulas which calculate the enclosure parameters are listed in Appendix B.

It is important to note the parallel combination of $$M_{AP}$$ and $$C_{AB}$$. This forms a Helmholtz resonator (click here for more information). Physically, the port functions as the “neck” of the resonator and the enclosure functions as the “cavity.” In this case, the resonator is driven from the piston directly on the cavity instead of the typical Helmholtz case where it is driven at the “neck.”  However, the same resonant behavior still occurs at the enclosure resonance frequency, $$f_{B}$$. At this frequency, the impedance seen by the loudspeaker diaphragm is large (see Figure 3 below). Thus, the load on the loudspeaker reduces the velocity flowing through its mechanical parameters, causing an anti-resonance condition where the displacement of the diaphragm is a minimum. Instead, the majority of the volume velocity is actually emitted by the port itself instead of the loudspeaker. When this impedance is reflected to the electrical circuit, it is proportional to $$1/Z$$, thus a minimum in the impedance seen by the voice coil is small. Figure 3 shows a plot of the impedance seen at the terminals of the loudspeaker. In this example, $$f_B$$ was found to be about 40 Hz, which corresponds to the null in the voice-coil impedance.

Figure 3. Impedances seen by the loudspeaker diaphragm and voice coil.

Quantitative Analysis of Port on Enclosure
The performance of the loudspeaker is first measured by its velocity response, which can be found directly from the equivalent circuit of the system. As the goal of most loudspeaker designs is to improve the bass response (leaving high-frequency production to a tweeter), low frequency approximations will be made as much as possible to simplify the analysis. First, the inductance of the voice coil, $$\it{L_E}$$, can be ignored as long as $$\omega \ll R_E/L_E$$. In a typical loudspeaker, $$\it{L_E}$$ is of the order of 1 mH, while $$\it{R_E}$$ is typically 8$$\Omega$$, thus an upper frequency limit is approximately 1 kHz for this approximation, which is certainly high enough for the frequency range of interest.

Another approximation involves the radiation impedance, $$\it{Z_{RAD}}$$. It can be shown [1] that this value is given by the following equation (in acoustical ohms):

$Z_{RAD} = \frac{\rho_0c}{\pi a^2}\left[\left(1 - \frac{J_1(2ka)}{ka}\right) + j\frac{H_1(2ka)}{ka}\right]$

Where $$J_1(x)$$ and $$H_1(x)$$ are types of Bessel functions. For small values of ka,

Hence, the low-frequency impedance on the loudspeaker is represented with an acoustic mass $$M_{A1}$$ [1]. For a simple analysis, $$R_E$$, $$M_{MD}$$, $$C_{MS}$$, and $$R_{MS}$$ (the transducer parameters, or Thiele-Small parameters) are converted to their acoustical equivalents. All conversions for all parameters are given in Appendix A. Then, the series masses, $$M_{AD}$$, $$M_{A1}$$, and $$M_{AB}$$, are lumped together to create $$M_{AC}$$. This new circuit is shown below.

Figure 4. Low-Frequency Equivalent Acoustic Circuit

Unlike sealed enclosure analysis, there are multiple sources of volume velocity that radiate to the outside environment. Hence, the diaphragm volume velocity, $$U_D$$, is not analyzed but rather $$U_0 = U_D + U_P + U_L$$. This essentially draws a “bubble” around the enclosure and treats the system as a source with volume velocity $$U_0$$. This “lumped” approach will only be valid for low frequencies, but previous approximations have already limited the analysis to such frequencies anyway. It can be seen from the circuit that the volume velocity flowing into the enclosure, $$U_B = -U_0$$, compresses the air inside the enclosure. Thus, the circuit model of Figure 3 is valid and the relationship relating input voltage, $$V_{IN}$$ to $$U_0$$ may be computed.

In order to make the equations easier to understand, several parameters are combined to form other parameter names. First, $$\omega_B$$ and $$\omega_S$$, the enclosure and loudspeaker resonance frequencies, respectively, are:

Based on the nature of the derivation, it is convenient to define the parameters $$\omega_0$$ and h, the Helmholtz tuning ratio:

A parameter known as the compliance ratio or volume ratio, $$\alpha$$, is given by:

$\alpha = \frac{C_{AS}}{C_{AB}} = \frac{V_{AS}}{V_{AB}}$|undefined

Other parameters are combined to form what are known as quality factors:

This notation allows for a simpler expression for the resulting transfer function [1]:

$\frac{U_0}{V_{IN}} = G(s) = \frac{(s^3/\omega_0^4)}{(s/\omega_0)^4+a_3(s/\omega_0)^3+a_2(s/\omega_0)^2+a_1(s/\omega_0)+1}$|undefined

where

Development of Low-Frequency Pressure Response
It can be shown [2] that for $$ka < 1/2$$, a loudspeaker behaves as a spherical source. Here, a represents the radius of the loudspeaker. For a 15” diameter loudspeaker in air, this low frequency limit is about 150 Hz. For smaller loudspeakers, this limit increases. This limit dominates the limit which ignores $$L_E$$, and is consistent with the limit that models $$Z_{RAD}$$ by $$M_{A1}$$.

Within this limit, the loudspeaker emits a volume velocity $$U_0$$, as determined in the previous section. For a simple spherical source with volume velocity $$U_0$$, the far-field pressure is given by [1]:

$p(r) \simeq j\omega\rho_0 U_0 \frac{e^{-jkr}}{4\pi r}$
 * undefined

It is possible to simply let $$r = 1$$ for this analysis without loss of generality because distance is only a function of the surroundings, not the loudspeaker. Also, because the transfer function magnitude is of primary interest, the exponential term, which has a unity magnitude, is omitted. Hence, the pressure response of the system is given by [1]:

$\frac{p}{V_{IN}} = \frac{\rho_0s}{4\pi}\frac{U_0}{V_{IN}} = \frac{\rho_0Bl}{4\pi S_DR_EM_AS}H(s)$
 * undefined

Where $$H(s) = sG(s)$$. In the following sections, design methods will focus on $$|H(s)|^2$$ rather than $$H(s)$$, which is given by:

This also implicitly ignores the constants in front of $$|H(s)|$$ since they simply scale the response and do not affect the shape of the frequency response curve.

Alignments
A popular way to determine the ideal parameters has been through the use of alignments. The concept of alignments is based upon filter theory. Filter development is a method of selecting the poles (and possibly zeros) of a transfer function to meet a particular design criterion. The criteria are the desired properties of a magnitude-squared transfer function, which in this case is $$|H(s)|^2$$. From any of the design criteria, the poles (and possibly zeros) of $$|H(s)|^2$$ are found, which can then be used to calculate the numerator and denominator. This is the “optimal” transfer function, which has coefficients that are matched to the parameters of $$|H(s)|^2$$ to compute the appropriate values that will yield a design that meets the criteria.

There are many different types of filter designs, each which have trade-offs associated with them. However, this design is limited because of the structure of $$|H(s)|^2$$. In particular, it has the structure of a fourth-order high-pass filter with all zeros at s = 0. Therefore, only those filter design methods which produce a low-pass filter with only poles will be acceptable methods to use. From the traditional set of algorithms, only Butterworth and Chebyshev low-pass filters have only poles. In addition, another type of filter called a quasi-Butterworth filter can also be used, which has similar properties to a Butterworth filter. These three algorithms are fairly simple, thus they are the most popular. When these low-pass filters are converted to high-pass filters, the $$s \rightarrow 1/s$$ transformation produces $$s^8$$ in the numerator.

More details regarding filter theory and these relationships can be found in numerous resources, including [5].

Butterworth Alignment
The Butterworth algorithm is designed to have a maximally flat pass band. Since the slope of a function corresponds to its derivatives, a flat function will have derivatives equal to zero. Since as flat of a pass band as possible is optimal, the ideal function will have as many derivatives equal to zero as possible at s = 0. Of course, if all derivatives were equal to zero, then the function would be a constant, which performs no filtering.

Often, it is better to examine what is called the loss function. Loss is the reciprocal of gain, thus

$

The loss function can be used to achieve the desired properties, then the desired gain function is recovered from the loss function.

Now, applying the desired Butterworth property of maximal pass-band flatness, the loss function is simply a polynomial with derivatives equal to zero at s = 0. At the same time, the original polynomial must be of degree eight (yielding a fourth-order function). However, derivatives one through seven can be equal to zero if [3]

$

With the high-pass transformation $$\Omega \rightarrow 1/\Omega$$,

$

It is convenient to define $$\Omega = \omega/\omega_{3dB}$$, since $$\Omega = 1 \Rightarrow |H(s)|^2 = 0.5$$ or -3 dB. This definition allows the matching of coefficients for the $$|H(s)|^2$$ describing the loudspeaker response when $$\omega_{3dB} = \omega_0$$. From this matching, the following design equations are obtained [1]:

Quasi-Butterworth Alignment
The quasi-Butterworth alignments do not have as well-defined of an algorithm when compared to the Butterworth alignment. The name “quasi-Butterworth” comes from the fact that the transfer functions for these responses appear similar to the Butterworth ones, with (in general) the addition of terms in the denominator. This will be illustrated below. While there are many types of quasi-Butterworth alignments, the simplest and most popular is the 3rd order alignment (QB3). The comparison of the QB3 magnitude-squared response against the 4th order Butterworth is shown below.

Notice that the case $$B = 0$$ is the Butterworth alignment. The reason that this QB alignment is called 3rd order is due to the fact that as B increases, the slope approaches 3 dec/dec instead of 4 dec/dec, as in 4th order Butterworth. This phenomenon can be seen in Figure 5.

Figure 5: 3rd-Order Quasi-Butterworth Response for $0.1 \leq B \leq 3$

Equating the system response $$|H(s)|^2$$ with $$|H_{QB3}(s)|^2$$, the equations guiding the design can be found [1]:

Chebyshev Alignment
The Chebyshev algorithm is an alternative to the Butterworth algorithm. For the Chebyshev response, the maximally-flat passband restriction is abandoned. Now, a ripple, or fluctuation is allowed in the pass band. This allows a steeper transition or roll-off to occur. In this type of application, the low-frequency response of the loudspeaker can be extended beyond what can be achieved by Butterworth-type filters. An example plot of a Chebyshev high-pass response with 0.5 dB of ripple against a Butterworth high-pass response for the same $$\omega_{3dB}$$ is shown below.

Figure 6: Chebyshev vs. Butterworth High-Pass Response.

The Chebyshev response is defined by [4]:

$

$$C_n(\Omega)$$ is called the Chebyshev polynomial and is defined by [4]:

Fortunately, Chebyshev polynomials satisfy a simple recursion formula [4]:

For more information on Chebyshev polynomials, see the Wolfram Mathworld: Chebyshev Polynomials page.

When applying the high-pass transformation to the 4th order form of $$|\hat{H}(j\Omega)|^2$$, the desired response has the form [1]:

$

The parameter $$\epsilon$$ determines the ripple. In particular, the magnitude of the ripple is $$10\rm{log}[1+\epsilon^2]$$ dB and can be chosen by the designer, similar to B in the quasi-Butterworth case. Using the recursion formula for $$C_n(x)$$,

$C_4\left(\frac{1}{\Omega}\right) = 8\left(\frac{1}{\Omega}\right)^4 - 8\left(\frac{1}{\Omega}\right)^2 + 1$

Applying this equation to $$|H(j\Omega)|^2$$ [1],

Thus, the design equations become [1]:

Choosing the Correct Alignment
With all the equations that have already been presented, the question naturally arises, “Which one should I choose?” Notice that the coefficients $$a_1$$, $$a_2$$, and $$a_3$$ are not simply related to the parameters of the system response. Certain combinations of parameters may indeed invalidate one or more of the alignments because they cannot realize the necessary coefficients. With this in mind, general guidelines have been developed to guide the selection of the appropriate alignment. This is very useful if one is designing an enclosure to suit a particular transducer that cannot be changed.

The general guideline for the Butterworth alignment focuses on $$Q_L$$ and $$Q_{TS}$$. Since the three coefficients $$a_1$$, $$a_2$$, and $$a_3$$ are a function of $$Q_L$$, $$Q_{TS}$$, h, and $$\alpha$$, fixing one of these parameters yields three equations that uniquely determine the other three. In the case where a particular transducer is already given, $$Q_{TS}$$ is essentially fixed. If the desired parameters of the enclosure are already known, then $$Q_L$$ is a better starting point.

In the case that the rigid requirements of the Butterworth alignment cannot be satisfied, the quasi-Butterworth alignment is often applied when $$Q_{TS}$$ is not large enough.. The addition of another parameter, B, allows more flexibility in the design.

For $$Q_{TS}$$ values that are too large for the Butterworth alignment, the Chebyshev alignment is typically chosen. However, the steep transition of the Chebyshev alignment may also be utilized to attempt to extend the bass response of the loudspeaker in the case where the transducer properties can be changed.

In addition to these three popular alignments, research continues in the area of developing new algorithms that can manipulate the low-frequency response of the bass-reflex enclosure. For example, a 5th order quasi-Butterworth alignment has been developed [6]. Another example [7] applies root-locus techniques to achieve results. In the modern age of high-powered computing, other researchers have focused their efforts in creating computerized optimization algorithms that can be modified to achieve a flatter response with sharp roll-off or introduce quasi-ripples which provide a boost in sub-bass frequencies [8].

Appendix B: Enclosure Parameter Formulas
Figure 7: Important dimensions of bass-reflex enclosure.

Based on these dimensions [1],

title: Acoustics/Bass-Reflex Enclosure Design


text:
False Friends of the Slavist/Map ljubiti

ljubiti


text:
Acoustics/Polymer-Film Acoustic Filters

Introduction
Acoustic filters are used in many devices such as mufflers, noise control materials (absorptive and reactive), and loudspeaker systems to name a few. Although the waves in simple (single-medium) acoustic filters usually travel in gases such as air and carbon-monoxide (in the case of automobile mufflers) or in materials such as fiberglass, polyvinylidene fluoride (PVDF) film, or polyethylene (Saran Wrap), there are also filters that couple two or three distinct media together to achieve a desired acoustic response. General information about basic acoustic filter design can be perused at the following wikibook page Acoustic Filter Design & Implementation. The focus of this article will be on acoustic filters that use multilayer air/polymer film-coupled media as its acoustic medium for sound waves to propagate through; concluding with an example of how these filters can be used to detect and extrapolate audio frequency information in high-frequency "carrier" waves that carry an audio signal. However, before getting into these specific type of acoustic filters, we need to briefly discuss how sound waves interact with the medium(media) in which it travels and how these factors can play a role when designing acoustic filters.

Changes in Media Properties Due to Sound Wave Characteristics
As with any system being designed, the filter response characteristics of an acoustic filter are tailored based on the frequency spectrum of the input signal and the desired output. The input signal may be infrasonic (frequencies below human hearing), sonic (frequencies within human hearing range), or ultrasonic (frequencies above human hearing range). In addition to the frequency content of the input signal, the density, and, thus, the characteristic impedance of the medium (media) being used in the acoustic filter must also be taken into account. In general, the characteristic impedance $$ Z_0 \,$$ for a particular medium is expressed as...      $$ Z_0 = \pm \rho_0 c  \,$$  $$ (Pa \cdot s/m) $$

where $$ \pm \rho_0 \,$$ = (equilibrium) density of medium  $$(kg/m^3)\,$$ $$ c \,$$ = speed of sound in medium $$ (m/s) \,$$

The characteristic impedance is important because this value simultaneously gives an idea of how fast or slow particles will travel as well as how much mass is "weighting down" the particles in the medium (per unit area or volume) when they are excited by a sound source. The speed in which sound travels in the medium needs to be taken into consideration because this factor can ultimately affect the time response of the filter (i.e. the output of the filter may not radiate or attenuate sound fast or slow enough if not designed properly). The intensity $$ I_A \,$$ of a sound wave is expressed as...

$$ I_A = \frac{1}{T}\int_{0}^{T} pu\quad dt = \pm \frac{P^2}{2\rho_0c} \,$$    $$(W/m^2) \,$$.

$$ I_A \,$$ is interpreted as the (time-averaged) rate of energy transmission of a sound wave through a unit area normal to the direction of propagation, and this parameter is also an important factor in acoustic filter design because the characteristic properties of the given medium can change relative to intensity of the sound wave traveling through it. In other words, the reaction of the particles (atoms or molecules) that make up the medium will respond differently when the intensity of the sound wave is very high or very small relative to the size of the control area (i.e. dimensions of the filter, in this case). Other properties such as the elasticity and mean propagation velocity (of a sound wave) can change in the acoustic medium as well, but focusing on frequency, impedance, and/or intensity in the design process usually takes care of these other parameters because most of them will inevitably be dependent on the aforementioned properties of the medium.

Why Coupled Acoustic Media in Acoustic Filters?
In acoustic transducers, media coupling is employed in acoustic transducers to either increase or decrease the impedance of the transducer, and, thus, control the intensity and speed of the signal acting on the transducer while converting the incident wave, or initial excitation sound wave, from one form of energy to another (e.g. converting acoustic energy to electrical energy). Specifically, the impedance of the transducer is augmented by inserting a solid structure (not necessarily rigid) between the transducer and the initial propagation medium (e.g. air). The reflective properties of the inserted medium is exploited to either increase or decrease the intensity and propagation speed of the incident sound wave. It is the ability to alter, and to some extent, control, the impedance of a propagation medium by (periodically) inserting (a) solid structure(s) such as thin, flexible films in the original medium (air) and its ability to concomitantly alter the frequency response of the original medium that makes use of multilayer media in acoustic filters attractive. The reflection factor and transmission factor $$ \hat{R} \,$$ and $$ \hat{T} \, $$, respectively, between two media, expressed as...

$$ \hat{R} = \frac{\text{pressure of reflected portion of incident wave}}{\text{pressure of incident wave}} = \frac{\rho c - Z_\mathrm{in}}{\rho c + Z_\mathrm{in}}$$

and $$ \hat{T} = \frac{\text{pressure of transmitted portion of incident wave}}{\text{pressure of incident wave}} = 1 + \hat{R}$$,

are the tangible values that tell how much of the incident wave is being reflected from and transmitted through the junction where the media meet. Note that $$ Z_{in} \,$$ is the (total) input impedance seen by the incident sound wave upon just entering an air-solid acoustic media layer. In the case of multiple air-columns as shown in Fig. 2, $$ Z_\mathrm{in} \,$$ is the aggregate impedance of each air-column layer seen by the incident wave at the input. Below in Fig. 1, a simple illustration explains what happens when an incident sound wave propagating in medium (1) and comes in contact with medium (2) at the junction of the both media (x=0), where the sound waves are represented by vectors.

As mentioned above, an example of three such successive air-solid acoustic media layers is shown in Fig. 2 and the electroacoustic equivalent circuit for Fig. 2 is shown in Fig. 3 where $$ L = \rho_s h_s \,$$ = (density of solid material)(thickness of solid material) = unit-area (or volume) mass, $$ Z = \rho c = \,$$ characteristic acoustic impedance of medium, and $$ \beta = k = \omega/c = \,$$ wavenumber. Note that in the case of a multilayer, coupled acoustic medium in an acoustic filter, the impedance of each air-solid section is calculated by using the following general purpose impedance ratio equation (also referred to as transfer matrices)...

$$ \frac{Z_a}{Z_0} = \frac{\left( \frac{Z_b}{Z_0} \right) + j\ \tan(kd)}{1 + j\ \left( \frac{Z_b}{Z_0} \right) \tan(kd)} \,$$

where $$ Z_b \,$$ is the (known) impedance at the edge of the solid of an air-solid layer (on the right) and $$ Z_a \,$$ is the (unknown) impedance at the edge of the air column of an air-solid layer.

Effects of High-Intensity, Ultrasonic Waves in Acoustic Media in Audio Frequency Spectrum
When an ultrasonic wave is used as a carrier to transmit audio frequencies, three audio effects are associated with extrapolating the audio frequency information from the carrier wave: (a) beating effects, (b) parametric array effects, and (c) radiation pressure.

Beating occurs when two ultrasonic waves with distinct frequencies $$ f_1 \,$$ and  $$ f_2 \,$$ propagate in the same direction, resulting in amplitude variations which consequently make the audio signal information go in and out of phase, or “beat”, at a frequency of  $$ f_1 - f_2 \,$$.

Parametric array effects occur when the intensity of an ultrasonic wave is so high in a particular medium that the high displacements of particles (atoms) per wave cycle changes properties of that medium so that it influences parameters like elasticity, density, propagation velocity, etc. in a non-linear fashion. The results of parametric array effects on modulated, high-intensity, ultrasonic waves in a particular medium (or coupled media) is the generation and propagation of audio frequency waves (not necessarily present in the original audio information) that are generated in a manner similar to the nonlinear process of amplitude demodulation commonly inherent in diode circuits (when diodes are forward biased).

Another audio effect that arises from high-intensity ultrasonic beams of sound is a static (DC) pressure called radiation pressure. Radiation pressure is similar to parametric array effects in that amplitude variations in the signal give rise to audible frequencies via amplitude demodulation. However, unlike parametric array effects, radiation pressure fluctuations that generate audible signals from amplitude demodulation can occur due to any low-frequency modulation and not just from pressure fluctuations occurring at the modulation frequency $$ \omega_M \,$$ or beating frequency  $$ f_1 - f_2 \,$$.

An Application of Coupled Media in Acoustic Filters
Figs. 1 - 3 were all from a research paper entitled New Type of Acoustics Filter Using Periodic Polymer Layers for Measuring Audio Signal Components Excited by Amplitude-Modulated High_Intensity Ultrasonic Waves submitted to the Audio Engineering Society (AES) by Minoru Todo, Primary Innovator at Measurement Specialties, Inc., in the October 2005 edition of the AES Journal. Figs. 4 and 5 below, also from this paper, are illustrations of test setups referred to in this paper. Specifically, Fig. 4 is a test setup used to measure the transmission (of an incident ultrasonic sound wave) through the acoustic filter described by Figs. 1 and 2. Fig. 5 is a block diagram of the test setup used for measuring radiation pressure, one of the audio effects mentioned in the previous section. It turns out that out of all of the audio effects mentioned in the previous section that are caused by high-intensity ultrasonic waves propagating in a medium, sound waves produced from radiated pressure are the hardest to detect when microphones and preamplifiers are used in the detection/receiver system. Although nonlinear noise artifacts occur due to overloading of the preamplifier present in the detection/receiver system, the bulk of the nonlinear noise comes from the inherent nonlinear noise properties of microphones. This is true because all microphones, even specialized measurement microphones designed for audio spectrum measurements that have sensitivity well beyond the threshold of hearing, have nonlinearities artifacts that (periodically) increase in magnitude with respect to increase at ultrasonic frequencies. These nonlinearities essentially mask the radiation pressure generated because the magnitude of these nonlinearities are orders of magnitude greater than the radiation pressure. The acoustic (low-pass) filter referred to in this paper was designed in order to filter out the "detrimental" ultrasonic wave that was inducing high nonlinear noise artifacts in the measurement microphones. The high-intensity, ultrasonic wave was producing radiation pressure (which is audible) within the initial acoustic medium (i.e. air). By filtering out the ultrasonic wave, the measurement microphone would only detect the audible radiation pressure that the ultrasonic wave was producing in air. Acoustic filters like these could possibly be used to detect/receive any high-intensity, ultrasonic signal that may carry audio information which may need to be extrapolated with an acceptable level of fidelity.

title: Acoustics/Polymer-Film Acoustic Filters


text:
Acoustics/Noise in Hydraulic Systems

Noise in Hydraulic Systems
Hydraulic systems are the most preferred source of power transmission in most of the industrial and mobile equipments due to their power density, compactness, flexibility, fast response and efficiency. The field hydraulics and pneumatics is also known as 'Fluid Power Technology'. Fluid power systems have a wide range of applications which include industrial, off-road vehicles, automotive systems, and aircraft. But, one of the main problems with the hydraulic systems is the noise generated by them. The health and safety issues relating to noise have been recognized for many years and legislation is now placing clear demands on manufacturers to reduce noise levels [1]. Hence, noise reduction in hydraulic systems demands lot of attention from the industrial as well as academic researchers. It needs a good understanding of how the noise is generated and propagated in a hydraulic system in order to reduce it.

Sound in fluids
The speed of sound in fluids can be determined using the following relation.

$$c = \sqrt {\frac{K}{\rho}} $$ where K - fluid bulk modulus, $$\rho $$- fluid density, c - velocity of sound

Typical value of bulk modulus range from 2e9 to 2.5e9 N/m2. For a particular oil, with a density of 889 kg/m3, speed of sound $$ c = \sqrt {\frac{2e9}{889}}= 1499.9 m/s $$

Source of Noise
The main source of noise in hydraulic systems is the pump which supplies the flow. Most of the pumps used are positive displacement pumps. Of the positive displacement pumps, axial piston swash plate type is mostly preferred due to their reliability and efficiency.

The noise generation in an axial piston pump can be classified under two categories (i) fluidborne noise and (ii) Structureborne noise

Fluidborne Noise (FBN)
Among the positive displacement pumps, highest levels of FBN are generated by axial piston pumps and lowest levels by screw pumps and in between these lie the external gear pump and vane pump [1]. The discussion in this page is mainly focused on axial piston swash plate type pumps. An axial piston pump has a fixed number of displacement chambers arranged in a circular pattern separated from each other by an angular pitch equal to $$ \phi = \frac {360}{n} $$ where n is the number of displacement chambers. As each chamber discharges a specific volume of fluid, the discharge at the pump outlet is sum of all the discharge from the individual chambers. The discontinuity in flow between adjacent chambers results in a kinematic flow ripple. The amplitude of the kinematic ripple can be theoretical determined given the size of the pump and the number of displacement chambers. The kinematic ripple is the main cause of the fluidborne noise. The kinematic ripples is a theoretical value. The actual flow ripple at the pump outlet is much larger than the theoretical value because the kinematic ripple is combined with a compressibility component which is due to the fluid compressibility. These ripples (also referred as flow pulsations) generated at the pump are transmitted through the pipe or flexible hose connected to the pump and travel to all parts of the hydraulic circuit.

The pump is considered an ideal flow source. The pressure in the system will be decided by resistance to the flow or otherwise known as system load. The flow pulsations result in pressure pulsations. The pressure pulsations are superimposed on the mean system pressure. Both the flow and pressure pulsations easily travel to all part of the circuit and affect the performance of the components like control valve and actuators in the system and make the component vibrate, sometimes even resonate. This vibration of system components adds to the noise generated by the flow pulsations. The transmission of FBN in the circuit is discussed under transmission below.

A typical axial piston pump with 9 pistons running at 1000 rpm can produce a sound pressure level of more than 70 dBs.

Structureborne Noise (SBN)
In swash plate type pumps, the main sources of the structureborne noise are the fluctuating forces and moments of the swash plate. These fluctuating forces arise as a result of the varying pressure inside the displacement chamber. As the displacing elements move from suction stroke to discharge stroke, the pressure varies accordingly from few bars to few hundred bars. These pressure changes are reflected on the displacement elements (in this case, pistons) as forces and these force are exerted on the swash plate causing the swash plate to vibrate. This vibration of the swash plate is the main cause of structureborne noise. There are other components in the system which also vibrate and lead to structureborne noise, but the swash is the major contributor.

''' Fig. 1 shows an exploded view of axial piston pump. Also the flow pulsations and the oscillating forces on the swash plate, which cause FBN and SBN respectively are shown for one revolution of the pump. '''

FBN
The transmission of FBN is a complex phenomenon. Over the past few decades, considerable amount of research had gone into mathematical modeling of pressure and flow transient in the circuit. This involves the solution of wave equations, with piping treated as a distributed parameter system known as a transmission line [1] & [3].

Lets consider a simple pump-pipe-loading valve circuit as shown in Fig. 2. The pressure and flow ripple at any location in the pipe can be described by the relations:

$$ \frac {}{} P = Ae^{-i k x} + Be^{+i k x}$$ .........(1)

$$ Q = \frac {1}{Z_{0}}(Ae^{-i k x} - Be^{+i k x})$$.....(2) where $$\frac {}{} A $$ and $$\frac {}{} B $$ are frequency dependent complex coefficients which are directly proportional to pump (source) flow ripple, but also functions of the source impedance $$\frac {}{} Z_{s}$$, characteristic impedance of the pipe $$\frac {}{} Z_{0}$$ and the termination impedance $$\frac {}{} Z_{t}$$. These impedances ,usually vary as the system operating pressure and flow rate changes, can be determined experimentally.

For complex systems with several system components, the pressure and flow ripples are estimated using the transformation matrix approach. For this, the system components can be treated as lumped impedances (a throttle valve or accumulator), or distributed impedances (flexible hose or silencer). Various software packages are available today to predict the pressure pulsations.

SBN
The transmission of SBN follows the classic source-path-noise model. The vibrations of the swash plate, the main cause of SBN, are transferred to the pump casing which encloses all the rotating group in the pump including displacement chambers (also known as cylinder block), pistons, and the swash plate. The pump case, apart from vibrating itself, transfers the vibration down to the mount on which the pump is mounted. The mount then passes the vibrations down to the main mounted structure or the vehicle. Thus the SBN is transferred from the swash plate to the main structure or vehicle via pumpcasing and mount.

Some of the machine structures, along the path of transmission, are good at transmitting this vibrational energy and they even resonate and reinforce it. By converting only a fraction of 1% of the pump structureborne noise into sound, a member in the transmission path could radiate more ABN than the pump itself [4].

Airborne noise (ABN)
Both FBN and SBN, impart high fatigue loads on the system components and make them vibrate. All of these vibrations are radiated as airborne noise and can be heard by a human operator. Also, the flow and pressure pulsations make the system components such as a control valve to resonate. This vibration of the particular component again radiates airborne noise.

Noise reduction
The reduction of the noise radiated from the hydraulic system can be approached in two ways.

(i) Reduction at Source - which is the reduction of noise at the pump. A large amount of open literature are available on the reduction techniques with some techniques focusing on reducing FBN at source and others focusing on SBN. Reduction in FBN and SBN at the source has a large influence on the ABN that is radiated. Even though, a lot of progress had been made in reducing the FBN and SBN separately, the problem of noise in hydraulic systems is not fully solved and lot need to be done. The reason is that the FBN and SBN are interrelated, in a sense that, if one tried to reduce the FBN at the pump, it tends to affect the SBN characteristics. Currently, one of the main researches in noise reduction in pumps, is a systematic approach in understanding the coupling between FBN and SBN and targeting them simultaneously instead of treating them as two separate sources. Such an unified approach, demands not only well trained researchers but also sophisticated computer based mathematical model of the pump which can accurately output the necessary results for optimization of pump design. The amplitude of fluid pulsations can be reduced, at the source, with the use of an hydraulic attenuator(5).

(ii) Reduction at Component level - which focuses on the reduction of noise from individual component like hose, control valve, pump mounts and fixtures. This can be accomplished by a suitable design modification of the component so that it radiates least amount of noise. Optimization using computer based models can be one of the ways.

Hydraulic System noise
Fig.3 Domain of hydraulic system noise generation and transmission (Figure recreated from [1])

title: Acoustics/Noise in Hydraulic Systems


text:
Acoustics/Basic Acoustics of the Marimba

Introduction
Like a xylophone, a marimba has octaves of wooden bars that are struck with mallets to produce tones. Unlike the harsh sound of a xylophone, a marimba produces a deep, rich tone. Marimbas are not uncommon and are played in most high school bands.

Now, while all the trumpet and flute and clarinet players are busy tuning up their instruments, the marimba player is back in the percussion section with her feet up just relaxing. This is a bit surprising, however, since the marimba is a melodic instrument that needs to be in tune to sound good. So what gives? Why is the marimba never tuned? How would you even go about tuning a marimba? To answer these questions, the acoustics behind (or within) a marimba must be understood.

Components of sound
What gives the marimba its unique sound? It can be boiled down to two components: the bars and the resonators. Typically, the bars are made of rosewood (or some synthetic version of wood). They are cut to size depending on what note is desired, then the tuning is refined by shaving wood from the underside of the bar.

Example
Rosewood bar, middle C, 1 cm thick

The equation that relates the length of the bar with the desired frequency comes from the theory of modeling a bar that is free at both ends. This theory yields the following equation:

$$ Length = \sqrt{\frac{3.011^2\cdot \pi \cdot t \cdot c}{8 \cdot \sqrt{12}\cdot f}}$$

where t is the thickness of the bar, c is the speed of sound in the bar, and f is the frequency of the note.

For rosewood, c = 5217 m/s. For middle C, f=262 Hz.

Therefore, to make a middle C key for a rosewood marimba, cut the bar to be:

$$ Length = \sqrt{\frac{3.011^2\cdot \pi \cdot .01 \cdot 5217}{8 \cdot \sqrt{12}\cdot 262}}= .45 m = 45 cm$$ ***

The resonators are made from metal (usually aluminum) and their lengths also differ depending on the desired note. It is important to know that each resonator is open at the top but closed by a stopper at the bottom end.

Example
Aluminum resonator, middle C

The equation that relates the length of the resonator with the desired frequency comes from modeling the resonator as a pipe that is driven at one end and closed at the other end. A "driven" pipe is one that has a source of excitation (in this case, the vibrating key) at one end. This model yields the following:

$$Length = \frac {c}{4\cdot f}$$

where c is the speed of sound in air and f is the frequency of the note.

For air, c = 343 m/s. For middle C, f = 262 Hz.

Therefore, to make a resonator for the middle C key, the resonator length should be:

$$ Length = \frac {343}{4 \cdot 262} = .327m = 32.7 cm $$

Resonator shape
The shape of the resonator is an important factor in determining the quality of sound that can be produced. The ideal shape is a sphere. This is modeled by the Helmholtz resonator. However, mounting big, round, beach ball-like resonators under the keys is typically impractical. The worst choices for resonators are square or oval tubes. These shapes amplify the non-harmonic pitches sometimes referred to as “junk pitches”. The round tube is typically chosen because it does the best job (aside from the sphere) at amplifying the desired harmonic and not much else.

As mentioned in the second example above, the resonator on a marimba can be modeled by a closed pipe. This model can be used to predict what type of sound (full and rich vs dull) the marimba will produce. Each pipe is a "quarter wave resonator" that amplifies the sound waves produced by of the bar. This means that in order to produce a full, rich sound, the length of the resonator must exactly match one-quarter of the wavelength. If the length is off, the marimba will produce a dull or off-key sound for that note.

Why would the marimba need tuning?
In the theoretical world where it is always 72 degrees with low humidity, a marimba would not need tuning. But, since weather can be a factor (especially for the marching band) marimbas do not always perform the same way. Hot and cold weather can wreak havoc on all kinds of percussion instruments, and the marimba is no exception. On hot days, the marimba tends to be sharp and for cold days it tends to be flat. This is the exact opposite of what happens to string instruments. Why? The tone of a string instrument depends mainly on the tension in the string, which decreases as the string expands with heat. The decrease in tension leads to a flat note. Marimbas on the other hand produce sound by moving air through the resonators. The speed at which this air is moved is the speed of sound, which varies proportionately with temperature! So, as the temperature increases, so does the speed of sound. From the equation given in example 2 from above, you can see that an increase in the speed of sound (c) means a longer pipe is needed to resonate the same note. If the length of the resonator is not increased, the note will sound sharp. Now, the heat can also cause the wooden bars to expand, but the effect of this expansion is insignificant compared to the effect of the change in the speed of sound.

Tuning myths
It is a common myth among percussionists that the marimba can be tuned by simply moving the resonators up or down (while the bars remain in the same position.) The thought behind this is that by moving the resonators down, for example, you are in effect lengthening them. While this may sound like sound reasoning, it actually does not hold true in practice. Judging by how the marimba is constructed (cutting bars and resonators to specific lengths), it seems that there are really two options to consider when looking to tune a marimba: shave some wood off the underside of the bars, or change the length of the resonator. For obvious reasons, shaving wood off the keys every time the weather changes is not a practical solution. Therefore, the only option left is to change the length of the resonator.

As mentioned above, each resonator is plugged by a stopper at the bottom end. So, by simply shoving the stopper farther up the pipe, you can shorten the resonator and sharpen the note. Conversely, pushing the stopper down the pipe can flatten the note. Most marimbas do not come with tunable resonators, so this process can be a little challenging. (Broomsticks and hammers are common tools of the trade.)

Example
Middle C Resonator lengthened by 1 cm

For ideal conditions, the length of the middle C (262 Hz) resonator should be 32.7 cm as shown in example 2. Therefore, the change in frequency for this resonator due to a change in length is given by:

$$\Delta Frequency = 262 Hz - \frac {c}{4\cdot (.327 + \Delta L)} $$

If the length is increased by 1 cm, the change in frequency will be:

$$\Delta Frequency = \frac {343}{4\cdot (.327 + .01)} - 262 Hz = 7.5 Hz$$

The acoustics behind the tuning a marimba go back to the design that each resonator is to be ¼ of the total wavelength of the desired note. When marimbas get out of tune, this length is no longer exactly equal to ¼ the wavelength due to the lengthening or shortening of the resonator as described above. Because the length has changed, resonance is no longer achieved, and the tone can become muffled or off-key.

Conclusions
Some marimba builders are now changing their designs to include tunable resonators. Since any leak in the end-seal will cause major loss of volume and richness of the tone, this is proving to be a very difficult task. At least now, though, armed with the acoustic background of their instruments, percussionists everywhere will now have something to do when the conductor says, “tune up!”

Links and References

 * 1)  http://www.gppercussion.com/html/resonators.html
 * 2)  http://www.mostlymarimba.com/
 * 3)  http://www.craftymusicteachers.com/bassmarimba/


text:
False Friends of the Slavist/Map lěkarьstvo

lekarьstvo


text:
Acoustics/Bessel Functions and the Kettledrum

Introduction
In class, we have begun to discuss the solutions of multidimensional wave equations. A particularly interesting aspect of these multidimensional solutions are those of bessel functions for circular boundary conditions. The practical application of these solutions is the kettledrum. This page will explore in qualitative and quantitative terms how the kettledrum works. More specifically, the kettledrum will be introduced as a circular membrane and its solution will be discussed with visuals (e.g. visualization of bessel functions, video of kettledrums and audio forms (wav files of kettledrums playing. In addition, links to more information about this material, including references will be included.

The math behind the kettledrum: the brief version
When one looks at how a kettledrum produces sound, one should look no farther than the drum head. The vibration of this circular membrane (and the air in the drum enclosure) is what produces the sound in this instrument. The mathematics behind this vibrating drum are relatively simple. If one looks at a small element of the drum head, it looks exactly like the mathematical model for a vibrating string (see:). The only difference is that there are two dimensions where there are forces on the element, the two dimensions that are planar to the drum. As this is the same situation, we have the same equation, except with another spatial term in the other planar dimension. This allows us to model the drum head using a helmholtz equation. The next step (solved in detail below) is to assume that the displacement of the drum head (in polar coordinates) is a product of two separate functions for theta and r. This allows us to turn the PDE into two ODES which are readily solved and applied to the situation of the kettledrum head. For more info, see below.

The math behind the kettledrum: the derivation
So starting with the trusty general Helmholtz equation:

 * $$\nabla^2\Psi+k^2\Psi=0.$$

Where $$k$$ is the wave number, the frequency of the forced oscillations divided by the speed of sound in the membrane.

Since we are dealing with a circular object, it makes sense to work in polar coordinates (in terms of radius and angle) instead of rectangular coordinates. For polar coordinates the Laplacian term of the Helmholtz relation ($$\nabla^2$$) becomes $$\frac{\partial^2 \Psi}{\partial r^2} + \frac{1}{r} \frac{\partial\Psi}{\partial r} +\frac{1}{r^2} \frac{\partial^2 \Psi}{\partial \theta^2}$$

Using the method of separation of variables (see Reference 3 for more info), we will assume a solution of the form
 * $$\Psi (r,\theta) = R(r) \Theta(\theta).$$

Substituting this result back into our trusty Helmholtz equation, then multiplying through by $$r^2/(R\Theta)$$ gives

 * $$\frac{1}{R} \left(r^2\frac{d^2 R}{dr^2} + r \frac{dR}{dr}\right) + k^2 r^2 = -\frac{1}{\Theta} \frac{d^2 \Theta}{d\theta^2},$$

where we moved the $$\theta$$-dependent terms to the right hand side. Since we separated the variables of the solution into two one-dimensional functions, the partial derivatives become ordinary derivatives. In order for the above equality to hold regardless of changes in $$r$$ and $$\theta$$, both sides must be equal to some constant. For simplicity, I will use $$\lambda^2$$ as this constant. This results in the following two equations:

 * $$ \frac{d^2 \Theta}{d\theta^2} = -\lambda^2 \Theta, $$

 * $$ r^2\frac{d^2 R}{dr^2} + r \frac{dR}{dr} + (k^2 r^2 - \lambda^2) R = 0. $$

The first of these equations readily seen as the standard second order ordinary differential equation which has a harmonic solution of sines and cosines with the frequency based on $$ \lambda $$. The second equation is what is known as Bessel's Equation. The solution to this equation is cryptically called Bessel functions of order $$ \lambda $$ of the first and second kind. These functions, while sounding very intimidating, are simply oscillatory functions of the radius times the wave number. Both sets of functions diminish as $$kr$$ becomes large, but are unbounded as $$kr$$ goes to zero for the Bessel functions of the second kind.

Now that we have the general solution to this equation, we can now model a infinite radius kettledrum head. However, since i have yet to see an infinite kettle drum, we need to constrain this solution of a vibrating membrane to a finite radius. We can do this by applying what we know about our circular membrane: along the edges of the kettledrum, the drum head is attached to the drum. This means that there can be no displacement of the membrane at the termination at the radius of the kettle drum. This boundary condition can be mathematically described as the following:

 * $$ R(a) = 0 $$

Where a is the arbitrary radius of the kettledrum. In addition to this boundary condition, the displacement of the drum head at the center must be finite. This second boundary condition removes the bessel function of the second kind from the solution. This reduces the $$R$$ part of our solution to:

 * $$ R(r) = AJ_{\lambda}(kr) $$

Where $$ J_{\lambda} $$ is a bessel function of the first kind of order $$ \lambda $$. Apply our other boundary condition at the radius of the drum requires that the wave number $$k$$ must have discrete values, ($$j_{mn}/a$$) which can be looked up. Combining all of these gives us our solution to how a drum head behaves (which is the real part of the following):

 * $$ y_{\lambda n}(r,\theta,t) = A_{\lambda n} J_{\lambda n}(k_{\lambda n} r)e^{j \lambda \theta+j w_{\lambda n} t} $$

The math behind the kettledrum: the entire drum
The above derivation is just for the drum head. An actual kettledrum has one side of this circular membrane surrounded by an enclosed cavity. This means that air is compressed in the cavity when the membrane is vibrating, adding more complications to the solution. In mathematical terms, this makes the partial differential equation non-homogeneous or in simpler terms, the right side of the Helmholtz equation does not equal zero. This result requires significantly more derivation, and will not be done here. If the reader cares to know more, these results are discussed in the two books under references 6 and 7.

Sites of interest
As one can see from the derivation above, the kettledrum is very interesting mathematically. However, it also has a rich historical music tradition in various places of the world. As this page's emphasis is on math, there are few links provided below that reference this rich history.

 * A discussion of Persian kettledrums: Kettle drums of Iran and other countries
 * A discussion of kettledrums in classical music: Kettle drum Lit.
 * A massive resource for kettledrum history, construction and technique" Vienna Symphonic Library