text:
Waves/Superposition

Superposition Principle
It is found empirically that as long as the amplitudes of waves in most media are small, two waves in the same physical location don't interact with each other. Thus, for example, two waves moving in the opposite direction simply pass through each other without their shapes or amplitudes being changed. When superimposed, the total wave displacement is just the sum of the displacements of the individual waves. This is called the superposition principle. At sufficiently large amplitude the superposition principle often breaks down -- interacting waves may scatter off of each other, lose amplitude, or change their form.

Interference is a consequence of the superposition principle. When two or more waves are superimposed, the net wave displacement is just the algebraic sum of the displacements of the individual waves. Since these displacements can be positive or negative, the net displacement can either be greater or less than the individual wave displacements. The former case is called constructive interference, while the latter is called destructive interference.

Figure 1.5: Superposition (lower panel) of two sine waves (shown individually in the upper panel) with equal amplitudes and wavenumbers $$k_1 = 4$$ and $$k_2 = 5$$

Let us see what happens when we superimpose two sine waves with different wavenumbers. Figure 1.5 shows the superposition of two waves with wavenumbers $$k_1 = 4$$ and $$k_2 = 5$$. Notice that the result is a wave with about the same wavelength as the two initial waves, but which varies in amplitude depending on whether the two sine waves are in or out of phase. When the waves are in phase, constructive interference is occurring, while destructive interference occurs where the waves are out of phase.

Figure 1.6: Superposition of two sine waves with equal amplitudes and wavenumbers $$k_1 = 10$$ and $$k_2 = 11$$

What happens when the wavenumbers of the two sine waves are changed? Figure 1.6 shows the result when $$k_1 = 10$$ and $$k_2 = 11$$. Notice that though the wavelength of the resultant wave is decreased, the locations where the amplitude is maximum have the same separation in $$x$$ as in figure 1.5.

Figure 1.7: Superposition of two sine waves with equal amplitudes and wavenumbers $$k_1 = 10$$ and $$k_2 = 12$$.

If we superimpose waves with $$k_1 = 10$$ and $$k_2 = 12$$, as is shown in figure 1.7, we see that the $$x$$ spacing of the regions of maximum amplitude has decreased by a factor of two. Thus, while the wavenumber of the resultant wave seems to be related to something like the average of the wavenumbers of the component waves, the spacing between regions of maximum wave amplitude appears to go inversely with the difference of the wavenumbers of the component waves. In other words, if $$k_1$$ and $$k_2$$ are close together, the amplitude maxima are far apart and vice versa.

Figure 1.8: Representation of the wavenumbers and amplitudes of two superimposed sine waves.

We can symbolically represent the sine waves that make up figures 1.5, 1.6, and 1.7 by a plot such as that shown in figure 1.8. The amplitudes and wavenumbers of each of the sine waves are indicated by vertical lines in this figure.

The regions of large wave amplitude are called wave packets. Wave packets will play a central role in what is to follow, so it is important that we acquire a good understanding of them. The wave packets produced by only two sine waves are not well separated along the $$x$$-axis. However, if we superimpose many waves, we can produce an isolated wave packet. For example, figure 1.9 shows the results of superimposing $$20$$ sine waves with wavenumbers $$k = 0.4 m$$, $$m = 1,2, \ldots, 20$$, where the amplitudes of the waves are largest for wavenumbers near $$k = 4$$.

Figure 1.9: Superposition of twenty sine waves with $$k_0 = 4$$ and $$\Delta k = 1$$.

In particular, we assume that the amplitude of each sine wave is proportional to $$\exp [-(k - k_0)^2 / \Delta k^2 ]$$, where $$k_0 = 4$$ and $$\Delta k = 1$$. The amplitudes of each of the sine waves making up the wave packet in figure 1.9 are shown schematically in figure 1.10.

Figure 1.10: Representation of the wavenumbers and amplitudes of 20 superimposed sine waves with $$k_0 = 4$$ and $$\Delta k = 1$$.

The quantity $$\Delta k$$ controls the distribution of the sine waves being superimposed -- only those waves with a wavenumber $$k$$ within approximately $$\Delta k$$ of the central wavenumber $$k_0$$ of the wave packet, i. e., for $$3 \le k \le 5$$ in this case, contribute significantly to the sum. If $$\Delta k$$ is changed to $$2$$, so that wavenumbers in the range $$2 \le k \le 6$$ contribute significantly, the wavepacket becomes narrower, as is shown in figures 1.11 and 1.12.

Figure 1.11: Superposition of twenty sine waves with $$k_0 = 4$$ and $$\Delta k = 2$$.

Figure 1.12: Representation of the wavenumbers and amplitudes of 20 superimposed sine waves with $$k_0 = 4$$ and $$\Delta k = 2$$.

$$\Delta k$$ is called the wavenumber spread of the wave packet, and it evidently plays a role similar to the difference in wavenumbers in the superposition of two sine waves -- the larger the wavenumber spread, the smaller the physical size of the wave packet. Furthermore, the wavenumber of the oscillations within the wave packet is given approximately by the central wavenumber.

We can better understand how wave packets work by mathematically analyzing the simple case of the superposition of two sine waves. Let us define $$k_0 = (k_1 + k_2 )/2$$ where $$k_1$$ and $$k_2$$ are the wavenumbers of the component waves. Furthermore let us set $$\Delta k = (k_2 - k_1 )/2$$. The quantities $$k_0$$ and $$\Delta k$$ are graphically illustrated in figure 1.8. We can write $$k_1 = k_0 - \Delta k$$ and $$k_2 = k_0 + \Delta k$$ and use the trigonometric identity $$\sin (a + b) = \sin (a) \cos (b) + \cos (a) \sin (b)$$ to find $$\sin (k_1 x) + \sin (k_2 x) = \sin [(k_0 - \Delta k)x] + \sin [(k_0 + \Delta k)x]$$ $$ = \sin (k_0 x) \cos (\Delta k x) - \cos (k_0 x) \sin(\Delta k x) +$$ $$\sin (k_0 x) \cos (\Delta k x) + \cos (k_0 x) \sin(\Delta k x) = 2 \sin (k_0 x) \cos ( \Delta k x) .$$ (2.17)

The sine factor on the bottom line of the above equation produces the oscillations within the wave packet, and as speculated earlier, this oscillation has a wavenumber $$k_0$$ equal to the average of the wavenumbers of the component waves. The cosine factor modulates this wave with a spacing between regions of maximum amplitude of
 * $$\Delta x = \pi / \Delta k . $$ (2.18)

Thus, as we observed in the earlier examples, the length of the wave packet $$\Delta x$$ is inversely related to the spread of the wavenumbers $$\Delta k$$ (which in this case is just the difference between the two wavenumbers) of the component waves. This relationship is central to the uncertainty principle of quantum mechanics.

title: Waves/Superposition


text:
Waves/Beats

Beats
Suppose two sound waves of slightly different frequencies impinge on your ear at the same time. The displacement perceived by your ear is the superposition of these two waves, with time dependence $$ A(t) = \sin ( \omega_1 t) + \sin ( \omega_2 t) = 2 \sin ( \omega_0 t) \cos ( \Delta \omega t), $$ (2.19)

where we now have $$\omega_0 = ( \omega_1 + \omega_2 )/2$$ and $$\Delta \omega = ( \omega_2 - \omega_1 )/2$$. What you actually hear is a tone with angular frequency $$\omega_0$$ which fades in and out with period $$T_{beat} = \pi / \Delta \omega = 2 \pi / ( \omega_2 - \omega_1 ) = 1 / ( f_2 - f_1 ). $$ (2.20)

The beat frequency is simply $$ f_{beat} = 1/T_{beat} = f_2 - f_1. $$ (2.21)

Note how beats are the time analog of wave packets -- the mathematics are the same except that frequency replaces wavenumber and time replaces space.

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

title: Waves/Beats


text:
Waves/Interferometers

Interferometers
An interferometer is a device which splits a beam of light into two sub-beams, shifts the phase of one sub-beam with respect to the other, and then superimposes the sub-beams so that they interfere constructively or destructively, depending on the magnitude of the phase shift between them. In this section we study the Michelson interferometer and interferometric effects in thin films.

The Michelson Interferometer
Figure 1.13: Sketch of a Michelson interferometer.

The American physicist Albert Michelson invented the optical interferometer illustrated in figure 1.13. The incoming beam is split into two beams by the half-silvered mirror. Each sub-beam reflects off of another mirror which returns it to the half-silvered mirror, where the two sub-beams recombine as shown. One of the reflecting mirrors is movable by a sensitive micrometer device, allowing the path length of the corresponding sub-beam, and hence the phase relationship between the two sub-beams, to be altered. As figure 1.13 shows, the difference in path length between the two sub-beams is $$2x$$ because the horizontal sub-beam traverses the path twice. Thus, constructive interference occurs when this path difference is an integral number of wavelengths, i. e.,
 * $$ 2x = m \lambda, m = 0, \pm 1, \pm 2, \ldots \quad \mbox{(Michelson interferometer)} $$ (2.22)

where $$\lambda $$ is the wavelength of the light and $$m$$ is an integer. Note that $$m$$ is the number of wavelengths that fits evenly into the distance $$2x$$.

title: Waves/Interferometers


text:
Waves/Thin Films

Thin Films
Figure 1.14: Plane light wave normally incident on a transparent thin film of thickness $$d$$ and index of refraction $$n > 1$$. Partial reflection occurs at the front surface of the film, resulting in beam A, and at the rear surface, resulting in beam B. Much of the wave passes completely through the film, as with C.

One of the most revealing examples of interference occurs when light interacts with a thin film of transparent material such as a soap bubble. Figure 1.14 shows how a plane wave normally incident on the film is partially reflected by the front and rear surfaces. The waves reflected off the front and rear surfaces of the film interfere with each other. The interference can be either constructive or destructive depending on the phase difference between the two reflected waves.

If the wavelength of the incoming wave is $$\lambda $$, one would naively expect constructive interference to occur between the A and B beams if $$2d$$ were an integral multiple of $$\lambda $$.

Two factors complicate this picture. First, the wavelength inside the film is not $$\lambda $$, but $$\lambda /n$$, where $$n$$ is the index of refraction of the film. Constructive interference would then occur if $$2d = m \lambda /n$$. Second, it turns out that an additional phase shift of half a wavelength occurs upon reflection when the wave is incident on material with a higher index of refraction than the medium in which the incident beam is immersed. This phase shift doesn't occur when light is reflected from a region with lower index of refraction than felt by the incident beam. Thus beam B doesn't acquire any additional phase shift upon reflection. As a consequence, constructive interference actually occurs when
 * $$ 2d = (m + 1/2) \lambda /n, m = 0,1,2, \ldots \quad \mbox{(constructive interference)} $$ (2.23)

while destructive interference results when
 * $$ 2d = m \lambda /n, m = 0,1,2, \ldots \quad \mbox{(destructive interference)} . $$ (2.24)

When we look at a soap bubble, we see bands of colors reflected back from a light source. What is the origin of these bands? Light from ordinary sources is generally a mixture of wavelengths ranging from roughly $$\lambda = 4.5 \times 10^{-7} \mbox{ m}$$ (violet light) to $$\lambda = 6.5 \times 10^{-7} \mbox{ m}$$ (red light). In between violet and red we also have blue, green, and yellow light, in that order. Because of the different wavelengths associated with different colors, it is clear that for a mixed light source we will have some colors interfering constructively while others interfere destructively. Those undergoing constructive interference will be visible in reflection, while those undergoing destructive interference will not.

Another factor enters as well. If the light is not normally incident on the film, the difference in the distances traveled between beams reflected off of the front and rear faces of the film will not be just twice the thickness of the film. To understand this case quantitatively, we need the concept of refraction, which will be developed later in the context of geometrical optics. However, it should be clear that different wavelengths will undergo constructive interference for different angles of incidence of the incoming light. Different portions of the thin film will in general be viewed at different angles, and will therefore exhibit different colors under reflection, resulting in the colorful patterns normally seen in soap bubbles.

title: Waves/Thin Films


text:
Waves/Derivatives

Math Tutorial -- Derivatives
Figure 1.15: Estimation of the derivative, which is the slope of the tangent line. When point B approaches point A, the slope of the line AB approaches the slope of the tangent to the curve at point A.

This section provides a quick introduction to the idea of the derivative. For a more detailed discussion and exploration of the differentiation and of Calculus, see Calculus and Differentiation.

Often we are interested in the slope of a line tangent to a function $$y(x)$$ at some value of $$x$$. This slope is called the derivative and is denoted $$dy/dx$$. Since a tangent line to the function can be defined at any point $$x$$, the derivative itself is a function of $$x$$:
 * $$ g(x) = \frac{d y(x)}{dx} . $$ (2.25)

As figure 1.15 illustrates, the slope of the tangent line at some point on the function may be approximated by the slope of a line connecting two points, A and B, set a finite distance apart on the curve:
 * $$ \frac{dy}{dx} \approx \frac{\Delta y}{\Delta x} . $$ (2.26)

As B is moved closer to A, the approximation becomes better. In the limit when B moves infinitely close to A, it is exact.

Table of Derivatives
Derivatives of some common functions are now given. In each case $$c$$ is a constant.

The product and chain rules are used to compute the derivatives of complex functions. For instance,

 * $$ \frac{d}{dx} ( \sin (x) \cos (x)) = \frac{d \sin (x)}{dx} \cos (x) + \sin (x) \frac{d \cos (x)}{dx} = \cos^2 (x) - \sin^2 (x) $$

and

 * $$ \frac{d}{dx} \log ( \sin (x) ) = \frac{1}{\sin (x)} \frac{d \sin (x)}{dx} = \frac{\cos (x)}{\sin (x)} . $$

title: Waves/Derivatives


text:
Waves/Group Velocity

Group Velocity
We now ask the following question: How fast do wave packets move? Surprisingly, we often find that wave packets move at a speed very different from the phase speed, $$c = \omega /k$$, of the wave composing the wave packet.

We shall find that the speed of motion of wave packets, referred to as the group velocity, is given by
 * $$u = \ 0.1 \mbox{m}^{-1}$$. The phase speed of deep ocean waves is $$c = (g/k)^{1/2}$$. However, since $$c \equiv \omega /k$$, we find the frequency of deep ocean waves to be $$\omega = (gk)^{1/2}$$. The group velocity is therefore $$u \equiv d \omega / dk = (g/k)^{1/2} / 2 = c/2$$. For the specified central wavenumber, we find that $$u \approx (9.8 \mbox{ m} \mbox{ s}^{-2} / 0.1 \mbox{ m}^{-1} )^{1/2} / 2 \approx 5 \mbox{ m} \mbox{ s}^{-1}$$. By contrast, the phase speed of deep ocean waves with this wavelength is $$c \approx 10 \mbox{ m} \mbox{ s}^{-1}$$.

Dispersive waves are waves in which the phase speed varies with wavenumber. It is easy to show that dispersive waves have unequal phase and group velocities, while these velocities are equal for non-dispersive waves.

Derivation of Group Velocity Formula
We now derive equation (1.36). It is easiest to do this for the simplest wave packets, namely those constructed out of the superposition of just two sine waves. We will proceed by adding two waves with full space and time dependence:
 * $$ A = \sin ( k_1 x - \omega_1 t ) + \sin ( k_2 x - \omega_2 t ) $$ (2.37)

After algebraic and trigonometric manipulations familiar from earlier sections, we find
 * $$ A = 2 \sin ( k_0 x - \omega_0 t ) \cos ( \Delta k x - \Delta \omega t ), $$ (2.38)

where as before we have $$k_0 = (k_1 + k_2 )/2$$, $$\omega_0 = ( \omega_1 + \omega_2 )/2$$, $$\Delta k = (k_2 - k_1 )/2$$, and $$\Delta \omega = ( \omega_2 - \omega_1 )/2$$. Again think of this as a sine wave of frequency $$\omega_0$$ and wavenumber $$k_0$$ modulated by a cosine function. In this case the modulation pattern moves with a speed so as to keep the argument of the cosine function constant:
 * $$ \Delta k x - \Delta \omega t = const. $$ (2.39)

Differentiating this with respect to $$t$$ while holding $$\Delta k$$ and $$\Delta \omega$$ constant yields
 * $$ u \equiv \frac{dx}{dt} = \frac{\Delta \omega}{\Delta k} . $$ (2.40)

In the limit in which the deltas become very small, this reduces to the derivative
 * $$ u = \frac{d \omega}{d k}, $$ (2.41)

which is the desired result.

title: Waves/Group Velocity


text:
Waves/1D Examples

Examples
We now illustrate some examples of phase speed and group velocity by showing the displacement resulting from the superposition of two sine waves, as given by equation (1.38), in the $$x$$-$$t$$ plane. This is an example of a spacetime diagram, of which we will see many examples later on.

Figure 1.16: Net displacement of the sum of two traveling sine waves plotted in the $$x - t$$ plane. The short vertical lines indicate where the displacement is large and positive, while the short horizontal lines indicate where it is large and negative. One wave has $$k = 4$$ and $$\omega = 4$$, while the other has $$k = 5$$ and $$\omega = 5$$. Thus, $$\Delta k = 5 - 4 = 1$$ and $$\Delta \omega = 5 - 4 = 1$$ and we have $$u = \Delta \omega / \Delta k = 1$$. Notice that the phase speed for the first sine wave is $$c_1 = 4/4 = 1$$ and for the second wave is $$c_2 = 5/5 = 1$$. Thus, $$c_1 = c_2 = u$$ in this case.

Figure 1.16 shows a non-dispersive case in which the phase speed equals the group velocity. The regions with vertical and horizontal hatching (short vertical or horizontal lines) indicate where the wave displacement is large and positive or large and negative. Large displacements indicate the location of wave packets. The positions of waves and wave packets at any given time may therefore be determined by drawing a horizontal line across the graph at the desired time and examining the variations in wave displacement along this line. The crests of the waves are indicated by regions of short vertical lines. Notice that as time increases, the crests move to the right. This corresponds to the motion of the waves within the wave packets. Note also that the wave packets, i. e., the broad regions of large positive and negative amplitudes, move to the right with increasing time as well.

Since velocity is distance moved $$\Delta x$$ divided by elapsed time $$\Delta t$$, the slope of a line in figure 1.16, $$\Delta t / \Delta x$$, is one over the velocity of whatever that line represents. The slopes of lines representing crests (the slanted lines, not the short horizontal and vertical lines) are the same as the slopes of lines representing wave packets in this case, which indicates that the two move at the same velocity. Since the speed of movement of wave crests is the phase speed and the speed of movement of wave packets is the group velocity, the two velocities are equal and the non-dispersive nature of this case is confirmed.

Figure 1.17: Net displacement of the sum of two traveling sine waves plotted in the $$x$$-$$t$$ plane. One wave has $$k = 4.5$$ and $$\omega = 4$$, while the other has $$k = 5.5$$ and $$\omega = 6$$. In this case $$\Delta k = 5.5 - 4.5 = 1$$ while $$\Delta \omega = 6 - 4 = 2$$, so the group velocity is $$u = \Delta \omega / \Delta k = 2/1 = 2$$. However, the phase speeds for the two waves are $$c_1 = 4/4.5 = 0.889$$ and $$c_2 = 6/5.5 = 1.091$$. The average of the two phase speeds is about $$0.989$$, so the group velocity is about twice the average phase speed in this case.

Figure 1.18: Net displacement of the sum of two traveling sine waves plotted in the $$x$$-$$t$$ plane. One wave has $$k = 4$$ and $$\omega = 5$$, while the other has $$k = 5$$ and $$\omega = 4$$. Can you figure out the group velocity and the average phase speed in this case? Do these velocities match the apparent phase and group speeds in the figure?

Figure 1.17 shows a dispersive wave in which the group velocity is twice the phase speed, while figure 1.18 shows a case in which the group velocity is actually opposite in sign to the phase speed. See if you can confirm that the phase and group velocities seen in each figure correspond to the values for these quantities calculated from the specified frequencies and wavenumbers.

title: Waves/1D Examples


text:
Waves/1D Problems

Problems

 * 1) Measure your pulse r ate. Compute the ordinary frequency of your heart beat in cycles per second. Compute the angular frequency in radians per second. Compute the period.
 * 2) An important wavelength for radio waves in radio astronomy is $$21 \mbox{ cm}$$. (This comes from neutral hydrogen.) Compute the wavenumber of this wave. Compute the ordinary and angular frequencies. (The speed of light is 3·108 m/s)
 * 3) Sketch the resultant wave obtained from superimposing the waves $$A = \sin (2x)$$ and $$B = \sin (3x)$$. By using the trigonometric identity given in equation (1.17), obtain a formula for $$A + B$$ in terms of $$\sin (5x/2)$$ and $$\cos (x/2)$$. Does the wave obtained from sketching this formula agree with your earlier sketch?
 * 4) Two sine waves with wavelengths $$\lambda_1$$ and $$\lambda_2$$ are superimposed, making wave packets of length $$L$$. If we wish to make $$L$$ larger, should we make $$\lambda_1$$ and $$\lambda_2$$ closer together or farther apart? Explain your reasoning.
 * 5) By examining figure 1.9 versus figure 1.10 and then figure 1.11 versus figure 1.12, determine whether equation (1.18) works at least in an approximate sense for isolated wave packets.
 * 6) The frequencies of the chromatic scale in music are given by
 * $$ f_i = f_0 2^{i/12} ,\quad i = 0,1,2, \ldots ,11, $$ (2.42)
 * where $$f_0$$ is a constant equal to the frequency of the lowest note in the scale.

Figure 1.19: Sketch of a police radar. Figure 1.20: Sketch of a Fabry-Perot interferometer.
 * 1) Compute $$f_1$$ through $$f_{11}$$ if $$f_0 = 440 \mbox{ Hz}$$ (the "A" note).
 * 2) Using the above results, what is the beat frequency between the "A" ($$i = 0$$) and "B" ($$i = 2$$) notes? (The frequencies are given here in cycles per second rather than radians per second.)
 * 3) Which pair of the above frequencies yields the smallest beat frequency? Explain your reasoning.
 * 4) Large ships in general cannot move faster than the phase speed of surface waves with a wavelength equal to twice the ship's length. This is because most of the propulsive force goes into making big waves under these conditions rather than accelerating the ship.
 * 5) How fast can a $$300 \mbox{ m}$$ long ship move in very deep water?
 * 6) As the ship moves into shallow water, does its maximum speed increase or decrease? Explain.
 * 7) Given the formula for refractive index of light quoted in this section, for what range of $$k$$ does the phase speed of light in a transparent material exceed the speed of light in a vacuum?
 * 8) A police radar works by splitting a beam of microwaves, part of which is reflected back to the radar from your car where it is made to interfere with the other part which travels a fixed path, as shown in figure 1.19.
 * 9) If the wavelength of the microwaves is $$\lambda $$, how far do you have to travel in your car for the interference between the two beams to go from constructive to destructive to constructive?
 * 10) If you are traveling toward the radar at speed $$v = 30 \mbox{ m} \mbox{ s}^{-1}$$, use the above result to determine the number of times per second constructive interference peaks will occur. Assume that $$\lambda = 3 \mbox{ cm}$$.
 * 1) Suppose you know the wavelength of light passing through a Michelson interferometer high accuracy. Describe how you could use the interferometer to measure the length of a small piece of material.
 * 2) A Fabry-Perot interferometer (see figure 1.20) consists of two parallel half-silvered mirrors placed a distance $$d$$ from each other as shown. The beam passing straight through interferes with the beam which reflects once off of both of the mirrored surfaces as shown. For wavelength $$\lambda $$, what values of $$d$$ result in constructive interference?
 * 3) A Fabry-Perot interferometer has spacing $$d = 2 \mbox{ cm}$$ between the glass plates, causing the direct and doubly reflected beams to interfere (see figure 1.20). As air is pumped out of the gap between the plates, the beams go through 23 cycles of constructive-destructive-constructive interference. If the wavelength of the light in the interfering beams is $$5 \times 10^{-7} \mbox{ m}$$, determine the index of refraction of the air initially in the interferometer.
 * 4) Measurements on a certain kind of wave reveal that the angular frequency of the wave varies with wavenumber as shown in the following table:

Figure 1.21: Sketch of a weird dispersion relation.
 * 1) Compute the phase speed of the wave for $$k = 3 \mbox{ m}^{-1}$$ and for $$k = 4 \mbox{ m}^{-1}$$.
 * 2) Estimate the group velocity for $$k = 3.5 \mbox{ m}^{-1}$$ using a finite difference approximation to the derivative.
 * 3) Suppose some type of wave has the (admittedly weird) dispersion relation shown in figure 1.21.
 * 4) For what values of $$k$$ is the phase speed of the wave positive?
 * 5) For what values of $$k$$ is the group velocity positive?
 * 1) Compute the group velocity for shallow water waves. Compare it with the phase speed of shallow water waves. (Hint: You first need to derive a formula for $$\omega (k)$$ from $$c(k)$$.)
 * 2) Repeat the above problem for deep water waves.
 * 3) Repeat for sound waves. What does this case have in common with shallow water waves?

title: Waves/1D Problems


text:
Waves

The wave is a physical phenomenon that is found in a variety of contexts, a perturbation in the medium. You undoubtedly know about ocean waves and have probably played with a stretched slinky toy, producing undulations which move rapidly along the slinky. Other examples of waves are sound, vibrations in solids, and light (photons have the particularity of sharing characteristics of both waves and particles).

Despite the vast differences in these different types of waves, they all consist of a fluctuation around an equilibrium position caused by a restoring force, and can be described by the same types of mathematical equations. Understanding these equations is a powerful tool because it will let you understand the basics of a wide variety of seemingly unrelated phenomena. The purpose of this section is to describe the kinematics of waves, i.e., to provide tools for describing the form and motion of all waves irrespective of their underlying physical mechanisms.

In this chapter we learn first about the basic properties of waves and introduce the most common type of wave, called the sinusoidal (sine) wave. In fact just about any type of wave can be expressed as a combination of sine functions using a technique proposed by Joseph Fourier. Examples of waves seen in the real world are presented. We then learn about the superposition principle, which allows us to construct complex wave patterns by superimposing sine waves. Using these ideas, we discuss the related ideas of beats and interferometry. Finally, the ideas of wave packets and group velocity are introduced.

Waves in one dimension

 * 1) The Mathematics of Waves
 * 2) Transverse, Longitudinal, and Torsional waves
 * 3) Sine Waves
 * 4) Types of Waves
 * 5) Sound Waves
 * 6) Light
 * 7) Superposition Principle
 * 8) Beats
 * 9) Interferometers
 * 10) Thin Films
 * 11) Derivatives
 * 12) Fourier Transforms
 * 13) Group Velocity

 * Examples
 * Problems
 * Solutions
 * Terminology

Waves in two and three dimensions

 * 1) Introduction
 * 2) Math:Vectors
 * 3) Plane Waves
 * 4) Superposition of Plane Waves
 * 5) Single slit Diffraction
 * 6) Double slit Diffraction
 * 7) Diffraction Grating

 * Problems

Geometrical optics

 * 1) Introduction
 * 2) Reflection and Refraction
 * 3) Total Internal Reflection
 * 4) Anisotropic Media
 * 5) Optical Instruments
 * 6) Fermat's Principle

 * Problems

title: Waves


text:
User talk:Eric119

Hi! I am glad someone filled in the things I was missing on the Calculus:Functions page, but I had a few questions about things I wasn't sure about. I might have modified some of them, but I wasn't sure why you did what you did, so I thought I might ask first:

 * In the definition of a function you referred to the formal definition of function (set of ordered pairs). Most people would completely miss the point, unless we went in more detail, which I don't think is wise.  What do you think?
 * "in the world of real numbers" is another (less) obscure reference. Anyone familar with complex numbers would understand what was meant without the parenthetical clause, but someone who wasn't familar wouldn't care.  Of course, that is no reason not to be precise, so leaving it in would still be appropriate.
 * In the example of adding, subtracting, etc. the functions f and g, you wrote "def" before the subtraction example. Why?  What do you mean?
 * I like the continuity section. I was going to wait until limits to introduce it, but you tied it in with graphing effectively.
 * I had left out a * 0 = 0 intentionally, since it is not an axiom of field theory (it can be proved easily). Of course, I never titled the list as "axioms of field theory", but rather as arithematic truths -- and a * 0 = 0 is certainly true.  Anyone reading it not familar with field theory would not notice one way or another.  So I am split in whether I think it should be included or not.  On second thought, the whole Algebra section is mostly redundant -- the only commonly missed fact is the distributive property (anyone who doesn't know a + 0 = a or a * 1 = a shouldn't be taking Calculus).

Conincidentally, my name happens to be Eric as well.

-- IntMan

Re: functions as ordered pairs -- Perhaps you're right. That little bit of information isn't necessary, as functions are rarely specified by giving a set of ordered pairs.

Re: complex numbers -- I hope people trying to take calculus have already been introduced to imaginary numbers in algebra or trig. Also, I remember that there is one method for solving differential equations that involves complex numbers (though I'm uncertain of the details), so if we think people won't know/remember them we should cover them in a review section (unless we are not covering that, but since there is a major section "Advanced Calculus" I assume we will be.)

Re: "def" -- It's just a typo. I meant nothing by it.

Re: continuity -- Thank you.

Re: a*0 -- Well, I thought a * 0 = 0 was an axiom, but apparently I misremembered. I definately agree that it should be removed.

Eric119 20:47 22 Jul 2003 (UTC)

About the Hello, World! Page, Sorry, I was being rude. Sometimes I do get carried away. I was probably just in a bad mood at the time. For example: using endl is just a preference because I don't like it when I write a program, and nothing shows up on the screen until the program exits. The return 0, is the only thing I really have a problem with. Point: You are right about the not having to return 0. Counter-Point: But since it is standard pratice not to rely on the system to free all of your ram you've allocated, should it also be right to rely on the system to return the 0?

Josh:Mrquick 23:13 29 Jul 2003 (UTC)

Well, I suppose that's a style issue, too. The implicit "return 0;" is guaranteed according to the C++ standard. You could say it's better to explictly state that there were no errors. Or you could say that by default we assume the program did not encounter errors.

Eric119 18:59 30 Jul 2003 (UTC)

About the calculus page: it didn't exist and was showing in red. See the Staff lounge. Apparently the database got confused and marked it as a nonexistent page with a history. Thanks for reverting it. Geoffrey 23:31, 11 Aug 2003 (UTC)

Hi Eric, sorry I didn't realize before that you've put up a new puzzle. I've moved it to our new subcategories under 'Arithmetic Puzzles'. Thomas

title: User talk:Eric119


text:
Organic Chemistry/Alkanes/Methane

=Methane= << Foundational concepts | Ethane >>

Methane, CH4 is the simplest organic molecule. It is a gas at standard temperature and pressure and is a carbon atom with four hydrogen atoms bonded to it in a tetrahedral shape.

This is a flattened, two-dimensional representation of methane that you will see commonly. The true three-dimensional form of methane does not have any 90 degree angles between bonded hydrogens. The bonds point to the four corners of a tetrahedron, forming cos−1(−1/3) ≈ 109.5 degree bond angles.

<< Foundational concepts | << Alkanes | Methane | Ethane >> | Introduction to reactions >>

title: Organic Chemistry/Alkanes/Methane


text:
Organic Chemistry/Alkanes/Ethane

=Ethane= << Methane | Propane thru decane >>

Ethane (/ˈɛθeɪn/ or /ˈiːθeɪn/) is a chemical compound, or hydrocarbon with a chemical formula of C2H6. At standard temperature and pressure, ethane is a colorless, odorless gas. Ethane is isolated on an industrial scale from natural gas, and as a byproduct of petroleum refining. Its chief use is as petrochemical feedstock for ethylene production. It could be described as two methane molecules attached to each other, less two hydrogens.

Number of hydrogens to carbons
This equation describes the relationship between the number of hydrogen and carbon atoms in alkanes:

 * H = 2C + 2

where "C" and "H" are used to represent the number of carbon and hydrogen atoms present in one molecule. If C = 2, then H = 6.

Many textbooks put this in the following format:

 * CNH2N+2

where "CN" and "H2N+2" represent the number of carbon and hydrogen atoms present in one molecule. If CN = 3, then H2N+2 = 2(3) + 2 = 8. (For this formula look to the "N" for the number, the "C" and the "H" letters themselves do not change.)

<< Organic chemistry | << Alkanes | << Methane | Ethane | Propane thru decane >> | Introduction to reactions >>

title: Organic Chemistry/Alkanes/Ethane


text:
User talk:Mrquick

Hello and welcome! Is Programming:Perl going to be chapter in the Programming textbook? If not then a link to the Wikipedia article on the Perl programming language article would be best - the last thing we want to do is recreate Wikipedia here! :-) --Maveric149 09:16 22 Jul 2003 (UTC)

Welcome! It will be as soon as someone who knows Perl (not me) decides to write it. Until that though, a link to Wikipedia would be acceptable; not a replacement though because a textbook is written differently than an encyclopedia. --Josh

 * Ah - I see now. Good everything is self-contained within the single book - that's what I like to see. Oh and you could use the Wikipedia article to start such a chapter: just make sure to say "From Wikipedia" in the edit summary - a mention on the "Authors page" would also be nice (doing both of those combined with the fact that we are all part of the Wikimedia family should be good enough for the GFDL...). --mav

Re: programmingc:ch1_cpp (an impossible name to remember, btw) Are you using an automated HTML syntax coloring program? I do not like the colors appearing in the main text (not the programs); you shouldn't include that text in your syntax-colorer input.

Also, the namespace problem can be fixed by leaving out the .h from iostream. Only in ancient (nonstandard) compilers does not exist (as opposed to ), and in all -compatible compilers the file's contents are in std. If a compiler does not support the non-h syntax (and the other thirty-odd standard headers), it's not a good one to learn from. Geoffrey 03:50 28 Jul 2003 (UTC)

I'll try that namespace thing... Yeah, that colored thing isn't that good. I wrote something myself to do the conversion, having things colored in the text didn't bother me, as long as it was actually a command (but if the consensus is not to do it, I won't). That "do" thing, I don't know why its not working? Take, "if", and "while", they don't just turn blue in the text, my program searches for the accompaying paranthesis, but for some reason it won't do that for "do". I can't figure out why... Its probably due to some weird mistake I made when I programmed the parser at ultra high speed... so I'm writing a new parser... Mrquick 18:09 28 Jul 2003 (UTC)

I just wanted to call you attention to a discussion I hope I've started at the staff lounge regarding namespaces. No disrespect, but I'm recommending against your approach, and I think we ought to keep a record of the our reasoning, so the Lounge archive will do it. LouI 18:10 1 Aug 2003 (UTC)

Is there some way I can or an admin have the server email me my password now, instead of waiting for I think a half of a month. I kind of forgot what it was :(. Josh

Lucky me, I figured it out, I didn't see the email myself a new password button. Mrquick 05:00, 15 Aug 2003 (UTC)

Hello :) Nice work in restructuring Computer programming, but I disliked your choice of the suffix "in programming", as it was very cumbersome for other titles.

By the way, if you've been active at the Wikipedia, the same syntax works here too. To create a tab indent, write a :, to create bullets, use successive * s, if you have an article in the form Namespace:Article, you can just write Namespace:Article and it'll show up without the Namespace: part. Leaving a space as the first character of the line forces the rest of the line to be in fixed-width font, perfect for code.

Thanks :) Dysprosia 09:55, 20 Aug 2003 (UTC)

Hello, It seems I've dissappeared for a bit. I just went off to college and the local college ethernet is down. After one day viruses killed it. Not fun. I should become active again as soon as I have a permanent, or semi-permanent, internet solution. Josh

title: User talk:Mrquick


text:
Optics

 * Do add new material and examples and make corrections. It all helps.

All help is welcome.

Optics
about this book

Contents

 * /Light/
 * /Nature of light/
 * /Waves/
 * /Evidence for wave properties of light/
 * /Evidence for light as a stream of particles/
 * /Features of a wave/
 * /Propagation of light/
 * /Fermat's Principle/
 * /Refraction/
 * /Total internal reflection/
 * /Fibre optics/
 * /Images/
 * Images
 * Real images
 * Virtual images
 * Curved mirrors
 * The mirror equation
 * /Lenses/
 * Introduction
 * Refraction at a spherical interface
 * A lens
 * Locating the image
 * The lensmaker’s equation
 * Converging and diverging lenses
 * Optical instruments using lenses
 * Single-lens systems
 * A magnifying glass
 * Compound optical systems
 * The refracting telescope
 * Interference and diffraction
 * Wave phenomena
 * /Huygens’ Principle/
 * Introduction
 * Interference
 * Diffraction
 * Summary of formulas in this chapter

 * Small angle approximation
 * Small angle approximation
 * Derivations for the Module 7 exam
 * Derivations for the Module 7 exam

X-ray optics

/Authors/

External Resources

 * Diffractive optics papers and patents.
 * Also available and  news and events.
 * Gradient-index optics reviews, theory and ideas

Optik Física/Óptica Optique Fysica/Optica Introdução à Física/Óptica Fysik A/Optik

title: Optics


text:
Wikibooks:Textbook planning

Note: See also Wikiversity page for discussion of Wikiversity / WikiU.

The following areas were recently suggested in the mailing list as important to resolve in regards to the textbook project.

When applicable, list your username to vote for (a) particular opinion(s). (This is a "vote" only in the sense of a show of support; no decision has been taken to count votes in any particular way, or even at all, to make final choices.) Write arguments for your position after the list of options; also add more options if you like. If there's not a list of options, but instead a list of points needing discussion, then you can list more points or give sublists of options for a particular point of discussion. Et cetera.

Name of the project

 * "Open Textbook Project" -- http://www.opentextbook.org - Karl Wick
 * "Free Textbook Project" --
 * "Wikimedia Textbooks" -- Kpjas
 * "Wikipedia Textbooks" --
 * "Wikibooks" (mav owns the urls and plans to transfer them to Wikimedia) -- mav, Toby Bartels, Kpjas, LouI, LittleDan, Thomas, Dhum Dhum
 * "Wikitext" -- Toby Bartels,

The URL could also be http://wikibooks.wikimedia.org/ (owned by the Foundation already). -- Toby Bartels 08:14 27 Jul 2003 (UTC)

We should avoid entering the philosophical debate between "open" and "free" in our name. -- Toby Bartels 08:14 27 Jul 2003 (UTC)

We should continue the tradition of having "wiki" in our name since that is what defines us the most - even more than the free license. --mav 19:44 28 Jul 2003 (UTC)
 * I agree completely with that one. Dhum Dhum 20:11 5 Aug 2003 (UTC)

We should not limit ourselves to being a textbook-only shop - all non-fiction books should be allowed. --mav 19:44 28 Jul 2003 (UTC)
 * Or even fiction books, that would be interesting LittleDan
 * If you are talking about stuff like Huck Finn with annotation and wikified words going to encyclopedia articles, then yes. But we should not be in the business of creating our own fictional worlds. That is a different project. --mav 17:24 1 Aug 2003 (UTC)

Types of content that are to be covered

 * Traditional textbook content -- Presumably everybody supports this!
 * Do realize that the effort may be futile, no matter the resulting quality. One would need to make a snapshot for some unusually brave (solidly tenured) professors to try, then supply an errata list and print many copies. A live wiki is too unstable to teach a class from. --AlbertCahalan 20:29, 13 Feb 2005 (UTC)
 * Non traditional textbooks, there are some subjects (especially in science on post-graduate levels) for which there are no traditional textbooks. In such a book its dynamics are not a problem. --IzI 18:07, 25 August 2005 (UTC)
 * Possible other book content:
 * How-to directions like cookbooks -- Toby Bartels,
 * Well, the envisioned Wikibooks' Cookbook might turn out to be a general book about cooking in different cultures with historical background and some conscious consumerism threads
 * The Cookbook is doing very well. It does suffer from problems with: deletionism, case-sensitivity and the Cookbook: prefix ruining normal wiki ease-of-use ( normal easy linking is unusable ), lack of Special:Specialpages specific to the Cookbook, and a sidebar Main_Page link that doesn't go to the main page of the cookbook. --AlbertCahalan 20:29, 13 Feb 2005 (UTC)
 * Manuals to use certain things (such as a software program)
 * Should be textbook-oriented in presentation. --Maveric149
 * A broad travel guide covering the entire world. -- Dhum Dhum
 * The people at www.wikitravel.org are going to fill that role. Our focus here should be in instructing students - travel guides are informative (like an encyclopedia), not instructive (like a textbook). --mav
 * Other non-fiction works
 * Fiction books. LittleDan
 * Something like a "Cliff's Notes" possibly? --Neolux 15:28 1 Aug 2003 (UTC)
 * Absolutely not newly-created fiction (that is a different project; "Wiktion"). But we should have public domain fiction of already established works (with annotation; such as the works of Mark Twain). We can also have Cliff's Notes-type extensive summaries of works. That would be neat. --mav 17:20 1 Aug 2003 (UTC)
 * Fiction study that does not go very deeply into criticism. Major themes that are universally accepted seem to be the order in textbooks I've dealt with. --Jdl32579 12:57, 14 Jun 2004 (UTC)
 * Remixes and compilations of free content (Wikipedia, ...). Sometimes all the knowledge is already freely available, all it has to be done is to organize it into a book (thought its not that simple). --IzI 18:07, 25 August 2005 (UTC)
 * I really favor -footnotes-, similar to those found along with biblical texts; I find myself having the need to use 'symbols' [e.g. (†), (‡), and (*)] to use for reference-like footnotes.RekonDog (talk) 10:10, 23 August 2009 (UTC)

Key policies

 * While a name hasn't been settled, we'll need a sort of "Wiki" policies for our developer interaction separate from any policies that affect the texts themselves:
 * A proposed outline of textbook considerations is available for Wiki development and discussion.
 * A short outline proposal for a description of processes is open for comment or update at textbook development process.

Timetable for setting things up

 * As it gets done
 * Schedules are for proprietary encyclopedias. LittleDan

Resources and techniques useful in creating educational content

 * Textbook Figures and Diagrams -- I've created a page for the discussion of resources to support the creation of figures and diagrams for wiki textbooks. --StevenBrewer 14:11, 17 October 2005 (UTC)
 * photographic images
 * diagrams
 * maps
 * chemical/mathematical formulae
 * Javascript quizzes
 * Practise problems in traditional format
 * Flash quizzes
 * I could make some, however, Flash is [en:Proprietary]. It's something we should think about before we dive in head-first. KirbyMeister 20:14, 7 May 2004 (UTC)
 * Java applets
 * cookies
 * flashcards: free version of Flashcard Exchange
 * flashcards, matching, wordsearch, and hangman from StudyStack.com
 * Request Tracker (List of requests) - Some of us can make useful drawings/diagrams/photos in subjects that we wouldn't be qualified to write textbooks for

GNU GPL Server Software to draw ideas from
Implemented on thousands of sites and in 30 languages. See website and development site.
 * Moodle - PHP software that aims to make quality online courses (eg distance education) easy to develop and conduct. Learning and development are guided using a social constructionist philosophy.

 * lon-capa might be a better model to support on-line courses. You can create extremely rich interactive problems and easily integrate many different kinds of resources. --StevenBrewer 14:11, 17 October 2005 (UTC)

Potential changes that need to be made to the software

 * Compiling many pages into one large document:
 * A standard book structure (sections) could be created that could be easily translated into LaTeX and further into PDF (with a table of contents and a cover picture). --IzI 17:47, 25 August 2005 (UTC)
 * Automating navigation links within textbooks:
 * Mav's suggestion on the mailing list for modular URLs -- mav,
 * It would be nice if when one comes to the end of a textbook page which one had accessed by clicking on a table of contents entry, one could go on to the next page of the book without having to go back to the table of contents and select the next page from there. On other websites one often sees this done with NEXT buttons; or PREVIOUS for going backwards. -- rew
 * Mav's suggestion as modified by Toby -- Toby Bartels, mav (I very much like Toby's improvements)
 * WikiGroups has already been developed with another php/GPL wiki; - this will allow us to have one TOC/Index per Wikibook but I think we can tweak that software to make it possible to have more than one TOC/Index per book (the trick would be to add the TOC/Index's name after a ? to every link on the TOC/Index page - then when you click on a link to a module, the WikiGroup code on that page will know which %trail% to dispaly. Those links in turn would have the TOC/Index's name after a ? but only for links in the %trail%).
 * Karl's idea to nest URLs in a way that reflects the structure of the book
 * Yet another proposal from Magnus Manske
 * In a related note an automatically generated index (cached!?) would make any textbook more useful m.linger
 * separation of books' contents from each other and from meta information (or rather a means to filter thing out)
 * This should be possible when categories are implemented for Wikipedia.
 * Wikipedia namespace moved to Wikibooks: or something like that when the name is settled

Proposal
The project wikibooks does not seem to be enjoying the same sort of growth that the original wikipedia did. There are probably a number of reasons for this. I shall give a few reasons I think are important and then follow up with some suggestions for the action that could be taken to remedy this.

1) Wikibooks requires a different logon to wikipedia. IMHO it would be better for the project if existing wikipedia developers could be automatically part of wikibooks.  This would allow more free movement of labour from wikipedia to wikibooks.  A strong motivating factor for people developing in wikipedia is the racking up of credit in the form of edited articles.  Why not allow create a contributions page that gives credit to all material worked on in any wiki project?

2) Wikibooks is not interconnected in the same way that wikipedia is. There is a clear need for a different structure to the books as they are meant to be (at least eventually) printed and should therefore have a linear style.  However, if links were made in such a way that they could deleted without interfering with the quality of the text then we could have the same interconnection that wikipedia has without that necessarily causing problems with the style of the books.

 * We need a way to designate sections as being developer areas (but within the actual text itself) We could produce a special heading "Developer's notes and links" which has a template (to ensure consistent usage.

 * We need a special link type that represents a temporary link for development purposes. This would allow the developer links and comments necessary to speed up development.  But at the same time would allow these links to be removed with ease when the wikibook is downloaded.  (Of course links just to wikipedia should be fairly easy to remove automatically although it might help if they appeared in a different colour to remind developers that that material reffered to will not be available to the readers of printed versions.

3) Also links can be made from relevent pages of wikipedia to wikibooks. This should draw in much more labour to liven up wikibooks.  This is one thing that I can implement on my own. So I shall start introducing a section on some selected wikipedia pages entitled "relevent wikibooks".  I may also start links to wikibooks in the main text of some articles in wikipedia.  However, without 2) we may find that people learn to ignore links to wikibooks as they are deadends.

Please comment on these proposals. 82.43.185.179 11:44, 27 Aug 2004 (UTC) (Barnaby dawson)

At least one wikibook is interconnected in the same way that wikipedia is. The Cookbook is highly self-connected. If anything, some connections need to be severed. At the bottom of most recipes you will find a Categories link. This should list all the categories in the cookbook. Unfortunately, it lists all sorts of other random unrelated stuff. Cookbook info is drowned out by the noise. --AlbertCahalan 20:40, 13 Feb 2005 (UTC)

Is #2 really the case? I tend to think that Wikibooks should be created to best serve the user. If that best way is by creating a non-linear book, so be it. Let's use the medium we have and not be tied to an old communication medium. liblamb 05:28, 14 Feb 2005 (UTC)

Learning a new topic needs to be linear (at least to an extent). For example, learning about ionic bonding in chemistry before learning about the periodic table would be very difficult. Coming back to a topic you have already learnt is when non-linear content is generally more useful (which is where GOOD appendices and indices come in). m.linger 02:06, 06 Jul 2005 (UTC)

 * Maybe the most important thing is increasing the ability for this 'free movement of labour' from wikipedia. There is vast untapped potential for all those experts to get interested in and excited about wikibooks. The two best ways, which I think have been floated around as nebulous proposals for some time, are as mentioned above - automatic or connected logon for existing wikipedia editors, and very simply, far more links to wikibooks for the click-happy wikipediholics! Is there anywhere on Wikipedia fairly heavily trafficked by people with the technical knowhow that could support or rubbish these suggestions? Mark Lewis 22:21, 14 Jun 2005 (UTC)

Internationalization
Yes, please!

Participants in textbook planning
You who are reading this, and anyone else.
 * maveric149
 * Karl Wick
 * Toby Bartels [I'm not active, although I still like my suggestion listed in .]
 * Kpjas
 * Cyp (Only made one real edit so far.)
 * Neolux
 * LouI
 * Josh
 * LittleDan
 * Thomas
 * Emperorbma
 * Zanimum
 * Jdl32579
 * Frankie Roberto - mostly interested in Linguistics and planning issues
 * 209.182.169.166 17:24, 16 Dec 2004 (UTC)
 * Yann
 * Nct
 * Everlong

See also: Wikibookians

Reference links

 * Reference works to use to build books
 * Media
 * Boilerplate request for permission

--Karl Wick 02:15 30 Jul 2003 (UTC)

title: Wikibooks:Textbook planning


text:
Computer Programming/Tools

Wikify
Wikify is a program written by Josh Ferguson specifically designed to turn a standard text file into a html/wiki format. It is being used to do the color syntax highlighting for the C++ code in the C++ section of this textbook.

Author's Statement: Wikify can not yet wikify itself, due to the fact that it has html codes in it, which confuses itself. I plan to rewrite it soon so it can wikify itself. Right now though, it works good with simple code. And it can be modified to work with other languages by changing the variables in it. Later on I'll probably move all the important variables into text files so it will be more easy to change things for other languages (i.e. by changing the text files, or extension checks), etc...

The code was written really fast, so the code isn't perfect, even though the author stubbornly maintains that it is close.

Code Snapshot as of 7-22-03

The escape sequences in the code give the wikiscript problems, so here is a link to it. Click on it, and it will take you to a page, where you can click on it, and it will come up.

This should be compilable by any standard C++ compiler.

title: Computer Programming/Tools


text:
BASIC Programming

= List of Pages =

 * 1) Introduction
 * 2) Normative BASIC
 * 3) Minimal BASIC (ANS X3.60-1978, ECMA Standard 55)
 * 4) Full BASIC (ANS X3.113-1987, ECMA Standard 116)
 * 5) Derivative BASIC
 * 6) Basics of BASIC
 * 7) Your First Program
 * 8) PRINT, CLS, and END
 * 9) Variables and Data Types
 * 10) User Input
 * 11) Documentation
 * 12) Control Structures
 * 13) IF...THEN...ELSEIF...ELSE
 * 14) WHILE...WEND
 * 15) FOR...NEXT
 * 16) DO...LOOP
 * 17) GOTO
 * 18) Intermediate BASIC
 * 19) /Random Number Generation/
 * 20) /Subroutines and Functions/
 * 21) Advanced BASIC
 * 22) External Libraries
 * 23) Related Wikibooks
 * 24) Chipmunk Basic – a traditional (vintage) terminal/command-line programming environment with baked-in support for a simple, old-fashioned, and easy-to-learn dialect of BASIC and compiled binary interpreters for MacOS
 * 25) Gambas – a Java-inspired (object-oriented) dialect of BASIC with its own robust IDE, complete with GUI form editor
 * 26) Introduction to BASIC – gives an extended history of BASIC and its variants
 * 27) Visual Basic Classic
 * 28) QBasic Note: The FreeBASIC used in this "Basic Programming" book is intended to be syntax compatible with Microsoft QuickBasic
 * 29) External links
 * 30) FreeBASIC Wiki Manual
 * 31) QB64 official website – a modern, extended BASIC programming language that retains QBasic/QuickBASIC 4.5 compatibility while compiling and freely distributing native interpreter binaries for Windows, Linux, and macOS

title: BASIC Programming


text:
Pascal Programming

Pascal is an influential computer programming language named after the mathematician Blaise Pascal. It was invented by Niklaus Wirth in 1968 as a research project into the nascent field of compiler theory. The backronym PASCAL standing for primary algorithmic scientific commercial application language highlights its suitability for computing tasks in science, making it certainly usable for general programming as well.

About

 * Target demographic
 * Adolescent and adult programming novices

 * Scope
 * Standard Pascal (ISO standard 7185) and selected modern extensions.

 * Description
 * This books will teach high-level programming, using the programming language Pascal.

 * Learning objectives
 * You can analyze trivial to medium difficult programming problems, take general software engineering principles into consideration, and write POC implementations in Pascal using its strengths and knowing its limits. This book will not make a senior-level programmer out of you, but you will definitely pass any college-level introductory CS classes.

 * Not covered here, but (possibly) in other Wikibooks
 * Computer architecture, low-level OS interactions, specific usage of high-level libraries such as.

 * Guidelines for co‑authors
 * American English spelling. Mathematical vocabulary, but explain words if they mean something special in mathematics.
 * Use Unextended Pascal, ISO 7185, as your base, and go from there.
 * Every example program is regarded and has to be by itself complete.

 * Responsible authors
 * These authors ensure the book follow a more or less uniform style and can be read from stem to stern with an acceptable degree of repetition. You are welcome to contribute individual chapters or sections without feeling responsible for the entire book.
 * Kai Burghardt (discuss • contribs) 00:00, 11 September 2021 (UTC)
 * you??

 * Structure
 * See, think, do: Expose the reader to beautiful code and challenge them.

Contents
Standard Pascal
 * Getting started
 * Beginning Pascal
 * Variables and Constants
 * Input and Output
 * Expressions and Branches
 * Routines
 * Enumerations
 * Sets
 * Arrays
 * Strings
 * Records
 * Pointers
 * Files
 * Scopes

Extensions Appendix
 * Schemata
 * Parameters
 * Complex numbers
 * Units
 * Object-oriented Programming
 * Exporting to libraries
 * Foreign Function Interfaces
 * Objective Pascal
 * Generics
 * Miscellaneous extensions
 * Preprocessor Functionality
 * Syntax Cheat Sheet
 * Appendix
 * Register
 * Example code

Alternative resources
Tutorials, Textbooks, and the like: References, Articles on certain topics:
 * “Learn Pascal tutorial” by Tao Yue
 * Doug Cooper: “Oh! Pascal!”
 * (available in multiple languages) Marco Cantù: “Essential Pascal” (see § “Getting Essential Pascal“)
 * (in German) Delphi Crashkurs
 * FreePascal Wiki
 * Chapter “Programming” in the “GNU Pascal Manual”

GNU-Pascal in Beispielen Programación en Pascal Programmation Pascal 파스칼 프로그래밍 Pascal פסקל Programmeren in Pascal Object Pascal Pascal Pascal

title: Pascal Programming


text:
Organic Chemistry/Foundational concepts of organic chemistry/Atomic structure/Octet rule and exceptions

<< Filling electron shells | Molecular orbitals >>

The octet rule refers to the tendency of atoms to prefer to have eight electrons in their valence shell.

The main exception to the rule is hydrogen, which is at lowest energy when it has two electrons in its valence shell.

Other notable exceptions are aluminum and boron, which can function well with six valence electrons; and some atoms beyond group three on the periodic table that can have over 8 electrons, including sulfur. Additionally, some noble gasses can form compounds when expanding their valence shell.

The other tendency of atoms with regard to their electrons is to maintain a neutral charge. Only the noble gasses have zero charge with filled valence octets. All of the other elements have a charge when they have eight electrons all to themselves. The result of these two guiding principles is the explanation for much of the reactivity and bonding that is observed within atoms; atoms seeking to share electrons in a way that minimizes charge while fulfilling an octet in the valence shell.

<< Foundational concepts | << Filling electron shells | Molecular orbitals >>


text:
Organic Chemistry/Foundational concepts of organic chemistry/Acids and bases/Proton donors and acceptors

<< Acids and bases | Electron donors and acceptors >>

The first and earliest definition of acids and bases was proposed in the 1800s by Swedish scientist Svante Arrhenius, who said that an acid was anything that dissolved in water to give up H+ ions. Such as hydrochloric acid and a base was anything that dissolved in water to give up OH- ions such a sodium hydroxide.

The Brønsted-Lowry definition of 1923 broadened this idea a bit:

 * an acid is any proton donor, and a base is any proton acceptor.

The focus of this definition is on donating and accepting protons, and is not limited to aqueous solution.

The Brønsted-Lowry definition of acids and bases is one of two definitions we commonly use. The second definition deals not with protons but with electrons, and has a slightly different emphasis.

<< Foundational concepts | << Resonance | << Acids and bases | Proton donors and acceptors | Alkanes >>


text:
Organic Chemistry/Foundational concepts of organic chemistry/Atomic structure/Molecular orbitals

In organic chemistry we look at the hybridization of electron orbitals in to something called molecular orbitals.

The s and p orbitals in a carbon atom combine into four hybridized orbitals that repel each other in a shape much like that of four balloons tied together.

Remember what a tetrahedron looks like ?

I knew you did ...


text:
Talk:Calculus/Functions

Kids Section
The final section about "Algebraic manipulation" is really strange, i think the author must assume that the reader of that advanced topic "Calculus" already have this very basic knowledge introduced in that section. i suggest omitting it to make the book more concise and aiming

Would it be worth noting in the Graphing Functions section that graphs are usually of the form y=mx+b, rather than the example which used x=my+b? i.e.:

Functions may be graphed by finding the value of f for various y and plotting the points (y, f(y)) in a Cartesian plane...

2(x+2)/2 = x+2. That's not incorrect. Maybe you meant 2x+2/2 ≠ 2 (x+2)/2 = x+2?

Is a function considered a variable or a rule? f(z) doesn't vary with x; f(x) doesn't vary with z. f(x) may be a variable, but f by itself is a rule or a mapping (or a "machine", as textbooks say). y is a dependent variable because it is understood to vary with x -- y(x) is never used.

What's the plus function doing? + is more common; plus(2,2) is almost never used except in Lisp or something. At least call them add, subtract, divide, multiply instead. Using plus(plus(x,y),z) is not the best way to show grouping because (x+y)+z is more common. Even you use 1/(x+2) at the bottom (instead of divide(1,plus(x,2))). --Geoffrey 15:37 23 Jul 2003 (UTC)

Points
You have some good points (and one misreading):

I did not write "2(x+2)/2 = x + 2", but rather "2(x+2)/2 = (x+2)/2", and the latter is incorrect.

As for the issue of a function being a variable or rule, I do not make a distinction in these informal pages between the 'rule' (i.e., function), or the result of the rule (i.e., the value of the function). The value of the function is indeed a variable. Making a big deal about distinguishing the two (very different) concepts may only confuse the matter, but an astute reader make take things 'too' literally and notice inconsistancies as you did. The best way of teaching functions is a matter of opinion. Arguably, one should teach functions in a nonstandard way to expose readers to different perspectives or methods, but also arguably, the standard way of teaching functions is standard because of its superiority.

The use of functions plus(x, y) and times(x, y) is to emphasize that the reader has already been doing functions for a long time, and to give the reader something firm to latch onto to relate abstract concepts of arbitrary functions to specific, familar functions. In particular, the composition of functions is confusing, but most people can compose the 'plus' and 'times' functions with ease. Of course, writing out 'plus' and 'times' is somewhat unnecessary.

At User Talk:Eric119 is some more discussion about Calculus/Functions.

If you see anything that needs modification, just go ahead and modify it!

-- IntMan

A graph is a locus of points in the Cartesian plane

I think that the majority of people who can understand the phrase "locus of points in the Cartesian plane" probably already know what a graph is. I think the fact that the majority of us have done maths to a university or higher level is causing us to use language that may be over-complex for our target audience.

--Imran 12:50 27 Jul 2003 (UTC)

 * Good point. Also, thanks for the exercises. -- IntMan

Hey, why take out my Example section in basic Calculus??

 * Reverted to get rid of the lines, but reverted too far. Sorry. 129.94.6.30 22:17, 11 Feb 2004 (UTC) (User:Dysprosia, logged out)

Ok, thanks :) The lines aren't formatting ppl want? --LWM
 * Horizontal rules aren't pretty, and should be used sparingly. Nice exercise, by the way... Dysprosia 20:49, 12 Feb 2004 (UTC)

- Could someone please clarify the following?:

 * If the language needs to be more clear here, it translates to:

 * Suppose there is a function $$f(x)=\sqrt{x}$$ whose domain is $$\{x|x>0|x\in\mathbb{R}\}$$.
 * }
 * The $$\sqrt{x}$$ basically asks what is the value of $$a$$ so that $$a^2=x$$, the domain encompasses values of $$x$$ such as $$0,2,\frac{1}{2},e,\pi,\text{etc.}$$ but not any negative numbers. As is written currently, this is clear. Musical Inquisit (discuss • contribs) 09:12, 19 February 2021 (UTC)
 * }
 * The $$\sqrt{x}$$ basically asks what is the value of $$a$$ so that $$a^2=x$$, the domain encompasses values of $$x$$ such as $$0,2,\frac{1}{2},e,\pi,\text{etc.}$$ but not any negative numbers. As is written currently, this is clear. Musical Inquisit (discuss • contribs) 09:12, 19 February 2021 (UTC)

- I had something written, but I retract the question. It was a really dumb question. I really did need to brush up my calculus. GEez

don't know meaning of symbols: I took algebra 2 not too long ago and i have no idea what some of these symbols mean. for example the last function in the table of examples that ends in "Takes an input and uses it as boundary values for an integration." the thing that sort of looks like a sideways 8, and the exact meaning of these [] when in this form: (a,b];[a,b). Any chance of adding the meaning of these? or at least tell me a place where i can find them?--V2os 04:04, 31 July 2005 (UTC)

 * I think the last function on the table will be explained later. The sideways 8 stands for infinity. (a,b] is interval notation (first hit on google: http://id.mind.net/~zona/mmts/miscellaneousMath/intervalNotation/intervalNotation.html). And for the Set Notation the funny R means real numbers, and the funny E means the thing on the left is an element of the set on the right (x is an element of the set of real numbers). (disclaimer: I didn't know most of this before today ;) 4.228.240.29

--

Hrmm... but what happens with:

When x is -3? Isn't the function undefined at that point?
 * Yes. Musical Inquisit (discuss • contribs) 09:12, 19 February 2021 (UTC)

-- "A function f(x) has an inverse function if and only if f(x) is one-to-one." Is this true? It has to be onto function as well, isn't it?
 * Imagine the following function $$y=f(x)=2x^{3}$$, where $$x\in\mathbb{Z}$$ and $$y\in\mathbb{R}$$. This function is obviously not surjective since $$y=\frac{1}{2}$$ cannot be found for any $$x\in\mathbb{Z}$$. However, this function is definitely invertible. Take any points $$(x,y)$$ and map them so that the points read $$(y,x)$$. This is given by $$f^{-1}(x)=\sqrt[3]{\frac{x}{2}}$$, with $$x\in\mathbb{R}$$ and $$y\in\mathbb{Z}$$. Hopefully, I made clear how a function does not also need to be onto. Musical Inquisit (discuss • contribs) 09:12, 19 February 2021 (UTC)

?
it seemse that wiki may have some problems... Look at the section on function manipulation. The formatting seems to have gone all wonky, and I have no idea why. THere is one part where it randomly substitutes a formula from the top of the page, instead of what is written there. If anyone has thoughts on how to fix, please do!

Problem sets
If this is supposed to be a textbook, where are the problem sets? It can be difficult to just read this and remember it all. I have half a mind to write a problem set. In fact, here's a few problems to get started (I'll leave the finishin' of it to someone more qualified... unless I get fed up):

Find the inverse of the following functions:

 * 1. $$f(x) = 3x^{2} + 4x$$
 * 2. $$f(x) = \frac{1+x}{2+x}$$

Evaluate:

 * 3.
 * $$(f\circ g)(x)$$ where
 * $$f(x) = \sqrt{x}$$ and
 * $$g(x) = 2x^{2}+3$$

216.215.128.84 (talk) 06:57, 15 December 2007 (UTC)

 * In this textbook, all exercises are part of a separate page. This has its own advantages and disadvantages, but that is the style this wikibooks went for. Musical Inquisit (discuss • contribs) 09:13, 19 February 2021 (UTC)

Length
Anyone else think this page is too long?Tiled (talk) 21:22, 17 November 2009 (UTC)

Translation left/right
Reading this, I got kind of a headache:

Horizontal translation by $$a$$ units left : $$f(x + a)$$

Horizontal translation by $$a$$ units right: $$f(x - a)$$

Is it possible that left and right have been interchanged? Please help me out ?

Andre anckaert (talk) 15:41, 15 January 2010 (UTC)
 * It's a little tricky to get your head around. Take f(x) = x2. Now, translate it left (you'll see): f(x + 1). What does x have to equal for y to equal zero (for the function to return zero)? It has to be -1, meaning, when y = 0, x must be -1, so the function was translated left. That's the best way I can think to explain it, hope it helped if you were still confused. DrSturm (talk) 16:38, 26 May 2010 (UTC)

important functions - quadratic
I am no mathematician and so will not consider editing this myself but the text as it is at present reads:

" A polynomial of the second degree. Its graph is a parabola, unless a = 0. (Don't worry if you don't know what this is.) "

I suspect the parenthesis text is reassuring the reader that knowledge of the parabola is not important, rather than not understanding when a=0 so it would be better written as:

"A polynomial of the second degree. Its graph is a parabola, (Don't worry if you don't know what a parabola is.) unless a = 0."

--JohnH99 (talk) 19:37, 18 April 2010 (UTC)

 * Your edit would be fine! Trust yourself and be bold, there are very few mathematicians around here.  Nothing invalidates the mathematics you know and every editor here (or on any wikimedia project) is on equal footing. Holding a mathematics degree gains you no additional credit or influence.  People are supposed to be judged solely on the quality of their edits, and I have found many non-mathematicians do a really great job editing math articles.  It seems they still remember how to explain the confusing parts, where if you've been studying it for 20 years, you forget which parts were confusing.   And if you make a mistake then (hopefully) someone will fix and and be civil in doing so. Thenub314 (talk) 20:09, 18 April 2010 (UTC)

Bijective and surjective
41.222.2.22 correctly points out these ideas are notable absent form this page and should probably be added. Thenub314 (talk) 16:44, 5 June 2010 (UTC)

 * I added these concepts, in the Modern Understanding of Functions section, so no need to worry. Musical Inquisit (discuss • contribs) 15:50, 31 March 2020 (UTC)

A function in everyday life has X, Y, and Z, not just X and Y
This is a function of two real variables (a location is described by two values - an x and a y ) which results in a vector (which is something that can be used to hold a direction and an intensity).

Real life is not 2 dimensional, therefore, in my mind anyway, a wind vector should have x, y and z parameters. --AaronEJ (discuss • contribs) 00:59, 29 July 2015 (UTC)

Formal definition
Well, saying that a function can be formally defined as, or in terms of, a concept of a "rule" is simply wrong on several counts.

First, a formal definition can only use the term "rule" if this term has already been formally defined for denoting a mathematical object, and in a way which does not rely on the concept of a "function" already. E.g. you cannot rely on "computation", since that formal concept relies on Turing Machines, and these are built out of functions.

Second, the image is misleading. If two functions f and g from R to R are defined by f(x)=x+x and g(x)=2x, then f and g are defined by different rules; one relies on the addition function, the other on the multiplication function. So if functions are the same as rules, then these are distinct functions. However, this is not the view actually taken in mathematics; they are identical functions, since their domains and ranges are identical, and f(x)=g(x) holds for all x in the common domain. Moreover, you will not be allowed to say that the derivative of the function that maps x to x^2 is equal to either of x+x or 2x, since derivative is defined by yet another rule, which involves a limit of a differential quotient.

It isn't intuitive either. When I stumble out of bed in the morning, find a pair of two identical looking socks, and master to put one on each foot, I know that in the end I will have successfully determined a function from the pair of socks to my pair of feet, and yet there is no rule whatsoever involved for determining this function. For similar reasons it doesn't even work well to explain informally that a function is something like a "rule". It blocks students from making any sense of the Choice Axiom, and this is not fortunate for those who want to go on to study Analysis and measure theory. 119.71.71.131 (discuss) 06:02, 4 October 2015 (UTC)

title: Talk:Calculus/Functions


text:
D Programming

The goal for this book is to create a complete, free, open-content, well-organized online book for the D programming language.

D is a programming language being designed as a successor to C++. Until this page gets better written and more informative, the D home can be found here.

Introduction
This book aims at beginners learning D language. It will cover all the language basics and some design aspects. In addition it will introduce topics like multi-threading, GUI programming and standard library to get you started with real-world applications.

Overview
To quote Walter Bright, the author of the D Programming Language: D is a general purpose systems and applications programming language. It is a higher level language than C++, but retains the ability to write high performance code and interface directly with the operating system API's and with hardware. D is well suited to writing medium to large scale million line programs with teams of developers. D is easy to learn, provides many capabilities to aid the programmer, and is well suited to aggressive compiler optimization technology.

D is not a scripting language, nor an interpreted language. It doesn't come with a VM, a religion, or an overriding philosophy. It's a practical language for practical programmers who need to get the job done quickly, reliably, and leave behind maintainable, easy to understand code.

D is the culmination of decades of experience implementing compilers for many diverse languages, and attempting to construct large projects using those languages. D draws inspiration from those other languages (most especially C++) and tempers it with experience and real world practicality.

D is a statically-typed, multi-paradigm language supporting imperative programming, object-oriented programming, and template metaprogramming. It also supports generics and design by contract.

Main features of D
D has many features not seen in C++, implementing design by contract, unit testing, true modules, automatic memory management (garbage collection), first class arrays, associative arrays, dynamic arrays, array slicing, nested functions, inner classes, closures (anonymous functions), and has a reengineered template syntax. D retains C++'s ability to do low-level coding, and adds to it with support for an integrated inline assembler. C++ multiple inheritance is replaced by single inheritance with interfaces and mixins. D's declaration, statement and expression syntax closely matches that of C++.

The inline assembler typifies the differentiation between D and application languages like Java and C#. An inline assembler allows a programmer to enter machine-specific assembly code alongside standard D code—a technique often used by systems programmers to access the low-level features of the processor needed to run programs that interface directly with the underlying hardware, such as operating systems and device drivers.

Built into the language is a documentation generator called Ddoc.

Memory management
Memory is usually managed with garbage collection, but specific objects can be finalized immediately when they go out of scope. Explicit memory management is possible using the overloaded operators new and delete, as well as simply calling C's malloc and free directly. It is also possible to disable garbage collection for individual objects, or even for the entire program if more control over memory management is desired.

Interaction with other systems
C's ABI (Application Binary Interface) is supported as well as all of C's fundamental and derived types, enabling direct access to existing C code and libraries. C's standard library is part of standard D.

In D 1.0, C++'s ABI is not supported, although it can access C++ code that is written to the C ABI, and can access C++ COM (Component Object Model) code. D 2 already supports some interaction with the C++ ABI.

Implementation
Current D implementations compile directly into native code for efficient execution.

Getting and installing D
The Digital Mars D compiler can be obtained from the digital mars website. http://www.digitalmars.com/d/download.html You will need the two files dmd.zip and dmc.zip. According to the manual both files should be extracted into a root directory or one without spaces or other special characters. The location of link.exe should then be added to the path. D programs can now be compiled by calling 'dmd'.

Win32: Example of configuration 1 (dmd)
Create a batch file "dmd_vars.bat" and move it to a directory that is included in the path:

Then:

 * Open a Command Window (cmd.exe)
 * Call "dmd_vars.bat" before compiling something
 * Call "dmd hello.d -ofhello_world" for example, which creates "hello_world.exe"

Note: This works well, even if you have other compilers/tools installed (which might also have link.exe/make.exe etc.)

Your first D programs
/First Program Examples/

With Tango library the classic hello world program is:

/Alternate Program Examples/

With the Phobos library the classic hello world program is:

Compilation
Compiling hello world:

Reference manual to D

 * The official website
 * http://www.digitalmars.com/d/
 * Other websites
 * http://www.prowiki.org/wiki4d/wiki.cgi?FrontPage
 * /Quick Reference Card/

All D features
This is an incomplete list of all D features. It is specifically created to show and teach the D programming language with lots of examples.

 * /Lexical/
 * /Modules/
 * /Declarations/
 * /Types/
 * /Properties/
 * /Attributes/
 * /Pragmas/
 * /Expressions/
 * /Statements/
 * /Arrays/
 * /Structs/
 * /Classes/
 * /Unions/
 * /Enums/
 * /Functions/
 * /Interfaces/
 * /Operator overloading/
 * /Templates/
 * /Mixins/
 * /Contracts/
 * /Conditional compilation/
 * /Exceptions/
 * /Garbage collection/
 * /Memory management/
 * /Floating point/
 * /Inline assembler/
 * /Documentation comments/
 * /Interfacing to C/
 * /Portability guide/
 * /Embedding D in HTML/
 * /Named character entities/
 * /Application Binary Interface/

D compilers

 * /GNU D Compiler/ (GDC)
 * /Digital Mars D compiler/ (DMD)
 * LLVM D Compiler (LDC)

D libraries

 * Libraries
 * Phobos - Standard D Runtime Library
 * Tango - D Runtime Library (beta)
 * Platform
 * Programming D in Windows
 * Programming D in Linux
 * Programming D in Macintosh
 * Programming D in xBSD

D GUI libraries

 * DFL - D Forms Library
 * gtkD - Wrapper for GTK+ (previously known as DUI)
 * wxD - Wrapper for wxWidgets
 * DWT - A D port of Java's SWT Toolkit
 * QtD - Wrapper for Qt Toolkit

Special

 * The D garbage collector
 * Using the Linux Realtime Application Interface (RTAI)

D言語 D Язык программирования D

title: D Programming


text:
Organic Chemistry/Alkanes/Propane thru decane

<< Ethane | Stereoisomers >>

Many more linear alkanes can be formed by adding one additional carbon to the end of a chain of carbons. Ethane is the shortest chain with two carbons but DNA is known to have carbon chains containing millions of linked carbons.

Alkanes are named based on the number of carbons in their longest chain:

Naming carbon chains up to ten

 * methane (1 carbon)CH4
 * ethane (2 carbons)C2H6
 * propane (3 carbons)C3H8
 * butane (4 carbons)C4H10
 * pentane (5 carbons)C5H12
 * hexane (6 carbons)C6H14
 * heptane (7 carbons)C7H16
 * octane (8 carbons)C8H18
 * nonane (9 carbons)C9H20
 * decane (10 carbons)C10H22

The general equation for Alkanes is CnH2n+2.(Substute the n with no.of carbons, eg:-Propane 3 C 'C3H2*3+2=C3H8' )

The suffixis on the first four are from an obscure system but you should be familiar with the rest.

Propane and butane are gases at standard temperature and pressure and are used commonly in lighters. Pentane on down the list are liquids at STP. Octane is the same as the octane in your gas tank.

You may also want to name hydrocarbon chains of more than ten carbons. Here is a list for your reference:

Naming longer carbon chains
........
 * triacontane (30 carbons)
 * tetracontane (40 carbons)
 * pentacontane (50 carbons)
 * hexacontane(60 carbons)

etc...

As the carbon chains get longer, molecules get relatively heavier and tend to move from being gases at Standard Temperature and Pressure to liquids to waxy solids.

Foundational concepts >> | << Alkanes | << Ethane | Propane thru decane | Drawing alkanes >> | Introduction to reactions >>