text:
Acoustic theory

Acoustic theory is a scientific field that relates to the description of sound waves. It derives from fluid dynamics. See acoustics for the engineering approach.

For sound waves of any magnitude of a disturbance in velocity, pressure, and density we have

\begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot \mathbf{v} + \nabla\cdot(\rho'\mathbf{v}) & = 0 \qquad \text{(Conservation of Mass)} \\ (\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0 \qquad \text{(Equation of Motion)} \end{align} $$

In the case that the fluctuations in velocity, density, and pressure are small, we can approximate these as

\begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot \mathbf{v} & = 0 \\ \frac{\partial \mathbf{v}}{\partial t} + \frac{1}{\rho_0}\nabla p'& = 0 \end{align} $$

Where $$\mathbf{v}(\mathbf{x},t)$$ is the perturbed velocity of the fluid, $$p_0$$ is the pressure of the fluid at rest, $$p'(\mathbf{x},t)$$ is the perturbed pressure of the system as a function of space and time, $$\rho_0$$ is the density of the fluid at rest, and $$\rho'(\mathbf{x}, t)$$ is the variance in the density of the fluid over space and time.

In the case that the velocity is irrotational ($$\nabla\times \mathbf{v} = 0$$), we then have the acoustic wave equation that describes the system:

\frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} - \nabla^2\phi = 0 $$

Where we have

\begin{align} \mathbf{v} & = -\nabla \phi \\ c^2 & = (\frac{\partial p}{\partial \rho})_s\\ p' & = \rho_0\frac{\partial \phi}{\partial t}\\ \rho' & = \frac{\rho_0}{c^2}\frac{\partial \phi}{\partial t} \end{align} $$

Derivation for a medium at rest
Starting with the Continuity Equation and the Euler Equation:

\begin{align} \frac{\partial \rho}{\partial t} +\nabla\cdot \rho\mathbf{v} & = 0 \\ \rho\frac{\partial \mathbf{v}}{\partial t} + \rho(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p & = 0 \end{align} $$

If we take small perturbations of a constant pressure and density:

\begin{align} \rho & = \rho_0+\rho' \\ p & = p_0 + p' \end{align} $$

Then the equations of the system are

\begin{align} \frac{\partial}{\partial t}(\rho_0+\rho') +\nabla\cdot (\rho_0+\rho')\mathbf{v} & = 0 \\ (\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla (p_0+p') & = 0 \end{align} $$

Noting that the equilibrium pressures and densities are constant, this simplifies to

\begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\nabla\cdot \rho'\mathbf{v} & = 0 \\ (\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0 \end{align} $$

A Moving Medium
Starting with

\begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{w}+\nabla\cdot \rho'\mathbf{w} & = 0 \\ (\rho_0+\rho')\frac{\partial \mathbf{w}}{\partial t} + (\rho_0+\rho')(\mathbf{w}\cdot\nabla)\mathbf{w} + \nabla p' & = 0 \end{align} $$

We can have these equations work for a moving medium by setting $$\mathbf{w} = \mathbf{u} + \mathbf{v}$$, where $$\mathbf{u}$$ is the constant velocity that the whole fluid is moving at before being disturbed (equivalent to a moving observer) and $$\mathbf{v}$$ is the fluid velocity.

In this case the equations look very similar:

\begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\mathbf{u}\cdot\nabla\rho' + \nabla\cdot \rho'\mathbf{v} & = 0 \\ (\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{u}\cdot\nabla)\mathbf{v} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0 \end{align} $$

Note that setting $$\mathbf{u} = 0$$ returns the equations at rest.

Linearized Waves
Starting with the above given equations of motion for a medium at rest:

\begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\nabla\cdot \rho'\mathbf{v} & = 0 \\ (\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0 \end{align} $$

Let us now take $$\mathbf{v},\rho',p'$$ to all be small quantities.

In the case that we keep terms to first order, for the continuity equation, we have the $$\rho'\mathbf{v}$$ term going to 0. This similarly applies for the density perturbation times the time derivative of the velocity. Moreover, the spatial components of the material derivative go to 0. We thus have, upon rearranging the equilibrium density:

\begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot \mathbf{v} & = 0 \\ \frac{\partial \mathbf{v}}{\partial t} + \frac{1}{\rho_0}\nabla p' & = 0 \end{align} $$

Next, given that our sound wave occurs in an ideal fluid, the motion is adiabatic, and then we can relate the small change in the pressure to the small change in the density by

p' = \left(\frac{\partial p}{\partial \rho_{0}}\right)_{s}\rho' $$

Under this condition, we see that we now have

\begin{align} \frac{\partial p'}{\partial t} +\rho_{0}\left(\frac{\partial p}{\partial \rho_0}\right)_{s}\nabla\cdot \mathbf{v} & = 0 \\ \frac{\partial \mathbf{v}}{\partial t} + \frac{1}{\rho_0}\nabla p' & = 0 \end{align} $$

Defining the speed of sound of the system:

c \equiv \sqrt{\left(\frac{\partial p}{\partial \rho_{0}}\right)_{s}} $$

Everything becomes

\begin{align} \frac{\partial p'}{\partial t} +\rho_0c^2\nabla\cdot \mathbf{v} & = 0 \\ \frac{\partial \mathbf{v}}{\partial t} + \frac{1}{\rho_0}\nabla p' & = 0 \end{align} $$

For Irrotational Fluids
In the case that the fluid is irrotational, that is $$\nabla\times\mathbf{v} = 0$$, we can then write $$\mathbf{v} = -\nabla\phi$$ and thus write our equations of motion as

\begin{align} \frac{\partial p'}{\partial t} -\rho_0c^2\nabla^2\phi & = 0 \\ -\nabla\frac{\partial\phi}{\partial t} + \frac{1}{\rho_0}\nabla p' & = 0 \end{align} $$

The second equation tells us that

p' = \rho_0 \frac{\partial \phi}{\partial t} $$

And the use of this equation in the continuity equation tells us that

\rho_0\frac{\partial^2 \phi}{\partial t} -\rho_0c^2\nabla^2\phi = 0 $$

This simplifies to

\frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} -\nabla^2\phi = 0 $$

Thus the velocity potential $$\phi$$ obeys the wave equation in the limit of small disturbances. The boundary conditions required to solve for the potential come from the fact that the velocity of the fluid must be 0 normal to the fixed surfaces of the system.

Taking the time derivative of this wave equation and multiplying all sides by the unperturbed density, and then using the fact that $$p' = \rho_0 \frac{\partial \phi}{\partial t}$$ tells us that

\frac{1}{c^2}\frac{\partial^2 p'}{\partial t^2} -\nabla^2p' = 0 $$

Similarly, we saw that $$p' = \left(\frac{\partial p}{\partial \rho_{0}}\right)_{s}\rho' = c^{2}\rho'$$. Thus we can multiply the above equation appropriately and see that

\frac{1}{c^2}\frac{\partial^2 \rho'}{\partial t^2} -\nabla^2\rho' = 0 $$

Thus, the velocity potential, pressure, and density all obey the wave equation. Moreover, we only need to solve one such equation to determine all other three. In particular, we have

\begin{align} \mathbf{v} & = -\nabla \phi \\ p' & = \rho_0 \frac{\partial \phi}{\partial t}\\ \rho' & = \frac{\rho_0}{c^2}\frac{\partial\phi}{\partial t} \end{align} $$

For a moving medium
Again, we can derive the small-disturbance limit for sound waves in a moving medium. Again, starting with

\begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\mathbf{u}\cdot\nabla\rho' + \nabla\cdot \rho'\mathbf{v} & = 0 \\ (\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{u}\cdot\nabla)\mathbf{v} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0 \end{align} $$

We can linearize these into

\begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\mathbf{u}\cdot\nabla\rho' & = 0 \\ \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{v} + \frac{1}{\rho_0}\nabla p' & = 0 \end{align} $$

For Irrotational Fluids in a Moving Medium
Given that we saw that

\begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\mathbf{u}\cdot\nabla\rho' & = 0 \\ \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{v} + \frac{1}{\rho_0}\nabla p' & = 0 \end{align} $$

If we make the previous assumptions of the fluid being ideal and the velocity being irrotational, then we have

\begin{align} p' & = \left(\frac{\partial p}{\partial \rho_{0}}\right)_{s}\rho' = c^{2}\rho' \\ \mathbf{v} & = -\nabla\phi \end{align} $$

Under these assumptions, our linearized sound equations become

\begin{align} \frac{1}{c^2}\frac{\partial p'}{\partial t} -\rho_0\nabla^2\phi+\frac{1}{c^2}\mathbf{u}\cdot\nabla p' & = 0 \\ -\frac{\partial}{\partial t}(\nabla\phi) - (\mathbf{u}\cdot\nabla)[\nabla\phi] + \frac{1}{\rho_0}\nabla p' & = 0 \end{align} $$

Importantly, since $$\mathbf{u}$$ is a constant, we have $$(\mathbf{u}\cdot\nabla)[\nabla\phi] = \nabla[(\mathbf{u}\cdot\nabla)\phi]$$, and then the second equation tells us that

\frac{1}{\rho_0} \nabla p' = \nabla\left[\frac{\partial\phi}{\partial t} + (\mathbf{u}\cdot\nabla)\phi\right] $$

Or just that

p' = \rho_{0}\left[\frac{\partial\phi}{\partial t} + (\mathbf{u}\cdot\nabla)\phi\right] $$

Now, when we use this relation with the fact that $$\frac{1}{c^2}\frac{\partial p'}{\partial t} -\rho_0\nabla^2\phi+\frac{1}{c^2}\mathbf{u}\cdot\nabla p' = 0$$, alongside cancelling and rearranging terms, we arrive at

\frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} - \nabla^2\phi + \frac{1}{c^2}\frac{\partial}{\partial t}[(\mathbf{u}\cdot\nabla)\phi] + \frac{1}{c^2}\frac{\partial}{\partial t}(\mathbf{u}\cdot\nabla\phi) + \frac{1}{c^2}\mathbf{u}\cdot\nabla[(\mathbf{u}\cdot\nabla)\phi] = 0 $$

We can write this in a familiar form as

\left[\frac{1}{c^2}\left(\frac{\partial}{\partial t} + \mathbf{u}\cdot\nabla\right)^{2} - \nabla^{2}\right]\phi = 0 $$

This differential equation must be solved with the appropriate boundary conditions. Note that setting $$\mathbf{u}=0$$ returns us the wave equation. Regardless, upon solving this equation for a moving medium, we then have

\begin{align} \mathbf{v} & = -\nabla \phi \\ p' & = \rho_{0}\left(\frac{\partial}{\partial t} + \mathbf{u}\cdot\nabla\right)\phi\\ \rho' & = \frac{\rho_{0}}{c^{2}}\left(\frac{\partial}{\partial t} + \mathbf{u}\cdot\nabla\right)\phi \end{align} $$

title: Acoustic theory


text:
Alexander Mackenzie (politician)

Alexander Mackenzie (January 28, 1822 – April 17, 1892) was a Canadian politician who served as the second prime minister of Canada, in office from 1873 to 1878.

Mackenzie was born in Logierait, Perthshire, Scotland. He left school at the age of 13, following his father's death, to help his widowed mother, and trained as a stonemason. Mackenzie immigrated to the Province of Canada when he was 19, settling in what became Ontario. His masonry business prospered, allowing him to pursue other interests – such as the editorship of a pro-Reformist newspaper called the Lambton Shield. Mackenzie was elected to the Legislative Assembly of the Province of Canada in 1862, as a supporter of George Brown.

In 1867, Mackenzie was elected to the new House of Commons of Canada for the Liberal Party. He became leader of the party (thus Leader of the Opposition) in mid-1873, and a few months later succeeded John A. Macdonald as prime minister, following Macdonald's resignation in the aftermath of the Pacific Scandal. Mackenzie and the Liberals won a clear majority at the 1874 election. He was popular among the general public for his humble background and consistent democratic principles.

As prime minister, Mackenzie continued the nation-building programme that had been begun by his predecessor. His government established the Supreme Court of Canada and Royal Military College of Canada, and created the District of Keewatin to better administer Canada's newly acquired western territories. However, it made little progress on the transcontinental railway, and struggled to deal with the aftermath of the Panic of 1873. At the 1878 election, Mackenzie's government suffered a landslide defeat. He remained leader of the Liberal Party for another two years, and continued on as a Member of Parliament (MP) until his death, due to a stroke.

Early life
Mackenzie was born on January 28, 1822, in Logierait, Perthshire, Scotland, the son of Mary Stewart (Fleming) and Alexander Mackenzie Sr. (born 1784) who were married in 1817. The site of his birthplace is known as Clais-'n-deoir (the Hollow of the Weeping), where families said their goodbyes as the convicted were led to nearby Gallows Hill. The house in which he was born was built by his father. He was the third of 10 boys, seven of whom survived infancy. Alexander Mackenzie Sr. was a carpenter and ship's joiner who had to move around frequently for work after the end of the Napoleonic Wars in 1815. Mackenzie's father died on March 7, 1836, and at the age of 13, Alexander Mackenzie Jr. was thus forced to end his formal education to help support his family. He apprenticed as a stonemason and met his future wife, Helen Neil, in Irvine, where her father was also a stonemason. The Neils were Baptist and shortly thereafter, Mackenzie converted from Presbyterianism to Baptist beliefs. Together with the Neils, he immigrated to Canada in 1842 to seek a better life. Mackenzie's faith was to link him to the increasingly influential temperance cause, particularly strong in Canada West where he lived, a constituency of which he was to represent in the Parliament of Canada.

The Neils and Mackenzie settled in Kingston, Ontario. The limestone in the area proved too hard for his stonemason tools, and not having money to buy new tools, Mackenzie took a job as a labourer constructing a building on Princess Street. The contractor on the job claimed financial difficulty, so Mackenzie accepted a promissory note for summer wages. The note later proved to be worthless. Subsequently, Mackenzie won a contract building a bomb-proof arch at Fort Henry. He later became a foreman on the construction of Kingston's four Martello Towers – Murney Tower, Fort Frederick, Cathcart Tower, and Shoal Tower. He was also a foreman on the construction of the Welland Canal and the Lachine Canal. While working on the Beauharnois Canal, a one-ton stone fell and crushed one of his legs. He recovered, but never regained the strength in that leg. While in Kingston, Mackenzie became a vocal opponent of religious and political entitlement and corruption in government.

Mackenzie married Helen Neil (1826–52) in 1845 and with her had three children, with only one girl, Mary, surviving infancy. Helen and he moved to Sarnia, Ontario (known as Canada West) in 1847 and Mary was born in 1848. They were soon joined from Scotland by the rest of Mackenzie's brothers and his mother. He began working as a general contractor, earning a reputation for being a hard-working, honest man, as well as having a working man's view on fiscal policy. Mackenzie helped construct many courthouses and jails across southern Ontario. A number of these still stand today, including the Sandwich Courthouse and Jail now known as the Mackenzie Hall Cultural Centre in Windsor, Ontario, and the Kent County Courthouse and Jail in Chatham, Ontario. He even bid, unsuccessfully, on the construction of the Parliament buildings in Ottawa in 1859. Helen died in 1852, finally succumbing to the effects of excessive doses of mercury-based calomel used to treat a fever while in Kingston. In 1853, he married Jane Sym (1825–93).

Early political involvement
Mackenzie involved himself in politics almost from the moment he arrived in Canada. He fought passionately for equality and the elimination of all forms of class distinction. In 1851, he became the secretary for the Reform Party for Lambton. After convincing him to run in Kent/Lambton, Mackenzie campaigned relentlessly for George Brown, owner of the Reformist paper The Globe in the 1851 election, helping Brown to win his first seat in the Legislative Assembly. Mackenzie and Brown remained the closest of friends and colleagues for the rest of their lives. In 1852, Mackenzie became editor of another reformist paper, the Lambton Shield. As an editor, Mackenzie was perhaps a little too vocal, leading the paper to a lawsuit for libel against the local conservative candidate. Because a key witness claimed Cabinet Confidence and would not testify, the paper lost the suit and was forced to fold due to financial hardship. After his brother, Hope Mackenzie, declined to run, Alexander was petitioned to run and won his first seat in the Legislative Assembly as a supporter of George Brown in 1861. When Brown resigned from the Great Coalition in 1865 over reciprocity negotiations with the United States, Mackenzie was invited to replace him as president of the council. Wary of Macdonald's motivations and true to his principles, Mackenzie declined.

He entered the House of Commons of Canada in 1867, representing the Lambton, Ontario, riding. No cohesive national Liberal Party of Canada existed at the time and with Brown not winning his seat, no official leader emerged. Mackenzie did not believe he was the best qualified for the position, and although he resisted offers of the position, he nevertheless sat as the de facto leader of the Official Opposition.

Mackenzie served as a Major in the 27th Lambton Battalion of Infantry from 1866 to 1874, serving on active duty during the Fenian Raids in 1870.

Prime Minister (1873–1878)
When the Macdonald government fell due to the Pacific Scandal in 1873, the Governor General, Lord Dufferin, called upon Mackenzie, who had been chosen as leader of the Liberal Party a few months earlier, to form a new government. Mackenzie formed a government and asked the Governor General to call an election for January 1874. The Liberals won a majority of the seats in the House of Commons having garnered 40% of the popular vote.

Mackenzie remained prime minister until the 1878 election when Macdonald's Conservatives returned to power with a majority government.

For a man of Mackenzie's humble origins to attain such a position was unusual in an age which generally offered such opportunity only to the privileged. Lord Dufferin expressed early misgivings about a stonemason taking over government, but on meeting Mackenzie, Dufferin revised his opinions:

"However narrow and inexperienced Mackenzie may be, I imagine he is a thoroughly upright, well-principled, and well-meaning man."

Mackenzie served concurrently as Minister of Public Works and oversaw the completion of the Parliament buildings. While drawing up the plans for the West Block, he included a circular staircase leading directly from his office to the outside of the building, which allowed him to escape the patronage-seekers waiting for him in his ante-chamber. Proving Dufferin's reflections on his character to be true, Mackenzie disliked intensely the patronage inherent in politics. Nevertheless, he found it a necessary evil to maintain party unity and ensure the loyalty of his fellow Liberals.

In keeping with his democratic ideals, Mackenzie refused the offer of a knighthood three times, and was thus the only one of Canada's first eight Prime Ministers not to be knighted. He also declined appointment to the UK Privy Council and hence does not bear the title "Right Honourable". His pride in his working class origins never left him. Once, while touring Fort Henry as prime minister, he asked the soldier accompanying him if he knew the thickness of the wall beside them. The embarrassed escort confessed that he didn't and Mackenzie replied, "I do. It is five feet, ten inches. I know, because I built it myself!"

As Prime Minister, Alexander Mackenzie strove to reform and simplify the machinery of government, achieving a remarkable record of reform legislation. He introduced the secret ballot; advised the creation of the Supreme Court of Canada; the establishment of the Royal Military College of Canada in Kingston in 1874 and the creation of the Office of the Auditor General in 1878. He completed the Intercolonial Railway, but struggled to progress on the national railway due to a worldwide economic depression, almost coming to blows with Governor General Lord Dufferin over imperial interference. Mackenzie stood up for the rights of Canada as a nation and fought for the supremacy of Parliament and honesty in government. Above all else, he was known and loved for his honesty and integrity.

However, his term was marked by economic depression that had grown out of the Panic of 1873, which Mackenzie's government was unable to alleviate. In 1874, Mackenzie negotiated a new free trade agreement with the United States, eliminating the high protective tariffs on Canadian goods in US markets. However, this action did not bolster the economy, and construction of the CPR slowed drastically due to lack of funding. In 1876, the Conservative opposition announced a National Policy of protective tariffs, which resonated with voters. When an election was held at the conclusion of Mackenzie's five-year term, the Conservatives were swept back into office in a landslide victory.

Supreme Court appointments
Mackenzie chose the following jurists to be appointed as justices of the Supreme Court of Canada by the Governor General:
 * Sir William Buell Richards (Chief Justice) – September 30, 1875
 * Télesphore Fournier – September 30, 1875
 * William Alexander Henry – September 30, 1875
 * Sir William Johnstone Ritchie – September 30, 1875
 * Sir Samuel Henry Strong – September 30, 1875
 * Jean-Thomas Taschereau – September 30, 1875
 * Sir Henri Elzéar Taschereau – October 7, 1878

Later life
After his government's defeat, Mackenzie remained Leader of the Opposition for another two years, until 1880. In 1881, he became the first president of The North American Life Assurance Company. He was soon struck with a mysterious ailment that sapped his strength and all but took his voice. Sitting in silence, he nevertheless remained an undefeated MP until his death in 1892 from a stroke that resulted from hitting his head during a fall. He died in Toronto and was buried in Lakeview Cemetery in Sarnia, Ontario.

Character
Mackenzie's first biography in 1892 referred to him as Canada's Stainless Statesman. He was a devout Baptist and teetotaller who found refuge in, and drew strength from, his family, friends, and faith. He was also a loyal friend and an incorrigible prankster (stuffed chimney on young in-laws; rolled boulder down Thunder Cape towards friend A. McKellar; burned Tory campaign placards in hotel woodstove early in morning). Unpretentious and down to earth, his public official austerity was in striking contrast to private compassion and giving nature. He was the soul of honour and integrity, a proud man who sought no recognition or personal enrichment and accepted gifts reluctantly. He preferred to follow than to lead (unreferenced – many times he refused leadership offers) and often found that duty outweighed the heavy burden of office. He was uncompromising on his principles, perhaps too much so. An historian at the time said, "He was, and ever will remain, the Sir Galahad of Canadian politics."

Very proud of his Scottish heritage, he was forever a Scot: "Nemo me impune lacessit" (no one attacks me with impunity). The Upper Canada rebellion leader W.L. Mackenzie said of him, "He is every whit a self-made, self-educated man. Has large mental capacity and indomitable energy." Canada's Governor General, Lord Dufferin, said he was "as pure as crystal, and as true as steel, with lots of common sense." A close friend, Chief Justice Sir Louis Davies, said he was "the best debater the House of Commons has ever known." A friend and colleague in Cabinet who went on to become prime minister of Canada, Sir Wilfrid Laurier, said he was "one of the truest and strongest characters to be met within Canadian history. He was endowed with a warm heart and a copious and rich fancy, though veiled by a somewhat reticent exterior, and he was of friends the most tender and true." Another friend and colleague, who went on to become premier of Ontario, Sir George Ross, said, "Mackenzie was sui generis a debater. His humorous sallies blistered like a blast from a flaming smelter. His sterling honesty is a great heritage, and will keep his memory green to all future generations." At his eulogy, Rev. Dr. Thomas compared him to the Duke of Wellington, who "stood four square, to all the winds that blow."

Newspapers around the world and in Canada had this to say about him. The London Times – the untiring energy, the business-like accuracy, the keen perception and reliable judgment, and above all the inflexible integrity, which marked his private life, he carried without abatement of one jot into his public career. The Westminster Review – a man, who although, through failing health and failing voice, he had virtually passed out of public life, yet retained to the last the affectionate veneration of the Canadian people as no other man of the time can be said to have done. The Charlottetown Patriot – in all that constitutes the real man, the honest statesman, the true patriot, the warm friend, and sincere Christian, he had few equals. Possessed of a clear intellect, a retentive memory, and a ready command of appropriate words, he was one of the most logical and powerful speakers we have ever heard. The St. John Telegraph – he was loved by the people and his political opponents were compelled to respect him even above their own chosen leader. As a statesman, he has had few equals. The Montreal Star – it is one of the very foremost architects of the Canadian nationality that we mourn. In the dark days of ’73, Canadians were in a state of panic, distrusting the stability of their newly-built Dominion; no one can tell what would have happened had not the stalwart form of Alexander Mackenzie lifted itself above the screaming, vociferating and denying mass of politicians, and all Canada felt at once, there was a man who could be trusted. The Toronto Globe – he was a man who loved the people and fought for their rights against privilege and monopoly in every form. The Philadelphia Record – Like Caesar, who twice refused a knightly crown, Alexander Mackenzie refused knighthood three times. Unlike Caesar, he owed his political overthrow to his incorruptible honesty and unswerving integrity.

Legacy
In their 1999 study of the Prime Ministers of Canada, which included the results of a survey of Canadian historians, J. L. Granatstein and Norman Hillmer found that Mackenzie was in 11th place just after John Sparrow David Thompson.

Namesakes
The following are named in honour of Alexander Mackenzie:
 * The Mackenzie Mountain Range in the Yukon and Northwest Territories
 * Mount Mackenzie, in the Selkirk Mountains of British Columbia
 * The Mackenzie Building, and the use of the Mackenzie tartan by the bands at the Royal Military College of Canada in Kingston, Ontario, "Alexander Mackenzie", the Royal Military College of Canada March for bagpipes, was composed in his honour by Pipe Major Don M. Carrigan, who was the College Pipe Major 1973 to 1985.
 * Mackenzie Hall in Windsor, Ontario
 * Alexander Mackenzie Scholarships in Economics and Political Science at McGill University and the University of Toronto
 * Alexander MacKenzie Park in Sarnia, Ontario
 * Alexander Mackenzie High School in Sarnia
 * Alexander Mackenzie Housing Co-Operative Inc. in Sarnia
 * Mackenzie Avenue, Ottawa, Ontario
 * Mackenzie Tower, West Block, Parliament Hill, Ottawa, Ontario

Other honours

 * A monument is dedicated to his tomb in Lakeview Cemetery, Sarnia, Ontario
 * "Honourable Alexander Mackenzie" (1964) by Lawren Harris, head of the Department of Fine Arts, Mount Allison University, now hangs in the Mackenzie Building, Royal Military College of Canada. The unveiling ceremony was performed by the Right Honourable Louis St. Laurent, a Canadian former Prime Minister, and the gift was accepted by the Commandant, Air Commodore L.J. Birchall. The painting was commissioned in memory of No. 244, Lieut.-Col, F.B. Wilson, O.B.E., her deceased husband, by Mrs, F.W. Dashwood. Also taking part in the ceremony was the Honourable Paul Hellyer, Minister of National Defence, President and Chancellor of the college. In attendance was Mrs. Burton R. Morgan of Ottawa, great-granddaughter of Alexander Mackenzie.
 * Burgess tickets presented to Alexander Mackenzie in Dundee, Dunkeld, Logierait, Irvine, and Perth Scotland

General sources

 * Alexander Mackenzie fonds at Library and Archives Canada
 * Alexander Mackenzie fonds at Library and Archives Canada

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

title: Alexander Mackenzie (politician)


text:
Talk:Algorithm

Where to cover "algorithms" as used in discussions about social media, big tech, etc?
I think this article should have some reference to the fact that "algorithms" are now being discussed in the context of social media platforms, "Big Tech", and related Internet technologies. As governments in Europe, the US, and other regions are discussing whether to regulate social platforms and companies such as Twitter, Facebook, and Google, the discussion often comes back to "algorithms". For instance, this is coming up quite often in discussions around Section 230 in the US. In Europe there is an EU activity of "Algorithm Awareness-Building." Just recently, two representatives introduced the "Protecting Americans from Dangerous Algorithms Act".

Given that many people may turn to Wikipedia to help understand what an "algorithm" is, I feel like there should be some mention of this usage on the page. But given the amount of detailed and academic info on the current page, I'm not sure how to best integrate this other content. Perhaps something brief in the lead paragraph and then a mention under "Informal definition"? Or a new section about "Algorithms in current politics"? Or a new section under "History" (although that seems mostly about the refinement of algorithms)? Any thoughts? - Dyork (talk) 17:25, 20 October 2020 (UTC)

 * I agree that there should be some mention of the social media usage on this page. However I think it's wrong to conflate that usage with the real meaning of the word "algorithm". In the social-media context, "algorithm" usually just means that some kind of processing has been done; it doesn't mean that the processing was in accordance with any kind of clear procedure, which is what an algorithm is supposed to be. I don't want this page to be about any old data-processing. That's not an algorithm. That's just what journos refer to as "algorithm". If we let that go, then we'll need a new term to refer to real algorithms: e.g. "Explicit procedure for performing transformations on data", which is a bit unwieldy.

 * Can we have our word back please, mister?

 * 12:30, 27 October 2021 (UTC)

 * Absolutely. I'm surprised there isn't a separate article Algorithm (social media) that's written for a much more lay audience. Social media algorithms are usually more complex and aren't talked about in terms of big-O or anything like that. Instead, it's on the social impact of various choices of what to hilight -- things like the filter bubble, algorithmic radicalization, and even allegations that some social media company's decisions can sway political elections. And there are many news articles about this topic. --Hirsutism (talk) 18:45, 24 August 2021 (UTC)

 * Or perhaps an article about the Social impact of algorithms, with a heavy emphasis on social media algorithms? Because there are articles about algorithms being used to hire and fire people, for insurance and policing, and several other places they can introduce bias, and those should probably also be addressed in the for-a-lay-audience article. --Hirsutism (talk) 20:18, 24 August 2021 (UTC)
 * "In the social-media context, "algorithm" usually just means that some kind of processing has been done; it doesn't mean that the processing was in accordance with any kind of clear procedure, which is what an algorithm is supposed to be. I don't want this page to be about any old data-processing. That's not an algorithm. That's just what journos refer to as "algorithm". If we let that go, then we'll need a new term to refer to real algorithms: e.g. "Explicit procedure for performing transformations on data", which is a bit unwieldy."
 * Thank you! I've been trying to understand what news/social media means by algorithms. But none of the actual definitions help. Now I see that that is because the popular use of algorithm is similar to the popular use of 'metaphysics.' It means something different and sillier.
 * We Social impact of algorithms: I see that there is already a wiki dealing with Algorithmic Bias that probably accounts for this. Jesseraekern (talk) 18:14, 2 December 2022 (UTC)
 * I've written a bit of an article at User:Hirsutism/Algorithm (social media). Could someone take a look at it, and tell me if they think the article should be expanded to cover all AI-with-a-social-impact, or should it focus only on social media algorithms? On the other side, there's quite a few articles about this already, I don't know whether we even need a new separate article, other than a short summary in this article? --Hirsutism (talk) 20:41, 29 August 2021 (UTC)

 * The "algorithms" that are discussed in the context of social media are usually not algorithms as discussed in this article; the social media "algorithms" that are derived from big datasets by training ML systems are not transparent, and can't even really be expressed. I think it's incorrect to call these systems "algorithms".

 * However I do appreciate that journalists and ML promoters do refer to these systems as "algorithms". Therefore I support the creation of a page for "algorithms" in a social media context. That might make it easier to keep this page focused on the subject of real algorithms.

 * MrDemeanour (talk) 08:56, 15 September 2021 (UTC)

 * Agreed: "algorithm" as used in a social-media context is an important concept, quite distinct from this article's focus. The content of Hirsutism's draft at User:Hirsutism/Algorithm (social media) looks like a good start, but I'd argue that social media algorithms are simply a special case of a recommender systems, both on the technical and on the social/behavioral side. The social and behavioral aspects of recommender systems are mostly covered in the filter bubble article, but that's only one aspect. I'm not sure how to organize this better.... --Macrakis (talk) 14:14, 27 October 2021 (UTC)

'Algorithm' as used by non-computer-scientists / non-experts to discuss recommender systems in social media (which, surely, are implemented by large numbers of algorithms) is a technically-incorrect use of the word almost entirely unrelated to this article. — Preceding unsigned comment added by 72.200.126.73 (talk) 12:52, 23 August 2022 (UTC)

Page tidy up needed.
This page is quite long and rambling and seems to have some content that could go to other pages such as the section on Euclid's algorithm and the conversation between Turing etc. This is a gateway concept to many related issues and it would be good if it was easier for the general reader to navigate. I can get started on some of this. Amanda Lawrence 01:16, 27 October 2021 (UTC)

 * I whole-heartedly agree that this article strays off topic in many places. In particular, there is a paragraph in the "Expressing algorithms" section that talks about Turing machines. I have no problem with that idea, but it seems to come out of nowhere to me as currently written. I would either delete it or add a transitional phrase to the beginning of the first sentence. But I'm not going to do that right now, since there is no need to fine-tune this section if it is likely to be significantly modified in line with some of the other discussion on this Talk page.
 * Mike-c-in-mv (talk) 23:44, 18 June 2023 (UTC)
 * @AmandaSLawrence 117.20.112.19 (talk) 00:25, 6 January 2024 (UTC)
 * The whole section on Euclid's algorithm apart from the first para could be cut which would really help the flow - the examples could be quite short if they have their own page. But I'm not sure if all that long text should be added to the Euclid page or just cut. Amanda Lawrence 11:12, 25 January 2024 (UTC)

March 24, 2023
As added by User:2601:196:180:dc0:7c5a:37cb:a8c6:32d7:

Someone, i guess(talk i guess|le edit list) 02:21, 25 March 2023 (UTC)

 * I think it's the end of the sentence, with text from the quote cut out. As in "Endofquotesentence. ...because reasons." with part of the quote removed. -- Roundish   ⋆  t c) 02:35, 25 March 2023 (UTC)

 * Exactly. I pulled the source. One box is to be singled out and called the starting point. We now further assume that a specific problem is to be given in symbolic form by a finite number of boxes being marked with a stroke. As I read MOS:ELLIPSES, we leave the period in the quotation mark, then do nonbreaking space, ellipsis, space, and pick back up with the text. The alternative would be to gloss the text, which would be something like (condensing) "...and called the starting point. [A] finite number of boxes [are] marked with a stroke." —C.Fred (talk) 02:45, 25 March 2023 (UTC)

 * Unrelated but I don't enjoy the fact this one HTML comment started an edit war. Someone, i guess(talk i guess|le edit list)  02:54, 25 March 2023 (UTC)
 * So are you saying you agree with the punctuation per the IP's edits and only meant to strike the inline comment? —C.Fred (talk) 02:59, 25 March 2023 (UTC)
 * Just checked - turns out this is exactly what happened lol. Still didnt warrant an edit war - they could've just redone the punctuation and removed their HTML comment and it could've been prevented Someone, i guess(talk i guess|le edit list)  03:03, 25 March 2023 (UTC)
 * The punctuation, in my opinion, seems to be correct seeing this conversation. I'll restore the latest version using ". ..." Someone, i guess(talk i guess|le edit list)  03:01, 25 March 2023 (UTC)

Algorithm defined as a finite sequence of instructions
I'm not sure if this is a good definition as algorithms may contain loops that sometimes may not stop (unless you close the program) and yet be correct, for instance a service that runs on the background Luiz Felipe de Barros Jordão Costa (talk) 01:40, 24 December 2023 (UTC)

Merge "Algorithmics" into this page
Anyone against merging Algorithmics into Algorithm? That page seems almost completely useless. Weebney (talk) 19:55, 23 January 2024 (UTC)

 * In favour - go for it. I don't think we need the illustration - not for the term anyway. Amanda Lawrence 10:57, 25 January 2024 (UTC)
 * It is finished. Weebney (talk) 06:02, 7 February 2024 (UTC)

Contrast with heuristics
While I understand the thought behind contrasting algorithms with heuristics, I think it makes the page longer than it needs to be. It doesn't add new information about algorithms nor does it clear up common misconceptions. It seems like more of a detour in what should be a concise article. Any thoughts on keeping or removing the heuristics paragraph? Clubspike2 (talk) 11:56, 25 June 2024 (UTC)

title: Talk:Algorithm