## Saturday, April 16, 2011

### Physical Fourier - The Harmonic Analyzer and the Differential Analyzer

 The Henrici Harmonic Analyzer

In the early 1800's, French mathematician Joseph Fourier found that any differentiable function can be represented to arbitrary accuracy by a sum of sine and cosine functions, no matter how complicated the function. For example, a periodic function f(x) can be represented by the sum of A sub-n times sin (nx) + B sub-n times cos (nx) for amplitudes A sub-n and B sub-n.

A harmonic analyzer is a physical device for determining the coefficients A sub-n and B sub-n. In 1876, Lord Kelvin, a British mathematical physicist, was first to invent the harmonic analyzer, for the analysis of curve traces related to ocean tidal observations. A paper with the curve of interest is wrapped around a main cylinder. The device is made to follow the curve, and the positions of various subcomponents are determined to give the desired coefficients. Kelvin writes that the " kinematic machine" predicts not merely "the times and heights of high water, but the depths of water at any instant, showing them by a continuous curve, for ... years in advance." Because the tides depend on the positions of the sun, moon, rotation of the earth. shape of the coastline, and sea floor profile, they can be quite complex.

In 1894, German mathematician Olaus Henrici (1840-1918) designed a Harmonic Analyzer for determining the harmonic components of complex sound waves such as those from musical instruments. The device employed several pulleys and glass spheres connected to measuring dials that gave the phase and amplitudes of 10 Fourier harmonic components.

In 1909, German engineer Otto Mader invented a harmonic analyzer that used gears and a pointer tracer to trace a curve; the different gears corresponded to harmonics. The Montgomery harmonic analyzer of 1938 used optical and photoelectric means for determining the harmonic content of a curve. H.C. Montgomery of Bell Labs wrote that the device "is especially adapted to the analysis of speech and music, since it operates directly from a conventional type of sound track on film."

 Stock Hollywood footage of a Differential Analyzer

DIFFERENTIAL ANALYZER

Differential equations play a crucial role in physics, engineering, chemistry, economics, and numerous other disciplines. These equations are relevant whenever a function expresses continuously changing quantities along with some rate of change, expressed as derivatives. Only the simplest differential equations yield solutions that are expressed in compact and explicit formulas with a finite number of basic functions like sines and Bessel Functions.

In 1927, American engineer Vannevar Bush and his colleagues developed a differential analyzer (DA), an analog computer with wheel-and-disk components that could solve, via integration methods, differential equations with several independent variables. The DA was among the first advanced computing devices to be used for practical applications.

Earlier versions of these kinds of devices had their roots in the work of Lord Kelvin and his harmonic analyzer in 1976. In the United States, researchers working at the Wright-Patterson Air force Base and Moore School of Electrical Engineering at the University of Pennsylvania built the DA devices, in part for creating artillery firing tables, prior to the invention of ENIAC (electronic numerical integrator and computer).

Over the years, the DA has had many applications, ranging from soil erosion studies and the building of blueprints for dams to the design of bombs used to destroy German dams during World War II. These devices have even been featured in science fiction films such as the 1956 classic Earth vs the Flying Saucers!

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### Earth vs the Flying Saucers (1956)

The differential analyzer is used to translate intercepted alien messages and learn of their plot to invade. Such a task is way beyond the capabilities of this machine, and the robotic handwriting output device is particularly amusing.

In his 1945 essay, "As We May Think," Bush described his vision of the memex, a futuristic machine that would enhance human memory by allowing humans to store and retrieve information linked by associations, in a manner similar to hypertext on the Web today. He wrote, "It is a far cry from the abacus to the modern keyboard accounting machine. It will be an equal step to the arithmetical machine of the future ..... Relief must be secured from laborious detailed manipulation of higher mathematics .... Man's spirit must be elevated ... "

From: "The MaTH bOOK", by Clifford A. Pickover