How can a string vibrate with a number of different frequencies at the same time? This problem occupied the minds of many
of the great mathematicians and musicians of the seventeenth and eighteenth century. Among the people whose work contributed to the solution of this problem are Marin Mersenne, Daniel Bernoulli, the Bach family, Jean-le-Rond d’Alembert, Leonhard Euler, and Jean Baptiste Joseph Fourier.
We discuss Fourier’s theory of
harmonic analysis. This is the decomposition of a
periodic wave into a (usually infinite) sum of sines and cosines. The frequencies involved are the integer multiples of the fundamental frequency of the periodic wave, and each has an amplitude which can be determined as an integral.
The presentation of periodic wave in terms of complex numbers is (usually infinite) sum of exponents. The exponent exp(e) is defined to be the inverse function of ln(x). So, y=exp(x) means the same as x=ln(y). And in this context: i*π=ln(1).
In harmonic analysis, presentation of periodic wave using real numbers is build up on arithmetics, while the presentation over complex numbers is ruled by algebra and geometry.
String wave propagation is described by wave equation as partial differential equation.