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The displacement, , of an elastic string under tension, ,
with a mass per unit length, , (or equivalently density) satisfies a wave
equation

(6.1) 
where . The general solution for an infitinte string is
given by d'Alembert's solution

(6.2) 
where and are arbitrary functions, representing arbitrary wave
shapes travelling with speeds and without change of shape.
Such waves may be decomposed into Fourier components of the form

(6.3) 
Writing the solution in this complex variable form is just a short
hand notation for the general solution
where and are constants. Note that we may rewrite
Whenever we have constant coefficients in the linear wave
equation we can look for solutions of the form (6.3). Then
substituting into (6.1) we replace
to obtain

(6.4) 
(6.4) is called the dispersion relation, which gives
the frequency, of a Fourier component in terms of its
wavenumber, and the wave speed. The phase speed
is
The period of the wave is and its wavelength is .
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Prof. Alan Hood
20001106