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# Waves

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 .

Subsections

Next: 2D Vibrations of a Up: Fundamentals Previous: Spherical Coordinates -
Prof. Alan Hood
2000-11-06