Consider a sound wave (that is, a longitudinal displacement wave) in an elastic medium with Young's modulusY (solid) or bulk modulusB (fluid) and unperturbed density ρ0. Suppose this wave is described by the wave function δx(x,f)> whereSx denotes the displacement of a point in the medium from its equilibrium position,x is position along the path of the wave, andt is time. The wave can also be regarded as a pressure wave, described by wave function δp(x,t), where δp denotes the change of pressure in the medium from its equilibrium value.
a) Find the relationship between δp(x,t) and δx(x,t), in general.
b) If the displacement wave is a pure sinusoidal function, with amplitudeA', wave numberk, and angular frequencyco,given by δx(x,t) = A cos (kx – ωt ), what is the corresponding pressure wave function, δp(x,i)? What is the amplitude of the pressure wave?
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