Acoustic waves and scalar wave equation

We now consider a simple model of seismic waves, which is widely used in applications. Let us assume that the earth can be treated as an acoustic medium and the influence of variations in density can be ignored. In this case the propagation of seismic waves in the earth can be described by the acoustic wave equation 

An acoustic wave in the time domain, P(r, t), can be represented as a superposition of waves in the frequency domain, p(r, w), using the inverse Fourier transform  

In the frequency domain, equation (13.54) can be written as the Helmholtz equation 

In concluding this section, I introduce a simple 1-D equation for acoustic waves. This equation arises in a case where the wavespeed, c, the pressure field, p, and the source field, / , are functions of only one coordinate, say, the vertical coordinate z. In practice this model can be used, for example, in acoustic well logging when one is interested in wave propagation along the borehole only. In this situation the general 3-D scalar Helmholtz equation (13.56) is reduced to the 1-D Helmholtz equation 

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