Acoustics

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Helmholtz equation

Since the velocity field $\underline v$ for acoustic waves is irrotational we can define an acoustic potential $?$ by:

$\underline v = \text{grad }\Phi$

Using the propagation equation of the previous paragraph, it is easy to obtain the new equation:

$\nabla ^2 \Phi - \frac{1}{{c_0 ^2 }}\frac{{\partial ^2 \Phi }}{{\partial t^2 }} = 0$

Applying the Fourier Transform, we get the widely used Helmoltz equation:

$\nabla ^2 \hat \Phi + k^2 \hat \Phi = 0$

where $k$ is the wave number associated with $?$ . Using this equation is often the easiest way to solve acoustical problems.