Acoustics

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The modal theory

This theory comes from the homogeneous Helmoltz equation $\nabla ^2 \hat \Phi + k^2 \hat \Phi = 0$ . Considering a simple geometry of a parallelepiped (L1,L2,L3), the solution of this problem is with separated variables�:

P (x, y, z) = X (x) Y (y) Z (z)

Hence each function X, Y and Z has this form�:

X (x) = A e ? i k x + B e i k x

With the boundary condition $\frac{{\partial P}} {{\partial x}} = 0$ , for x=0 and x=L1 (idem in the other directions), the expression of pressure is�:

$P\left( {x,y,z} \right) = C\cos \left( {\frac{{m\pi x}} {{L1}}} \right)\cos \left( {\frac{{n\pi y}} {{L2}}} \right)\cos \left( {\frac{{p\pi z}} {{L3}}} \right)$

$k^2 = \left( {\frac{{m\pi }}{{L1}}} \right)^2 + \left( {\frac{{n\pi }}{{L2}}} \right)^2 + \left( {\frac{{p\pi }}{{L3}}} \right)^2$

where m,n,p are whole numbers

It is a three-dimensional stationary wave. Acoustic modes appear with their modal frequencies and their modal forms. With a non-homogeneous problem, a problem with an acoustic source Q in r0, the final pressure in r is the sum of the contribution of all the modes described above.

The modal density $\frac{{dN}}{{df}}$ is the number of modal frequencies contained in a range of 1Hz. It depends on the frequency f, the volume of the room V and the speed of sound c0�:

$\frac{{dN}}{{df}} \simeq \frac{{4\pi V}}{{c_0^2 }}f^2$

The modal density depends on the square frequency, so it increase rapidly with the frequency. At a certain level of frequency, the modes are not distinguished and the modal theory is no longer relevant.