In this video we will calculate the acoustic eigenmodes and eigenfrequencies of a room. This simple example has very important practical applications in acoustics. The system we consider here is a parallelepipedic volume of dimensions alpha, beta and gamma. In this volume, as you have seen in a previous video, the pressure obeys the wave equation d^2p/dt^2 - c^2 Laplacian(p) = 0. This is for the internal volume. Additionally, on the six rigid boundaries, the normal velocity component has to vanish. This writes u.n = 0. Once again, in order to convert this condition into a condition for the pressure, we make use of the momentum conservation in a still non viscous fluid for small fluid movements, which links the acceleration to the pressure gradient. Expressed in terms of the pressure, the boundary condition for walls at x=0 and x=alpha then writes dp/dx = 0. Similarly, for the walls at y=0 and beta, it writes dp/dy = 0. On the two last walls at z = 0 and gamma, the boundary condition is dp/dz = 0. Here are finally the equations and boundary conditions that the pressure has to satisfy in this rectangular box. We will now use the variable separation principle to look for solutions of this equations in the form of a product of a function of x only, a function of y only, a function of z only, and a function of time only. If we put the solution written in this form into the wave equation, we obtain an equation in the form of a sum of terms where only one of the function is derived: f'', X'', Y'' and Z''. Next, we divide this equation by XYZf, put X/X'' on the left, and everything else on the right. We obtain on the left something that depends only on x, and on the right something that do not depend on x. Consequently, it must be constant. Let us call this constant -k_x^2. The boundary condition at x = 0 and alpha is dp/dx = 0. This imposes that the derivative of the function capital X vanishes at x = 0 and alpha. The solution is X = cos(k_x x), with k_x = n pi/2 alpha, n being a positive integer. Let us now look at the function Y. The equation can be put in a form where there is only Y''/Y on the left, and the rest on the right. Again it must be constant, and we call this constant k_y^2. The boundary condition takes a similar form as for X: Y' has to vanish at y = zero and beta. The solution of this is cos(k_y y), with k_y = m pi/2 beta, m being an integer. And finally, exactly the same process can be applied to solve the equation for Z, which is found to be cos(k_z z), with k_z = l pi/2 gamma, l being an integer. There is now one last function that is still unknown. It is the function of time f. As before, f''/f can be put on the left, and the rest on the right. A constant then appear, which is choosen to be -omega^2. The solution of this equation is f = A cos(omega t) + B sin(omega t). The constants A and B depend on the initial state of the system at t = 0. The frequancy omega of this harmonic function equals c sqrt(k_x^2+k_y^2+k_z^2). There exists an infinite number of possible values of omega, one for each set of the integer numbers n, m and l. Here is finally the solution we have found for the pressure. It is a triple infinite sum of products of a function of space and an harmonic function of time. The function of space is the eigenmode. It is a product of a function of x, a function of y, and a function of z. And the frequency omega n m l is recalled here. Let us now take a look at the shape of the eigenmodes. The graphs on the right show the isopressure surfaces of the first six eigenmodes. For instance the mode 100 has only variation of the pressure along x, as here the pressure does not depend on the other space variables. Similarly, the mode 010 depends only on y. The mode 110 depends on x and y. The mode 200 depends only on x and has a wavelength that equals half that of the 100 mode. The mode 210 has the same x dependency as the mode 200 and the same y dependency as the mode 010. And finally, the mode 001 is the first mode that depends on z. So, we have calculated the acoustic modes of a rectangular room and their associated eigenfrequencies. What is it useful for? For instance, it can have strong consequences on the equilibrium and quality of sound diffusion by loudspeakers in such rooms. If one place loudspeakers in a room, the frequency spectrum of the sound received sound changes as one walks around. This change in the frequency spectrum is directly related the respective values of the eigenmodes at the various positions. This is a very important effect one has to consider when designing a room for sound mixing tasks for example. These rooms and their sound diffusion systems have to as neutral as possible. Hence the influence of room modes has to be minimized as much as possible. This is why such rooms are generally not rectangular, and covered by absorbing or diffusing panels. We have seen how to calculate the acoustic eigenmodes and eigenfrequencies of a room. These modes are responsible for the fact that one can hear strong changes in the sound when moving in a room and has to be considered seriously when designing rooms for sound reproduction!
