- #1
Marcus95
- 50
- 2
Homework Statement
Show that the possible resonance frequencies in a 3D box with side a are constant multiples of ##(l^2+m^2+n^2)^{1/2}##, where l, m and n are integers. Assume that the box with sides a is filled with a gas in which the speed of sound is constant. Hence show that the number of different resonancec frequencies less than v is ##\approx \frac{4\pi a^3 v^3}{9\sqrt{3}c^3}##.
Homework Equations
##c = \frac{\omega}{k}##
##f = \frac{\omega}{2\pi}##
The Attempt at a Solution
I solved the first part by assuming that the variables could be decoupled and that the waveequation is on the form: ##\Psi(x,y,z,t) = X(x) Y(y) Z(z) cos(\omega t)## where ##X(x) = A_xcos(k_xx) + B_xsin(k_xx)## and equally for Y and Z. Using the boundary conditions that no particle motion should occur at the walls, I ended up with:
##\Psi(x,y,z,t) = C sin(k_xx) sin(k_yy) sin(k_zz) cos(\omega t) ##
where ##k_x = \frac{\pi n}{a}##, ##k_y = \frac{\pi m}{a}##, ##k_z = \frac{\pi l}{a}##
This ultimately lead to:
##f= \frac{c}{2a} (l^2+m^2+n^2)^{1/2} ## as was to be shown.
However, I am completely stuck on the last part. I esentially end up with the inequality:
##(l^2+m^2+n^2)^{1/2} < \frac{2av}{c} ##
but from here I have no idea how to progress to find the number of solutions.