Number of Different resonances in a closed Box

[Marcus95]

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.

 

[BvU]

What is the volume of a sphere with radius $2af\over c$ ?

 

[Marcus95]

What is the volume of a sphere with radius $2af\over c$ ?

The volume is $\frac{32\pi a^3f^3}{3c^3}$ which is close to what we want to prove, but I don't see how a sphere comes into the number of integer solutions in a cubic box...

 

[Marcus95]

What is the volume of a sphere with radius $2af\over c$ ?

Is it related to somekind of point density?
$V=\frac{4\pi r^3}{3}=\frac{32\pi a^3f^3}{3c^3}=16\cdot3\sqrt{3}\frac{4\pi a^3f^3}{9\sqrt{3}c^3}=\frac{1}{\rho}N$ ?
I still don't see why we get a sphere in a cubic box though and how we are supposed to know $\rho$.

 

[BvU]

Not in the box with physical dimensions a, but in the space of l, m, n... where a radius is $2af\over c$

 

[Marcus95]

Not in the box with physical dimensions a, but in the space of l, m, n... where a radius is $2af\over c$

I am sorry, but I still don't have any idea how to solve this.

 

[BvU]

You are after the number of

resonance frequencies less than v

which is roughly the number of grid points in $l,m,n$ space for which $l^2+m^2+n^2 \le$ a certain number, the (radius of a sphere)²