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Number of Different resonances in a closed Box

  1. Jan 26, 2017 #1
    1. The problem statement, all variables and given/known data
    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}##.

    2. Relevant equations
    ##c = \frac{\omega}{k}##
    ##f = \frac{\omega}{2\pi}##

    3. 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 occure 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.
     
  2. jcsd
  3. Jan 26, 2017 #2

    BvU

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    What is the volume of a sphere with radius ##2af\over c## ?
     
  4. Jan 27, 2017 #3
    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 in to the number of integer solutions in a cubic box...
     
  5. Jan 29, 2017 #4
    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##.
     
  6. Jan 29, 2017 #5

    BvU

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    Not in the box with physical dimensions a, but in the space of l, m, n... where a radius is ##2af\over c##
     
  7. Jan 29, 2017 #6
    I am sorry, but I still don't have any idea how to solve this.
     
  8. Jan 30, 2017 #7

    BvU

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    You are after the number of
    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
     
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