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Homework Statement:

The radial oscillations of an ideal gas in a spherical cavity of radius r' are governed by the spherical wave equation subject to the boundary condition ψ(r',t)=0. ψ(r,t) is the radial displacement and v is the speed of sound. Show that the general solution of this equation is written
$$ \psi(r,t) = \Sigma_{i=1,\infty} sinc(i\frac{ir\pi}{r'})cos(\omega_{i}t\phi_{i})$$
where
$$ sinc(x) = \frac{sin(x)}{x} $$
$$ \omega_{i} = i \pi \frac{v}{r'} $$
Homework Equations:
 $$ \frac{\partial^2 \psi}{\partial t^2} = v^{2}(\frac{\partial^2 \psi}{\partial r^2} + \frac{2}{r} \frac{\partial \psi}{\partial r}) $$
Since the spherical wave equation is linear, the general solution is a summation of all normal modes.
To find the particular solution for a given value of i, we can try using the method of separation of variables.
$$ ψ(r,t)=R(r)T(t)ψ(r,t)=R(r)T(t) $$
Plug this separable solution into the spherical wave equation.
$$ \frac{1}{T}\frac{\partial^{2}T}{\partial t^{2}} = \frac{v^{2}}{R} \frac{\partial^{2}R}{\partial r^{2}} + \frac{2 v^{2}}{rR} \frac{\partial R}{\partial r} $$
$$ R(r)=Acos(k_{i}r)+Bsin(k_{i}r) $$
$$ T(t)=Ccos(ω_{i}t)+Dsin(ω_{i}t) $$
Given the initial condition ψ(r',t)=0,
$$ 0=Acos(kir′)+Bsin(kir′)=Acos(kir′)+Bsin(kir′) $$
To find the particular solution for a given value of i, we can try using the method of separation of variables.
$$ ψ(r,t)=R(r)T(t)ψ(r,t)=R(r)T(t) $$
Plug this separable solution into the spherical wave equation.
$$ \frac{1}{T}\frac{\partial^{2}T}{\partial t^{2}} = \frac{v^{2}}{R} \frac{\partial^{2}R}{\partial r^{2}} + \frac{2 v^{2}}{rR} \frac{\partial R}{\partial r} $$
$$ R(r)=Acos(k_{i}r)+Bsin(k_{i}r) $$
$$ T(t)=Ccos(ω_{i}t)+Dsin(ω_{i}t) $$
Given the initial condition ψ(r',t)=0,
$$ 0=Acos(kir′)+Bsin(kir′)=Acos(kir′)+Bsin(kir′) $$
Only one boundary condition is given, but I wonder if there is some other boundary condition for spherical waves that would allow us to ascertain more information, or if there is a superior method. I am trying to figure out how to proceed form here.
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