To solve the given problem, we need to determine the type of sound wave generated by a spherical loudspeaker located at the center of a room with sound-absorbing walls.
Given information:
- The loudspeaker is spherical and centered at the origin of our coordinate system ((\vec{r} = 0)).
- The radius of the sphere is (R).
- The pressure on the surface of the sphere oscillates according to ( p_0 \cos(\omega t) ).
We are instructed to use the following form for the wave function: [ \psi(\vec{r}, t) = \chi(|\vec{r}|, t) ]
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Determine the Spherical Wave Equation: Since the problem involves spherical symmetry, the wave function (\psi(\vec{r}, t)) only depends on the radial distance (r = |\vec{r}|) and time (t). The general form of the wave equation in spherical coordinates (with spherical symmetry) is: [ \left( \frac{\partial^2}{\partial r^2} + \frac{2}{r} \frac{\partial}{\partial r} - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \right) \chi(r, t) = 0 ]
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Boundary Condition: The pressure oscillation at the surface of the sphere ((r = R)) is given by: [ p(R, t) = p_0 \cos(\omega t) ] Since pressure is related to the wave function, we can assume: [ \chi(R, t) = p_0 \cos(\omega t) ]
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Solution to the Spherical Wave Equation: The solution to the spherical wave equation for a time-harmonic source is: [ \chi(r, t) = \frac{A}{r} \cos\left( \omega t - kr \right) ] where (A) is an amplitude constant, (k = \frac{\omega}{c}) is the wave number, and (c) is the speed of sound.
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Applying Boundary Conditions: At (r = R), we have: [ \chi(R, t) = \frac{A}{R} \cos\left( \omega t - kR \right) = p_0 \cos(\omega t) ] This gives: [ \frac{A}{R} = p_0 \quad \text{and} \quad kR = 0 \Rightarrow k = \frac{\omega}{c} ] So, (A = p_0 R).
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Final Wave Function: Substituting (A) back into the solution, we get: [ \chi(r, t) = \frac{p_0 R}{r} \cos\left( \omega t - \frac{\omega r}{c} \right) ]
Hence, the wave function describing the sound wave generated by the spherical loudspeaker is: [ \psi(\vec{r}, t) = \frac{p_0 R}{|\vec{r}|} \cos\left( \omega t - \frac{\omega |\vec{r}|}{c} \right) ]
This describes a spherical sound wave emanating from the source at the center of the room, with amplitude decreasing as (\frac{1}{r}) and a phase shift corresponding to the distance traveled by the wave.
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