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Mathematics LibreTexts

6.6: Spherically Symmetric Vibrations

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Another application of spherical harmonics is a vibrating spherical membrane, such as a balloon. Just as for the two-dimensional membranes encountered earlier, we let u(θ,ϕ,t) represent the vibrations of the surface about a fixed radius obeying the wave equation, utt=c22u, and satisfying the initial conditions

u(θ,ϕ,0)=f(θ,ϕ),ut(θ,ϕ,0)=g(θ,ϕ).

In spherical coordinates, we have (for ρ=r= constant.)

utt=c2r2(1sinθθ(sinθuθ)+1sin2θ2uϕ2),

where u=u(θ,ϕ,t).

clipboard_e4c98696a0a832a43e4fef1f6f6a488d2.png
Figure 6.6.1: A vibrating sphere of radius r with the initial conditions

u(θ,ϕ,0)=f(θ,ϕ),ut(θ,ϕ,0)=g(θ,ϕ).

The boundary conditions are given by the periodic boundary conditions

u(θ,0,t)=u(θ,2π,t),uϕ(θ,0,t)=uϕ(θ,2π,t),

where 0<t, and 0<θ<π, and that u=u(θ,ϕ,t) should remain bounded.

Noting that the wave equation takes the form

utt=c2r2Lu, where LYm=(+1)Ym

for the spherical harmonics Ym(θ,ϕ)=Pm(cosθ)eimϕ, then we can seek product solutions of the form

um(θ,ϕ,t)=T(t)Ym(θ,ϕ).

Inserting this form into the wave equation in spherical coordinates, we find

TYm=c2r2T(t)(+1)Ym,

or

T+(+1)c2r2T(t)

The solutions of this equation are easily found as

T(t)=Acosωt+Bsinωt,ω=(+1)cr.

Therefore, the product solutions are given by

um(θ,ϕ,t)=[Acosωt+Bsinωt]Ym(θ,ϕ)

for =0,1,,m=,+1,,.

clipboard_e851a08ac55d6cfa2ce134d3d6bee5df1.png
Figure 6.6.2: Modes for a vibrating spherical membrane:

Row 1:(1,0),(1,1);Row 2:(2,0),(2,1),(2,2);Row 3:(3,0),(3,1),(3,2),(3,3).

The general solution is found as

u(θ,ϕ,t)==0m=[Amcosωt+Bmsinωt]Ym(θ,ϕ).

An interesting problem is to consider hitting the balloon with a velocity impulse while at rest. An example of such a solution is shown in Figure 6.6.3. In this images several modes are excited after the impulse.

clipboard_ecc7e5bb2f37b79bd45af9afb28bedb63.png
Figure 6.6.3: A moment captured from a simulation of a spherical membrane after hit with a velocity impulse.

This page titled 6.6: Spherically Symmetric Vibrations is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Russell Herman via source content that was edited to the style and standards of the LibreTexts platform.

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