cos
cos
cos
,
= 
cos 
− 
− 
+ 
+ 
− 
140 |
6 Acoustic Waves from Spherical Sources |
Let’s assume that the spherical source of radius a is uniformly dilating and radiating spherical waves. The vibration u(a, t) at the surface of this pulsating sphere (r ¼ a) has an amplitude of Ua, an angular frequency of ω, and a phase of θo. The velocity at the surface of the sphere provides the boundary condition for solving the velocity at any point outside the sphere:
uðr ¼ a, tÞ ¼ Ua cos ðωt þ θoÞ
At the surface of the sphere (r ¼ a), the velocity u(x, t) of the radiated wave must equal to the velocity u(a, t) of the surface of the sphere. Based on this boundary condition at the surface of the sphere, velocity and sound pressure at any point outside of the sphere can be derived and are shown below before validation:
rpðr, tÞ ¼ A cos ðωt kr þ θÞ
ruðr, tÞ ¼ |
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A |
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cos ðωt kr þ θ ϕÞ |
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ρoc cos ðϕÞ |
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where: |
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A ¼ Uaρoc cos ðϕaÞa |
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θ ¼ ka þ θo þ ϕa |
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ka |
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ϕa ¼ tan 1 |
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! cos ðϕaÞ ¼ |
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ka |
1 þ ðkaÞ2 |
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1 |
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q |
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kr |
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ϕ ¼ tan 1 |
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kr |
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1 þ ðkrÞ
The following section will derive the above pressure and velocity radiating from the sphere based on the boundary condition at the surface of the sphere.
From the previous section, the general solution for the velocity of the spherical wave is:
A
uðr, tÞ ¼ rρoc cos ðϕÞ cos ðωt kr þ θ ϕÞ
where:
tan ðϕÞ ¼ kr1
To calculate the unknown constants θ and A, substitute the boundary conditions of velocity on the surface (r ¼ a) of the sphere into the above general solution as shown below: