TAM310hw09 - b. Find the R equation and show that it has...

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TAM 310 Advanced Engineering Analysis I Spring 2008 Prof.R.Rand Homework No.9 (Due Tuesday April 1) 1. The radial vibrations of a gas contained in a rigid spherical shell of radius unity is governed by the PDE: 2 ψ ∂t 2 = 1 ρ 2 ∂ρ p ρ 2 ∂ψ ∂ρ P (1) with the boundary condition ∂ψ ∂ρ = 0 at ρ = 1 (2) [Here ψ is a “velocity potential”, and eq.(2) says that the radial velocity at the boundary is zero.] a. Using separation of variables with ψ = T ( t ) R ( ρ ), show that the equation on T is of the form: d 2 T dt 2 + ω 2 T = 0 (3) Note that the separation constant ω 2 , which is an eigenvalue of the R equation, physically rep- resents a frequency of a mode of vibration.
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Unformatted text preview: b. Find the R equation and show that it has the general solution R = c 1 sin + c 2 cos . c. Find a numerical value for the smallest frequency of free vibration . 2. The vibrations of a rectangular membrane are governed by the PDE: 2 u t 2 = 2 u x 2 + 2 u y 2 , x 1 , y 2 (4) The four edges of the membrane are held xed with displacement u = 0. Use separation of variables to nd the rst 5 lowest frequencies of free vibration. Reference: Edwards and Penney, Section 10.5....
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