Explanation Given: TheKlein-Gordon equation is ∇ 2 u = ( 1 v 2 ) ∂ 2 u ∂ t 2 + λ 2 u . Calculation: The Klein-Gordon equation in polar form. ∇ 2 u = ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r + 1 r 2 ∂ 2 u ∂ θ 2 − − − ( i ) Let us assume that u = F ( r , θ ) T ( t ) is a solution and substitute it in the equation ( i ) . Divide both sides by u . 1 F ( ∂ 2 F ∂ r 2 ) + 1 F ⋅ 1 r ( ∂ F ∂ r ) + 1 F ⋅ 1 r 2 ( ∂ 2 F ∂ r 2 ) = 1 T ( 1 v 2 ) ∂ 2 T ∂ t 2 + λ 2 − − − ( i i ) Use a separation constant − n 2 which separate the equation ( i i ) in two parts. ∂ 2 F ∂ r 2 + 1 r ( ∂ F ∂ r ) + 1 r 2 ( ∂ 2 F ∂ r 2 ) + ( n 2 − λ 2 ) F = 0 − − − ( i i i ) ∂ 2 T ∂ t 2 + n 2 v 2 λ 2 = 0 − − − ( i v ) From equation ( i v ) . ∂ 2 T ∂ t 2 = − n 2 v 2 λ 2 ⇒ T = { sin ( n v t ) cos ( n v t ) F ( r , θ ) is expressed as F ( r , θ ) = R ( r ) Θ ( θ ) . Substitute it in equation ( i i i ) . Multiply both sides by r 2 F . r 2 R ⋅ ∂ 2 R ∂ r 2 + r R ( ∂ R ∂ r ) + 1 Θ ( ∂ 2 Θ ∂ θ 2 ) + ( n 2 − λ 2 ) r 2 = 0 − − − ( v ) Use a separation constant k 2 . 1 Θ ( ∂ 2 Θ ∂ θ 2 ) = − k 2 ⇒ Θ = { sin ( k θ ) cos ( k θ ) Substitute k into the equation ( v ) and multiply both sides by R . r 2 ⋅ ∂ 2 R ∂ r 2 + r ( ∂ R ∂ r ) − k 2 R + ( n 2 − λ 2 ) r 2 R = 0 − − − ( v i ) Use change of variables z = ( k 2 − λ 2 ) 1 2 r

Mathematical Methods in the Physical Sciences

3rd Edition

ISBN: 9780471198260

Author: Mary L. Boas

Publisher: Wiley, John & Sons, Incorporated

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Chapter 13.10, Problem 26MP

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Find the characteristic frequencies of the circular membrane.

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Chapter 13 Solutions

Mathematical Methods in the Physical Sciences

Ch. 13.1 - Assume from electrostatics the equations E=/0 and...Ch. 13.1 - Show that the expression u=sin(xvt) describing a...Ch. 13.1 - Assume from electrodynamics the following...Ch. 13.1 - Obtain the heat flow equation (1.3) as follows:...Ch. 13.2 - After you find the series solution of a problem,...Ch. 13.2 - T=0,0x10,100,10x20. Solve the semi-infinite plate...Ch. 13.2 - Solve the semi-infinite plate problem if the...Ch. 13.2 - Solve the semi-infinite plate problem if the...Ch. 13.2 - Show that the solutions of (2.5) can also be...Ch. 13.2 - Show that the series in (2.12) can be summed to...

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Find the characteristic frequencies of the circular membrane.

Solution Summary: The author explains the Klein-Gordon equation in polar form.

Find the characteristic frequencies of the circular membrane.

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Chapter 13 Solutions