Explanation Given: The given wave equation is ∂ 2 u ∂ x 2 = ∂ 2 u ∂ t 2 + 2 β ∂ u ∂ t at 0 < β < 1 , t > 0 and string is stretched to the x -axis at x = 0 and x = π . Calculation: The given wave equation is, ∂ 2 u ∂ x 2 = ∂ 2 u ∂ t 2 + 2 β ∂ u ∂ t Consider the equation, u ( x , t ) = X ( x ) T ( t ) (1) Separate the variables of the above equation, X ″ X = − λ and T ″ T + 2 β T ′ T = − λ where λ = α 2 The equation of the variable X is, X ″ + λ X = 0 (2) The equation of the variable T is, T ″ + 2 β T ′ + λ T = 0 (3) The general solution of the equation (2) is, X ( x ) = c 1 sin α x + c 2 cos α x (4) At given condition x = 0 and x = π , X ( 0 ) = 0 and X ( π ) = 0 Substitute X ( 0 ) = 0 in equation (4), X ( 0 ) = c 1 sin 0 + c 2 cos 0 0 = 0 + c 2 c 2 = 0 Substitute c 2 = 0 and X ( π ) = 0 in equation (4), X ( π ) = c 1 sin α π + 0 c 1 sin α π = 0 For non-trivial solution c 1 ≠ 0 , sin α π = 0 α = n Substitute the values of c 2 and α in equation (4), X ( x ) = c 1 sin n x (5) The general solution of the equation (3) is, T ( t ) = e − β t ( c 3 cos n 2 − β 2 t + c 4 sin n 2 − β 2 t ) (6) Substitute the values of X ( x ) and T ( t ) from equation (5) and (6) in equation (1), u ( x , t ) = e − β t c 1 sin n x ( c 3 cos n 2 − β 2 t + c 4 sin n 2 − β 2 t ) = e − β t ( c 1 c 3 cos n 2 − β 2 t + c 1 c 4 sin n 2 − β 2

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 13.4, Problem 15E

To determine

The displacement u(x,t) when the string starts from the initial displacement.

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Chapter 13.4, Problem 14E

Chapter 13.4, Problem 16E

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Chapter 13 Solutions

Advanced Engineering Mathematics

Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

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The displacement u ( x , t ) when the string starts from the initial displacement.

Solution Summary: The author calculates the wave equation u(x,t) when the string starts from the initial displacement.

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The displacement u(x,t) when the string starts from the initial displacement.

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Chapter 13 Solutions