Explanation Given: A string is tied to the x -axis at x = 0 and x = L and its initial displacement is u ( x , 0 ) = f ( x ) when 0 < x < L . Calculation: The wave equation is, a 2 ∂ 2 u ∂ x 2 = ∂ 2 u ∂ t 2 Consider the equation, u ( x , t ) = X ( x ) T ( t ) (1) Separate the variables of the above equation, X ″ X = − λ and T ″ α 2 T = − λ The equation of the variable X is, X ″ + λ X = 0 (2) The equation of the variable T is, T ″ + α 2 λ T = 0 (3) The general solution of the equation (2) is, X ( x ) = c 1 cos α x + c 2 sin α x (4) At given condition x = 0 and x = L , X ( 0 ) = 0 and X ( L ) = 0 Substitute X ( 0 ) = 0 in equation (4), X ( 0 ) = c 1 cos 0 + c 2 sin 0 0 = c 1 + 0 c 1 = 0 Substitute c 1 = 0 and X ( L ) = 0 in equation (4), X ( L ) = 0 + c 2 sin α L c 2 sin α L = 0 For non trivial solution c 2 ≠ 0 , sin α L = 0 α = n π L Substitute the values of c 1 and α in equation (4), X ( x ) = c 2 sin n π L x (5) The general solution of the equation (3) is, T ( t ) = c 3 cos n π a L t + c 4 sin n π a L t (6) Substitute the values of X ( x ) and T ( t ) from equation (5) and (6) in equation (1), u ( x , t ) = c 2 sin n π L x ( c 3 cos n π a L t + c 4 sin n π a L t ) = ( c 2 c 3 cos n π a L t + c 2 c 4 sin n π a L t ) sin n π L x Rewrite the term c 2 c 3 = A n and c 2 c 4 = B n in the above equation, u ( x , t ) = ∑ n = 1 ∞ ( A n cos n π a L t + B n sin n π a L t ) sin n π L x (7) The value of A n is, A n = 2 L ∫ 0 L f ( x ) sin n π L x d x (8) The value of B n is, B n = 2 n π a ∫ 0 L g ( x ) sin n π L x d x (9) The values of f ( x ) and g ( x ) are, f ( x ) = u ( x , 0 ) and g ( x ) = ∂ u ∂ t | t = 0 At the given conditions u ( x , 0 ) = f ( x ) and ∂ u ∂ t | t = 0 = 0 , So, the value of g ( x ) is 0

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 13.4, Problem 9E

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The value of the function u(x,t) when the string is released from the rest.

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Chapter 13.4, Problem 8E

Chapter 13.4, Problem 10E

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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 value of the function u ( x , t ) when the string is released from the rest.

Solution Summary: The author calculates the value of the function u(x,t) when the string is released from the rest.

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The value of the function u(x,t) when the string is released from the rest.

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Chapter 13 Solutions