Explanation Given: The given wave equation is a 2 ∂ 2 u ∂ x 2 = ∂ 2 u ∂ t 2 and the given conditions are u ( 0 , t ) = 0 , u ( 1 , t ) = 0 , u ( x , 0 ) = x ( 1 − x ) and ∂ u ∂ t | t = 0 = x ( 1 − x ) when t > 0 and 0 < x < 1 . Calculation: The given 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 u ( 0 , t ) = 0 and u ( 1 , t ) = 0 , X ( 0 ) = 0 and X ( 1 ) = 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 ( 1 ) = 0 in equation (4), X ( 1 ) = 0 + c 2 sin α c 2 sin α = 0 For non trivial solution c 2 ≠ 0 , sin α = 0 α = n π Substitute the values of c 1 and α in equation (4), X ( x ) = c 2 sin n π x (5) The general solution of the equation (3) is, T ( t ) = c 3 cos n π a t + c 4 sin n π a 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 x ( c 3 cos n π a t + c 4 sin n π a t ) = ( c 2 c 3 cos n π a t + c 2 c 4 sin n π a t ) sin n π 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 t + B n sin n π a t ) sin n π x (7) The value of A n is, A n

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 13.4, Problem 5E

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The wave equation under given conditions.

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