Explanation Given: The given wave equation is a 2 ∂ 2 u ∂ x 2 = ∂ 2 u ∂ t 2 and the given conditions are ∂ u ∂ t | x = 0 = 0 , ∂ u ∂ t | x = L = 0 , u ( x , 0 ) = f ( x ) and ∂ u ∂ t | t = 0 = g ( x ) when t > 0 and 0 < x < L at boundary conditions x = 0 and x = L . Calculation: The given wave equation is, a 2 ∂ 2 u ∂ x 2 = ∂ 2 u ∂ t 2 The displacement equation is, u ( x , t ) = u 0 ( x , t ) + u 1 ( x , t ) (1) Consider the equation, u 1 ( x , t ) = X 1 ( x ) T 1 ( t ) (2) Separate the variables of the above equation, X ″ 1 X 1 = − λ and T ″ 1 α 2 T 1 = − λ where λ = α 2 The equation of the variable X 1 is, X ″ 1 + λ X 1 = 0 The general solution of the above equation is, X 1 ( x ) = c 1 cos α x + c 2 sin α x At given condition ∂ u ∂ t | x = 0 = 0 and ∂ u ∂ t | x = L = 0 , c 2 = 0 For non trivial solution c 1 ≠ 0 and α = n π L Substitute the values of c 2 and α in X 1 ( x ) , X 1 ( x ) = c 1 cos n π L x The equation of the variable T 1 is, T ″ 1 + a 2 λ T 1 = 0 The general solution of the above equation is, T 1 ( t ) = c 3 cos n π a L t + c 4 sin n π a L t Substitute the values of X 1 ( x ) and T 1 ( t ) in equation (2), u 1 ( x , t ) = c 1 cos n π L x ( c 3 cos n π a L t + c 4 sin n π a L t ) = ( c 1 c 3 cos n π a L t + c 1 c 4 sin n π a L t ) cos n π L x Rewrite the terms c 1 c 3 = A n and c 1 c 4 = B n in the above equation, u 1 ( x , t ) = ∑ n = 1 ∞ ( A n cos n π a L t + B n sin n π a L t ) cos n π L x (3) Consider the equation, u 0 ( x , t ) = X 0 ( x ) T 0 ( t ) (4) When λ = 0 separate the varia

6th Edition

ISBN: 9781284127003

Author: ZILL

Publisher: JONES+BARTLETT LEARNING,LLC (CC)

See similar textbooks

Videos

Question

Chapter 13.4, Problem 12E

To determine

The displacement u(x,t) of vibrating string.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.4, Problem 11E

Chapter 13.4, Problem 13E

Students have asked these similar questions

Problems Determine I, and Iy for the cross-sections 6.11 See the steel tables in the Appendix Table A3. |x= = " -2x8 STEEL PLATE CENTERED -W14x82

Problems Construct shear and moment diagrams using the equilib- rium method. Indicate the magnitudes of Vmax and Mmax. 7.4 8=2K/FT. W=2K/FT. 10' 10'

Problems Determine I, and Iy for the cross-sections 6.8 89 2" 1011 2" ส 10 |x= ly=

Chapter 13 Solutions

EBK 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

Knowledge Booster

$Advanced Math$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

The displacement u ( x , t ) of vibrating string.

Solution Summary: The author calculates the displacement of a vibrating string by using the wave equation a2partial.

Videos

The displacement u(x,t) of vibrating string.

Want to see the full answer?

Chapter 13 Solutions