Explanation Given: The given wave equation is a 2 ( ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 ) = ∂ 2 u ∂ t 2 and the given boundary conditions are u ( 0 , y , t ) = 0 , u ( π , y , t ) = 0 , u ( x , 0 , t ) = 0 , u ( x , π , t ) = 0 , u ( x , y , 0 ) = x y ( x − π ) ( y − π ) and ∂ u ∂ t | t = 0 = 0 . Calculation: The given wave equation is, a 2 ( ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 ) = ∂ 2 u ∂ t 2 (1) Consider the equation, u ( x , y , t ) = X ( x ) Y ( y ) T ( t ) (2) Separate the variables of equation (2), X ″ X = − α 2 and − Y ″ Y + T ″ a 2 T = − α 2 The equation of the variable X is, X ″ + α 2 X = 0 (3) The equation of the variable Y and T is, Y ″ Y = T ″ a 2 T + α 2 (4) Separate the variables of equation (4), Y ″ Y = − β 2 and T ″ a 2 T + α 2 = − β 2 The equation of the variable Y is, Y ″ + β 2 Y = 0 (5) The equation of the variable T is, T ″ + a 2 ( α 2 + β 2 ) T = 0 (6) The general solution of the equation (3) is, X ( x ) = c 1 cos α x + c 2 sin α x (7) The general solution of the equation (5) is, Y ( y ) = c 3 cos β y + c 4 sin β y (8) The general solution of the equation (6) is, T ( t ) = c 5 cos a α 2 + β 2 t + c 6 sin a α 2 + β 2 t (9) At boundary condition u ( 0 , y , t ) = 0 , X ( 0 ) = 0 X ( x ) = c 1 cos 0 + c 2 sin 0 c 1 = 0 At boundary condition u ( x , 0 , t ) = 0 , Y ( 0 ) = 0 Y ( y ) = c 3 cos 0 + c 4 sin 0 c 3 = 0 At boundary conditions u ( π , y , t ) = 0 and u ( x , π , t ) = 0 , X ( π ) = 0 and Y ( π ) = 0 α = m and β = n Substitute c 1 = 0 and α = m in equation (7), X ( x ) = c 2 sin m x (10) Substitute c 3 = 0 and β = n in equation (8), Y ( y ) = c 4 sin n y (11) Substitute α = m and β = n in equation (9), T

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

See similar textbooks

Videos

Question

Chapter 13.8, Problem 3E

To determine

The solution of the wave equation under given boundary conditions.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.8, Problem 2E

Chapter 13.8, Problem 4E

Students have asked these similar questions

a) x(t) = rect(t − 3) b) x(t) = −3t rect(t) . c) x(t) = 2te 3u1(t) d) x(t) = e−2|t| 2. Sketch the magnitude and phase spectrum for the four signals in Problem (1).

G(x) = dt 1+√t (x > 0). Find G' (9)

What is the area of this figure? 7 mi 3 mi 8 mi 5 mi 2 mi 6 mi 3 mi 9 mi

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

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 solution of the wave equation under given boundary conditions.

Solution Summary: The author explains the solution of the wave equation under given boundary conditions.

Videos

The solution of the wave equation under given boundary conditions.

Want to see the full answer?

Chapter 13 Solutions