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# Consider the Greens function G(x, t; x1, t1) for the two-dimensional wave equation as

ISBN: 9780321797056 284

## Solution for problem 11.2.12 Chapter 11.2

Applied Partial Differential Equations with Fourier Series and Boundary Value Problems | 5th Edition

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Problem 11.2.12

Consider the Greens function G(x, t; x1, t1) for the two-dimensional wave equation as the solution of the following three-dimensional wave equation: 2u t2 = c22u + Q(x, t) u(x, 0) = 0 u t (x, 0) = 0 Q(x, t) = (x x1)(y y1)(t t1). We will solve for the two-dimensional Greens function by this method of descent (descending from three dimensions to two dimensions). *(a) Solve for G(x, t; x1, t1) using the general solution of the three-dimensional wave equation. Here, the source Q(x, t) may be interpreted either as a point source in two dimensions or as a line source in three dimensions. [Hint: dz0 may be evaluated by introducing the three-dimensional distance from the point source, 2 = (x x1) 2 + (y y1) 2 + (z z0) 2.] (b) Show that G is a function only of the elapsed time tt1 and the two-dimensional distance r from the line source, r2 = (x x1) 2 + (y y1) 2. (c) Where is the effect of an impulse felt after a time has elapsed? Compare to the one- and three-dimensional problems. (d) Sketch G for t t1 fixed. (e) Sketch G for r fixed.

Step-by-Step Solution:
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- r f4-rinpQ t nl \) 5 > r{xLta ctx t-,) xzt> .-/ r cl t< . Q x c{r Ll c'{ tt f\ (-1ur 'a-- )5 tit ) e...

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