Question

Let a, c be positive constants and assume that a/ 2πc is a positive integer. Consider the equation U_tt + au_t = c^2U_xx , which represents a damped version of the wave equation (telegrapher’s equation). Assuming Dirichlet boundary conditions u(0, t) = u(1, t) = 0, on the infinite strip 0 ≤ x ≤ 1, t ≥ 0, with initial conditions u(x, 0) = f(x), ut(x, 0) = 0, complete the following:

(a) Find all separable solutions (of the form u = T X) to the telegrapher’s equation that satisfy the Dirichlet boundary conditions.

(b) Write down a series solution for the initial/boundary value problem. Be sure to specify how to calculate the series coefficients of this solution.

(c) Briefly describe how the solutions to the telegrapher’s equation differ from the solutions to the standard wave equation (a = 0). Does the change in solution have any expected features?

Answer 1

Doubt in any step then comment below...i will explain you..

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Please thumbs up for this solution...thanks..

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now part c..

As if a not equal to 0 ...then we have exponential term "e^(-at/2) in solution ..this dcereases wave exponentially and tends solution to 0 ..

But if a= 0 ...so our solution continous move in a wave with same amplitude ....and the solution not goes to 0 for larger value of t ...

So this is main difference which is depend on a ..