Boundary conditions don’t apply in the equation’s region of validity

1. The problem statement, all variables and given/known data
A tight string lies along the positive x-axis when unperturbed. Its displacement from the x-axis is denoted by [itex]y(x, t)[/itex]. It is attached to a boundary at [itex]x = 0[/itex]. The condition at the boundary is
[tex]y+\alpha \frac{\partial y}{\partial x} =0[/tex] where [itex]\alpha[/itex] is a constant.

Write the displacement as the sum of an incident wave and reflected wave,
[tex]y(x, t) = e^{−ikx−i\omega t} + re^{ikx−i\omega t},\qquad x > 0,[/tex] and compute the reflection coefficient, [itex]r[/itex]. Writing [itex]r = |r|e^{i\phi}[/itex], show that [itex]|r| = 1[/itex] and find [itex]\phi[/itex].

2. Relevant equations

3. The attempt at a solution
Since the boundary condition applies at [itex]x=0[/itex] and the equation given is only valid for [itex]x>0[/itex] I can’t use that, so what equation should I use?

(If you just apply the condition that the incident and reflected wave are equal at [itex]x=0[/itex], since there is no transmission, you get what I believe is the desired result, but how would one go about this problem with the method it wants)

Leave a comment

Your email address will not be published.


Show Buttons
Hide Buttons