- #1

- 748

- 8

## Homework Statement

A string is sewn in a 2D membrane. What is the amplitude of the reflected wave? All the parameters you need in order to get to the result are known.

## Homework Equations

## The Attempt at a Solution

Ok, so we have a 2D membrane, with a string at ##x=0## along the the ##y## axis.

Now the wave coming with wave vector ##\vec k_0=k_0(cos\alpha ,sin\alpha )## will reflect with ##\vec k_1=k_1(-cos\gamma , sin \gamma )## and what goes through is ##\vec k_2 =k_2 (cos \beta, sin\beta )##.

Meaning on the left we have $$z_l(x,y,t)=e^{i(\vec k_0\vec r-\omega t)}+re^{i(\vec k_1\vec r-\omega t)}$$ and on the right hand side we have $$z_r(x,y,t)=te^{i(\vec k_2\vec r-\omega t)}$$

Up to this point, I am quite positive everything is ok. Now following steps:

First boundary condition is $$z_l(0,y,t)=z_r(0,y,t)$$ but again I have problems with the not-so-obvious second boundary condition.

I would say it is simply $$\rho z_{tt}=F(\frac{\partial }{\partial x}z_r-\frac{\partial }{\partial x}z_l)$$ if ##\rho## is the density of the string and ##F## the tension of the membrane.

BUT this boundary condition brings me to a wrong solution. :/ As if I was missing some terms in the second boundary condition. Could anyone help?