A string sewn in a 2D membrane

[skrat]

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?

 

[TSny]

Your work looks good to me so far.

 

[skrat]

Ok, than maybe the solutions to this problem are wrong.
The solution says (without any explanation of the symbols or anything else) that the second boundary condition is $$\rho z_{tt}=F\frac{\partial ^2}{\partial y^2}z_r+\gamma
(\frac{\partial }{\partial x}z_r-\frac{\partial }{\partial x}z_l)$$
I assume ##\gamma ## stands for surface tension. That is the reason why I posted this question, because I have no idea (nor does anybody around me) what the term ##F\frac{\partial ^2}{\partial y^2}z_r## is...

But if you can't see it either, than this has to be a mistake or maybe we simplified the problem a bit too much?

EDIT: Please note that the symbol ##F## I used in this post has absolutely NO relation to the ##F## I used in the first post.

 

[TSny]

Ok, than maybe the solutions to this problem are wrong.
The solution says (without any explanation of the symbols or anything else) that the second boundary condition is $$\rho z_{tt}=F\frac{\partial ^2}{\partial y^2}z_r+\gamma
(\frac{\partial }{\partial x}z_r-\frac{\partial }{\partial x}z_l)$$
I assume ##\gamma ## stands for surface tension. That is the reason why I posted this question, because I have no idea (nor does anybody around me) what the term ##F\frac{\partial ^2}{\partial y^2}z_r## is...

But if you can't see it either, than this has to be a mistake or maybe we simplified the problem a bit too much?

Ah, I was mistaken and overlooked something. It's interesting! I guess the string is assumed to have some tension F. Thus, the curvature of the string leads to an additional force term involving the second derivative with respect to distance (y) along the string. Recall the derivation of the wave equation for a string.

 

[skrat]

I guess the string is assumed to have some tension F. Thus, the curvature of the string leads to an addition force term involving the second derivative with respect to distance (y) along the string. Recall the derivation of the wave equation for a string.

Oh, that's great! One day I hope to be experienced (or smart enough) to find those little mistakes by myself... :/
Of course the wave equation for a string in general form is ##\rho u_{tt}=Fu_{xx}## which exactly explains the term in second boundary condition.

That is great TSny, thank you for your help!