Solution: Find Stationary State of 2D Membrane w/ Coin on Top

[skrat]

Homework Statement

Find stationary state of a circular membrane, on which a coin is put.

Homework Equations

The Attempt at a Solution

I am not sure about the second boundary condition, so if any of you has the time, please check the following solution:
$$u_{tt}=c^2\nabla ^2u+\rho g$$ Where $u$ is the defelction from a horizontal $x$ axis, $\rho$ is the density of the coin and $g$ is the gravity. In polar coordinates this means $$\frac 1 r \frac{\partial }{\partial r}(r\frac{\partial }{\partial r}u)=-\frac{\rho g}{c^2}$$ Solving this PDE should give me $$u(r)=-\frac{\rho g}{4c^2}r^2+Aln(r)+B$$ Now the first boundary condition is kind of the obvious one: $$u(r=R)=0=-\frac{\rho g}{4c^2}R^2+Aln(R)+B$$ if $R$ is the radius of the membrane.
Now the second boundary condition is a bit confusing for me... But here is my idea:
If I write Newton's law for the coin with radius $a$, than the condition should be (I guess): $$mg=F_0\frac{du}{dr}$$ where I already did an approximation that the membrane will deflect by a really small angle $\sin \varphi \approx \varphi =\frac{du}{dr}$. Where $F_0$ should be the tension on the membrane. - and this is the most confusing part, because in order to determine constants $A$ and $B$ I defined a new constant $F_0$ which is also unknown.

.. Hmmm? Is this ok or is the second boundary condition a complete nonsense?

 

[TSny]

Have you stated everything that is given in the problem? I would think that you would need to assume that the "surface tension" of the membrane is given. The surface tension is related to your constant F_o.

I believe you want to find the stationary displacement of the membrane between r = a and r = R. In this region, what is the density of the coin?

 

[skrat]

I stated almost everything yes. - The problem also says that I if I need any parameters (such as: coin radius and its mass, etc...) I should just assume that they are all known. But yeah, I think you are right. If I add surface tension as one of the parameters of the problem, than my idea in the first post should be just fine.

I believe you want to find the stationary displacement of the membrane between r = a and r = R. In this region, what is the density of the coin?

Yes.

 

[TSny]

In the equation $u_{tt}=c^2\nabla ^2u+\rho g$ should the last term be there? Are there any external forces acting on the membrane between r = a and r = R?

 

[skrat]

Hmm, good point.
Not if I would be interested in the stationary displacement of the membrane between $r=a$ and $r=R$. Thank you!

 

[TSny]

Right, that last term should not be there. You can also check that the term does not have the same dimensions as the other terms.

 

[skrat]

Ok, hold on a second...

The general form is $u_{tt}=c^2\nabla ^2u +f(x,t)$. And let's say that I want to know the displacement of the membrane from $r=0$ to $r=R$. Ok, of course the displacement from $r=0$ to $r=a$ will be constant so this really isn't a new problem, but my question here is:
Until you mentioned it, I was sure that the last term is ok. (Despite the fact that I don't need it.) How would have to change it?

The last term should (as far as I know) describe the external forces on the membrane. In this case, that's the coin with its mass. So $\rho g$. But like you said, the dimensions don't match

 

[TSny]

$f(x,t)$ is the force per unit area (or pressure) divided by the mass per unit area.

 

[skrat]

One (hopefully last) question. I am sorry to bring this up a few days later, but.

The question is am.. how to say... it is not directly related to this problem, but it I will be more thoroughly if I post it here.
In the first post I wrote a PDE $$\frac 1 r \frac{\partial }{\partial r}(r\frac{\partial }{\partial r}u)=-\frac{\rho g}{c^2}$$ which I later decided to solve by integrating both sides. However I also have all sorts of problem with $$\nabla ^2 u(r,t)= f(r,t)$$ where as a solution are Bessel and Neuman functions.

So my question here is... How do I know when to integrate and when the solution is Bessel function? Both problems are in polar coordinates, both problems have a nice symmetry that depends only on $r$ ... I can't see any difference between the problems - yet the solutions couldn't be me more different.