# A coin on 2D membrane

## Homework Statement

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

## 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
Homework Helper
Gold Member
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 Fo.

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?

Chestermiller
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
Homework Helper
Gold Member
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?

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
Homework Helper
Gold Member
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.

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
Homework Helper
Gold Member
##f(x,t)## is the force per unit area (or pressure) divided by the mass per unit area.

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.