Free boundary conditions on vibrating rectangular membranes

[abilolado]

I've been trying to come up with wave equations to describe the motion on vibrating rectangular (more specifically, square) membranes. However, most paper I find assume fixed edges.
What are the boundary conditions I need to apply to the 2D wave equations in order to have an free boundary in a finite membrane of side L centered at the origin?
By free boundary I just mean not fixed, as in, an incoming wave will be reflected on the same original side, rather then the opposite side on a fixed boundary.
Thank you in advance

 

[Orodruin]

Did you try deriving them directly from variational principles?

 

[abilolado]

Did you try deriving them directly from variational principles?

Can you be more specific? As in Euler-Lagrange equations?
I've been doing it from the classical wave equation

Did you try deriving them directly from variational principles?

Can you be more specific?
As in Euler-Lagrange equation?
I've been getting the equation of motion (generalized, wihout the boundary conditions) from the classical wave equation $∇^2U=\frac {1}{c^2} \frac {∂^2U}{∂t^2}$
Im unaware if there is another way.

 

[Orodruin]

Can you be more specific? As in Euler-Lagrange equations?
I've been doing it from the classical wave equation

Can you be more specific?
As in Euler-Lagrange equation?
I've been getting the equation of motion (generalized, wihout the boundary conditions) from the classical wave equation $∇^2U=\frac {1}{c^2} \frac {∂^2U}{∂t^2}$
Im unaware if there is another way.

You cannot derive the boundary conditions of an equation from the equation itself. However, you can use the same physical principle to derive the free boundary conditions as you can use to derive the wave equation, e.g., Hamilton's principle.

The alternative is to look at the force balance at the membrane boundary and I would say it is significantly more cumbersome.

 

[abilolado]

You cannot derive the boundary conditions of an equation from the equation itself. However, you can use the same physical principle to derive the free boundary conditions as you can use to derive the wave equation, e.g., Hamilton's principle.

The alternative is to look at the force balance at the membrane boundary and I would say it is significantly more cumbersome.

I know I cannot derive the boundary conditions from the equation itself, that is not my question, the question is, what are the boundary conditions for a non-fixed boundary.
As in, for a fixed boundary I would state that $U(x,0,t) = 0, U(x,L,t=0), U(0,y,t)=0, and U(L,y,t)=0$
What would I have to state for a non-fixed boundary?

 

[Orodruin]

What would I have to state for a non-fixed boundary?

Have you done the case of a vibrating string? What did you do to find the free boundary conditions for that?

 

[abilolado]

Have you done the case of a vibrating string? What did you do to find the free boundary conditions for that?

I have not.
How would I "find" boundary conditions, as you say?
The ones I used as an example just come from simple intuition, making the boundaries of a square of side L, always having an amplitude of 0 for any time.
I don't know what that would be for a non-fixed boundary, neither for a string.

 

[abilolado]

I've seen, in some places, a boundary condition for an open end of a string is $\frac {∂U}{∂x} = 0$. I do not understand why that is, but, by this logic, would an equivalent for a 2D membrane be $\frac {∂U}{∂x} + \frac {∂U}{∂y} = 0$?
In other words, is $∇U(0,y,t)=0$, $∇U(L,y,t)=0$, $∇U(x,0,t)=0$ and $∇U(x,L,t)=0$ the boundary condition for a non-fixed boundary membrane? if so, why is that?

 

[Orodruin]

I've seen, in some places, a boundary condition for an open end of a string is $\frac {∂U}{∂x} = 0$. I do not understand why that is, but, by this logic, would an equivalent for a 2D membrane be $\frac {∂U}{∂x} + \frac {∂U}{∂y} = 0$?
In other words, is $∇U(0,y,t)=0$, $∇U(L,y,t)=0$, $∇U(x,0,t)=0$ and $∇U(x,L,t)=0$ the boundary condition for a non-fixed boundary membrane? if so, why is that?

No, that is not the correct generalisation. The correct generalisation is that the derivative in the normal direction is zero.

I would suggest to study the derivation of this in the string case first. You can do so by considering force equilibrium at the string end-point. (Or, simpler if you know how, by posing natural boundary conditions using Lagrange mechanics.)

 

[abilolado]

any pointers for books or papers that address similar issues?

 

[Andy Resnick]

I've been trying to come up with wave equations to describe the motion on vibrating rectangular (more specifically, square) membranes. However, most paper I find assume fixed edges.
What are the boundary conditions I need to apply to the 2D wave equations in order to have an free boundary in a finite membrane of side L centered at the origin?
By free boundary I just mean not fixed, as in, an incoming wave will be reflected on the same original side, rather then the opposite side on a fixed boundary.
Thank you in advance

For 1-D beams, the free end boundary conditions are that the bending moment and shear force vanish: Y''(L) = 0 and Y'''(L) = 0. Similar expressions are used for simply supported plates:

https://en.wikipedia.org/wiki/Bending_of_plates

 

[Orodruin]

You should be able to find the derivation for the string without much trouble.

I would recommend my book, but it will not be published for another year ...

For 1-D beams, the free end boundary conditions are that the bending moment and shear force vanish: Y''(L) = 0 and Y'''(L) = 0. Similar expressions are used for simply supported plates:

https://en.wikipedia.org/wiki/Bending_of_plates

This is not a beam, it is a string/membrane satisfying the wave equation - not the beam equation.

 

[Andy Resnick]

This is not a beam, it is a string/membrane satisfying the wave equation - not the beam equation.

Did you not check the link? Or even read the link caption?

 

[Orodruin]

Did you not check the link? Or even read the link caption?

Yes. Did you even read the question?

You only get boundary conditions on the second and third derivatives if you are dealing with differential equations that contain higher order spatial derivatives.

 

[abilolado]

I believe I've solved it.
The boundary conditions for a non-fixed end on a vibrating string is $\frac {∂U(L,t)}{∂x}=0$
Similarly, if one considers a square membrane as a series of vibrating string crossed orthogonally (which I believe is a valid assumption, since that is how computer simulations generally treat vibrating membranes, and they give fairly approximate results), we can get the following conditions:
$\frac {∂U(0,y,t)}{∂x}=0$
$\frac {∂U(L_x,y,t)}{∂x}=0$
$\frac {∂U(x,0,t)}{∂y}=0$
$\frac {∂U(x,L_y,t)}{∂y}=0$
which yields the following answer for the wave function:
$\sum_{m=0}^∞ \sum_{n=0}^∞ [A cos(k t)+B sin(k t)]cos(\frac {m π}{L_x} x)cos(\frac {n π}{L_y} y)$
Which agrees with Xiao answer here http://rudar.ruc.dk/bitstream/1800/5190/1/Chladni Pattern.pdf
Also, the equation for a fixed boundary is similar, only with $sin$ instead of $cos$. which makes sense.
I believe this thread is closed, thank you all for the help!

 

[Orodruin]

Since you have figured it out in a different way, let me just mention the method of deriving it through variational calculus. The Lagrangian density corresponding to the string is given by
$$
\mathcal L = \frac 12 (\rho u_t^2 - S u_x^2)
$$
where $\rho$ is the linear density and $S$ the tension. The variation of the term involving the tension is of the form
$$
- S \int_0^\ell u_x\, \delta u_x \, dx = - S [u_x(\ell,t) \delta u(\ell,t) - u_x(0,t) \delta u(0,t)] + S \int_0^\ell u_{xx} \delta u\, dx
$$
where we have used partial integration. The integral on the right-hand side is what goes into giving the $u_{xx}$ term in the wave equation whereas there are two possibilities for the boundary terms to vanish:

1. If you have fixed boundary conditions, the variations are fixed to be zero at the boundary and the boundary terms vanish.
2. If you have free boundary conditions, the variations are arbitrary. Thus, in order to make sure that the variation of the action is zero - i.e., your solution is one of stationary action - it is necessary that the solution satisfies the boundary conditions $u_x(\ell,t) = u_x(0,t) = 0$.

You can easily generalise this to more dimensions, and you will obtain that the free boundary conditions are conditions on the normal derivatives on the boundary.

 

[Andy Resnick]

Yes. Did you even read the question?

You only get boundary conditions on the second and third derivatives if you are dealing with differential equations that contain higher order spatial derivatives.

Such as is the case for an isotropic, homogeneous plate under pure bending. Given that the plate has to be supported *somewhere*, the OP is a solved problem (Edit: Chladni plates and) http://old.utcluj.ro/download/doctorat/Rezumat_Fetea_Marius.pdf. The link I originally posted has the relevant expressions as well- please don't get defensive.

 

[Orodruin]

Such as is the case for an isotropic, homogeneous plate under pure bending. The OP is a solved problem, please don't get defensive.

For the third time - there is no resistance to bending here! The OP is dealing with the wave equation on a membrane - not a plate with rigidity. I therefore find it confusing and misleading to the OP to start talking about plates and higher order derivatives and I do not understand how you think that this benefits the OP.

Of course, if you do consider objects with rigidity you can still apply the Lagrangian formalism just taking into account that the potential energy density in the beam/plate due to bending is proportional to the square of the second derivative (for small deviations from the rest state). The free boundary conditions follow directly from the same argumentation and you never have to look at the force or torque balance - you just have to do an additional partial integration.

 

[abilolado]

I'm still an undergrad in physics and this is all becoming a bit too advanced for me.
Not that I'm giving up on it, solving the wave equations for a membrane is surely a start and I might jump to the plates once I fully understand the simpler concepts.
I appreciate all the help from all of you guys! I'll keep this thread tabbed for when I start thinking about plates and all.

 

[Orodruin]

I'm still an undergrad in physics and this is all becoming a bit too advanced for me.
Not that I'm giving up on it, solving the wave equations for a membrane is surely a start and I might jump to the plates once I fully understand the simpler concepts.
I appreciate all the help from all of you guys! I'll keep this thread tabbed for when I start thinking about plates and all.

Don't worry, you did what you set out to do and created your own intuitive image of the correct boundary conditions. I would simply skip all of the mentions about plates and beams for now.

A word of advice for the future: Do not select "A" for your thread levels (until you have knowledge equivalent to a grad student in the subject). The thread level is intended for getting replies at an appropriate level. Since you are an undergraduate, the appropriate level would be "I".

 

[RamonCarrillo]

I've been trying to come up with wave equations to describe the motion on vibrating rectangular (more specifically, square) membranes. However, most paper I find assume fixed edges.
What are the boundary conditions I need to apply to the 2D wave equations in order to have an free boundary in a finite membrane of side L centered at the origin?
By free boundary I just mean not fixed, as in, an incoming wave will be reflected on the same original side, rather then the opposite side on a fixed boundary.
Thank you in advance

The Book of Fetter and Walecka (Theoretical Mechanics of Particles and continua, page 277 [Dover edition] ) discuss the solution for one of the edges Free, I think you can extrapole from this method.