Connecting wave equation with a non-constant force only in bounds

[latot]

TL;DR Summary

How to apply a specific force only in the bounds of the wave equation

Hi hi, I'm confused about how to mix this two concepts, actually the wave equation:

$\frac {\partial^2 u} {\partial t^2} = v_x^2 \frac {\partial^2 u} {\partial x^2} + v_y^2\frac {\partial^2 u} {\partial y^2} + force$

The equation will apply the rule all over the space, but I have the next conditions that I don't know how to mix:

• Inside the bounds there is no external force, so, only apply the wave equation alone
• In the bounds there is a force applied all the time (not necessary constant), let's say is a vector over u, x, and y.

I'm confused how to mix this two physics concepts to one.

My idea for now is use Dirac Delta in the bounds with a force vector, but I don't know if is valid or the only way.

There is a point about the bounds, and is their nature, I don't know very well how construct this in the model, here two examples:

• If out of the bounds there is nothings the wave should not be reflected
• If we consider in the bounds there is a wall the wave will collide and will be reflected, there we have other system

Thx.

 

[Valerie Park]

The way to mix these two concepts is to use boundary conditions. Boundary conditions are conditions that are imposed on the solution (in this case, the wave equation) at the boundaries of the domain. For example, if you have a wall in the bounds, then you could impose a Dirichlet boundary condition, which states that the solution must equal a certain value on the boundary. If there is nothing outside the bounds, then you could impose a Neumann boundary condition, which states that the normal derivative of the solution must equal a certain value on the boundary. In order to incorporate the external force into the model, you would need to include it in the boundary conditions.