how to simulate wave interference [closed]

Question

I need to simulate wave interference with reflection from surfaces.

What formulas I need to use?

What differential equation I need to solve? - Could someone help me out?

Answer 1

There are many different fluid systems that can produce waves and the associated interference: surface water waves, gas dynamics, etc. Each of these are treated very differently from a computational point of view - each using different numerical methods.

For your fairly poorly defined question, I would suggest you look at ideal gas dynamics. That is non-dissipative, non-viscous gas dynamics. The system of equations used to study such flows are quasi-linear hyperbolic system of PDEs, or conservation laws. In one-dimension (which will be the simplest case you want to consider as a starter) they take the generic form

$$\partial_{t} \mathbf{U} + \mathbf{\mathbf{A}}(\mathbf{U})\partial_{x}\mathbf{U} = 0,$$

where $\mathbf{U} = (\rho, \rho u, e)^{\mathsf{T}}$ is the vector of conserved quantities (for mass, momentum and energy) and $\mathbf{\mathbf{A}}(\mathbf{U})$ is the Jacobian matrix for the system. The Eigenvalues of the Jacobian matrix provide the wavespeeds for the given system under consideration. The above equation can be recast using the vector of conservative fluxed $\mathbf{F}(\mathbf{U})$ via

$$\partial_{t} \mathbf{U} + \partial_{x} \mathbf{F}(\mathbf{U}) = 0.$$

Now, for one-dimensional gas dynamics this system can be solved analytically, but it is not easy and it is mathematically involved. In two dimensions there are exact solutions but they require an iterative solver. The most common way to get solutions for such systems is to use something called Godunov's theorem and associated Riemann Solver, these methods provide a very fast and accurate wave of resolving the flow AND wave solutions for such systems.

Even to create your own one-dimensional solver for gas dynamics will not be easy. For two dimensions (with the reflection from walls that you mention above) will be another level up. My advice would be to first read this paper (from a masters thesis). This will provide a tutorial for creating your own one-dimensional solver for gas dynamics.

Good luck.