Independent solutions of scalar wave equations

[Karthiksrao]

Hi,

This has been bothering me for a while now.. The scalar wave equation is a 2nd order differential equation. So we would expect two independent solutions for it.

However when you try to find the solution of the scalar wave equation (in spherical coordinates) by employing the separation of variables we would end up getting a series summation to infinite terms of (legendre polynomials)*(bessels)*(Trigonometric ) functions.

How do you find the *two* independent solutions from this infinite summation series ?

Thanks

 

[mikeph]

I guess you are trying this in three dimensions from the coordinate system, I can't figure out why you are not trying it in cartesian coordinates though.

Split it up and use d'Alembert's solution in all three dimensions to get a nicer answer?

 

[Karthiksrao]

I am looking at the 3D scalar wave equation in spherical coordinates which is a well discussed problem in electromagnetic theory. But thanks.

 

[cortiver]

Second-order linear ordinary differential equations (ODEs) have two linearly independent solutions. The wave equation is a second order partial differential equation (PDE) and will have infinitely many independent solutions (e.g. u(x,y,z,t) = cos(k(x-ct)) is a solution for any real k). If you like you can think of the wave equation as an infinite number of coupled ODEs, one for each point in space.