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.

 

[PJC01]

for your question. The question you have raised is a common confusion among students studying the scalar wave equation. Let me try to clarify it for you.

The scalar wave equation is indeed a second order differential equation, which means that we would expect two independent solutions. However, when solving it in spherical coordinates using the separation of variables method, we end up with a series summation that may seem infinite.

But this does not mean that we only have one solution. In fact, the infinite series represents a general solution, which can be expressed as a linear combination of two independent solutions. These two solutions can be chosen as any two solutions that satisfy the boundary conditions of the problem at hand.

To better understand this, let's take an example. Consider the simple harmonic oscillator equation: d^2x/dt^2 + ω^2x = 0. This is a second order differential equation with two independent solutions: x1 = sin(ωt) and x2 = cos(ωt). However, if we solve this equation using the separation of variables method, we end up with an infinite series: x = ∑(An*sin(nωt) + Bn*cos(nωt)). This does not mean that we only have one solution, but rather that we have an infinite number of solutions that can be expressed as a linear combination of x1 and x2.

Similarly, in the scalar wave equation, the infinite series represents a general solution that can be expressed as a linear combination of two independent solutions. These two solutions can be chosen based on the boundary conditions of the problem, and will result in a unique solution for that specific problem.

I hope this helps clarify your confusion. If you have any further questions, please don't hesitate to ask.