Inhomogeneous wave equation: RHS orthogonal to homogeneous solutions

[ollielgg]

TL;DR Summary

Brillouin states that an inhomogeneous hyperbolic PDE has a finite solution only if the RHS is orthogonal to the homogenous solutions

Hi, I've been reading Brillouin's 'Wave Propagation in Periodic Media'.

About the following equation
$$\nabla^2u_1+\frac{\omega^2_0}{V_0}u_1 = R(r)$$
Brillouin states that "it is well known that such an equation possesses a finite solution only if the right-hand term is orthogonal to all solutions of the homogeneous equation:"
$$\iint_{\text{all space}} u_1^*R(r) dr = 0$$

This is not a property of hyperbolic PDEs I've come across before. I wasn't able to find anything in my PDE textbooks. Would anyone be able to suggest why this is the case? I would be very appreciative.

 

[jasonRF]

It has nothing to do with the class of PDEs; it is true because the wave operator is self-adjoint. If it weren't, then the right-hand term needs to be orthogonal to solutions of the homogeneous adjoint equation. To show it is necessary, assume $L x = a$ has a solution, where the operator $L$ has adjoint $L^\dagger$, and let $z$ be any solution of $L^\dagger z = 0$. Then we have
$$
0 = \left\langle x, L^\dagger z \right\rangle = \left\langle L x, z \right\rangle = \left\langle a, z \right\rangle
$$
where $\left\langle a, z \right\rangle$ indicates the inner product of $a$ and $z$.

jason

 

[ollielgg]

Thanks a lot! This is a big help.

 

[jasonRF]

You are welcome. Also, welcome to Physics Forums! I didn't notice that was your first post.

I hope you enjoy Brillouin's book. I have looked at it on a number of occasions and it looks pretty interesting and has lots of pretty pictures of Brillouin zones! But I have never taken the time to work through it.

Jason