Inhomogeneous wave equation: RHS orthogonal to homogeneous solutions

In summary, the conversation discusses an equation from Brillouin's book on wave propagation in periodic media. The equation has a finite solution only if the right-hand term is orthogonal to all solutions of the homogeneous equation. This is due to the self-adjoint nature of the wave operator. The conversation also includes a welcome to a new member and a recommendation to enjoy Brillouin's book.
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ollielgg
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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.
 
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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
 
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Thanks a lot! This is a big help.
 
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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
 
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1. What is an inhomogeneous wave equation?

An inhomogeneous wave equation is a type of partial differential equation that describes the propagation of a wave in a medium where the properties of the medium are not uniform. This means that the wave equation has a term on the right-hand side (RHS) that represents the non-uniformity of the medium.

2. What does it mean for the RHS of the inhomogeneous wave equation to be orthogonal to homogeneous solutions?

This means that the non-uniformity represented by the RHS does not affect the solutions of the wave equation that are homogeneous, or do not vary with respect to the spatial coordinates. In other words, the non-uniformity does not change the shape or behavior of the wave.

3. How is the inhomogeneous wave equation solved?

The inhomogeneous wave equation can be solved using a variety of methods, such as separation of variables, Fourier transforms, or Green's functions. The specific method used depends on the form of the equation and the boundary conditions.

4. What are some real-world applications of the inhomogeneous wave equation?

The inhomogeneous wave equation has many applications in physics and engineering, including describing the behavior of waves in non-uniform media such as sound waves in the atmosphere, electromagnetic waves in optical fibers, and seismic waves in the Earth's crust.

5. Are there any limitations to the inhomogeneous wave equation?

Like any mathematical model, the inhomogeneous wave equation has its limitations. It assumes that the medium is continuous and that the wave is propagating in only one direction. It also does not take into account factors such as dispersion, dissipation, and nonlinearity, which may affect the behavior of real waves.

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