Proving a wave satisfies the Helmholtz equation

[Matt Chu]

Homework Statement

Consider a harmonic wave given by

$$\Psi (x, t) = U(x, y, z) e^{-i \omega t}$$

where ##U(x, y, z)## is called the complex amplitude. Show that ##U## satisfies the Helmholtz equation:

$$ (\nabla + k^2) U (x, y, z) = 0 $$

Homework Equations

Everything important already in the problem.

The Attempt at a Solution

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The first thing I attempted to do was to express ##U## in terms of ##\Psi## and ##e^{-i \omega t}##. This led me to a long set of derivations that in no way gave me anything remotely close to zero. I'm confused as to how to solve this, as the ##k## component of the Helmholtz equation seems to be problematic; it seems the only way to prove that the whole expression equals zero would be if ##U = 0##.

 

[Orodruin]

What differential equation should the wave satisfy?