# Proving general solution of Helmholtz equation

1. Oct 30, 2013

### physicslove2

1. The problem statement, all variables and given/known data

Prove that F(k•r -ωt) is a solution of the Helmholtz equation, provided that ω/k = 1/(µε)1/2, where k = (kx, ky, kz) is the wave-vector and r is the position vector. In F(k•r -ωt), “k•r –ωt” is the argument and F is any vector function.

2. Relevant equations
Helmholtz Equation: ∇2A+k2A=0

3. The attempt at a solution
I first completed the dot of k and r thus,
F(xkx+ yky+zkz)

I am a little confused at this point, if I plug it into the equation I don't see how it would be 0

Last edited: Oct 30, 2013
2. Oct 30, 2013

### SteamKing

Staff Emeritus
It's not clear what you've done, since you haven't shown your work.

3. Oct 30, 2013

### physicslove2

I completed the dot between k and r within F.

k = (kx, ky, kz)
TA said to use r=(x,y,z)

thus the function becomes
F(xkx+ yky+zkz)

2F(xkx+ yky+zkz)=0, right?

but then how would k2F(xkx+ yky+zkz)=0?

4. Oct 30, 2013

### BruceW

2F(xkx+ yky+zkz)=0 is not right. Because then, as you say, you would need k2F=0, which is the trivial solution.

It is a kind of weird problem in the first place, since the Helmholtz equation does not care about time dependence. Are you sure the problem wasn't to show that $F(k\cdot r -\omega t)$ is a solution of the wave equation, and to go on to prove that it is also a solution of the Helmholtz equation?

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