Proving general solution of Helmholtz equation

[physicslove2]

Homework Statement

Prove that F(k•r -ωt) is a solution of the Helmholtz equation, provided that ω/k = 1/(µε)^1/2, where k = (k_x, k_y, k_z) 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.

Homework Equations

Helmholtz Equation: ∇²A+k²A=0

The Attempt at a Solution

I first completed the dot of k and r thus,
F(xk_x+ yk_y+zk_z)I am a little confused at this point, if I plug it into the equation I don't see how it would be 0

 

[SteamKing]

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

 

[physicslove2]

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

I completed the dot between k and r within F.

k = (k_x, k_y, k_z)
TA said to use r=(x,y,z)

thus the function becomes
F(xk_x+ yk_y+zk_z)

∇²F(xk_x+ yk_y+zk_z)=0, right?but then how would k²F(xk_x+ yk_y+zk_z)=0?

 

[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?