# Is Helmholtz equation a Poisson Equation?

• yungman
In summary, the Helmholtz equation and Poisson's equation are both second order PDEs, but the latter is more general as the function f can be any value, while in the Helmholtz equation, f is equal to -ku. The separation of variables method can be used to solve certain types of the Helmholtz equation. While they have similarities, one is not a subset or form of the other.

#### yungman

Helmholtz equation:##\nabla^2 u=-ku## is the same form of ##\nabla^2 u=f##.

So is helmholtz equation a form of Poisson Equation?

They're both second order PDEs, but the Poisson f is a more general function, not necessarily related to the unknown function u. If the function f is 0, then the Poisson equation reduces to the Laplace equation.

In the solution of certain types of the Helmholtz equation, the separation of variables can be utilized.

http://en.wikipedia.org/wiki/Helmholtz_equation

http://en.wikipedia.org/wiki/Poisson's_equation

The generality of 'f' in Poisson's equation makes it trickier to solve than Laplace.

Thanks for the reply, I understand the difference between the two. But Helmholtz is also in form of Poisson, only when ##f=-ku##. So, can I say Helmholtz is a subset or one form of Poission Equation?

Thanks

Last edited:
yungman said:
Thanks for the reply, I understand the difference between the two. But Helmholtz is also in form of Poisson, only when ##f=-k\nabla^2 u##. So, can I say Helmholtz is a subset or one form of Poission Equation?

Thanks

I think you mean when f = -ku

SteamKing said:
I think you mean when f = -ku

Yes, my bad. What is FWIW?

Thanks

FWIW = For What It's Worth

• 1 person
Leaving the chat speak aside, generally speaking the only connection between Poisson's equation and Helmholtz equation is that they are both elliptic 2nd order linear PDEs. One is not a particular case of the other, as posts 2 and especially 3,4 above insinuate.

Thanks everyone.