What is Helmholtz equation: Definition and 44 Discussions

In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation:





∇

2


f
=
−

k

2


f


{\displaystyle \nabla ^{2}f=-k^{2}f}
where ∇2 is the Laplace operator (or "Laplacian"), k2 is the eigenvalue, and f is the (eigen)function. When the equation is applied to waves, k is known as the wave number. The Helmholtz equation has a variety of applications in physics, including the wave equation and the diffusion equation, and it has uses in other sciences.

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  1. bob012345

    I Helmholtz Equation in Cartesian Coordinates

    So given the Helmholtz equation $$\nabla^2 u(x,y,z) + k^2u(x,y,z)=0$$ we do the separation of variables $$u=u_x(x)u_y(y)u_z(z)= u_xu_yu_z$$ and ##k^2 = k_x^2 + k_y^2 +k_z^2## giving three separate equations; $$\nabla^2_x u_x+ k_x^2 u_x=0$$ $$\nabla^2_y u_y+ k_y^2 u_y=0$$ $$\nabla^2_z u_z+ k_z^2...
  2. redtree

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  3. hilbert2

    A Ground state energy of a particle-in-a-box in coordinate scaling

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  4. P

    Confirming Green's function for homogeneous Helmholtz equation (3D)

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  5. Matt Chu

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  6. Matt Chu

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  7. DrClaude

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    Homework Statement Show that $$ G(x,x') = \left\{ \begin{array}{ll} \frac{1}{2ik} e^{i k (x-x')} & x > x' \\ \frac{1}{2ik} e^{-i k (x-x')} & x < x' \end{array} \right. $$ is a Green's function for the 1D Helmholtz equation, i.e., $$ \left( \frac{\partial^2}{\partial x^2} + k^2 \right) G(x,x') =...
  8. F

    I Wavevector k in Helmholtz Equation

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  9. T

    A Observations Helmholtz equation is unable to explain

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  10. Vajhe

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  11. S

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  12. J

    When can I use Helmholtz equation for electromagnetics

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  13. A

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  14. A

    Have a software that solves Helmholtz equation, can I use it for Poisson?

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  15. E

    Helmholtz equation and Multislice approach

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  16. G

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  17. Y

    Question on using solution from Helmholtz equation in Poisson equation

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  18. Y

    Help in deriving series expansion of Helmholtz Equation.

    Orthogonality of spherical bessel functions Homework Statement Proof of orthogonality of spherical bessel functions The book gave \int_{0}^{a}\int_{0}^{2\pi}\int_{0}^{\pi} j_{n} (\lambda_{n,j}r) j_{n'} (\lambda_{n',j'}r) Y_{n,m}(\theta,\phi)\overline{Y}_{n',m'}(\theta,\phi) \sin\theta...
  19. P

    Proving general solution of Helmholtz equation

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  20. Y

    Is Helmholtz equation a Poisson Equation?

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  21. M

    NH Modified Helmholtz Equation with Robin Boundary Condition

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  22. W

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  23. T

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  24. B

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  25. B

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  26. J

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  27. Q

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  28. R

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  29. A

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  30. Demon117

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  31. S

    PDE : Can not solve Helmholtz equation

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  32. W

    Showing that an equation satisfied the helmholtz equation

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  33. H

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    Homework Statement In 2D: (del2 + k2) u(r,theta) = 0 with b.c u(R,theta) = f(theta) Starting from the general solution (by separation of variables) show that the solution can be rewritten as: \int K(r,theta,theta')*f(theta') dtheta' from 0 to 2*Pi Homework Equations The general solution is...
  34. T

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  35. Y

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  36. X

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  37. H

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  38. L

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  39. S

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  41. M

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  42. M

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  43. J

    Solving Helmholtz Equation in a Hollow Cylinder

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  44. U

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