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:






{\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

    I Derivation of the Helmholtz equation

    I am trying to understand the Helmholtz equation, where the Helmholtz equation can be considered as the time-independent form of the wave equation. It seems to me that the Helmholtz equation can be derived from the Fourier transform, such that it is part of a larger set of equations of varying...
  3. hilbert2

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

    The energy spectrum of a particle in 1D box is known to be ##E_n = \frac{h^2 n^2}{8mL^2}##, with ##L## the width of the potential well. In 3D, the ground state energy of both cubic and spherical boxes is also proportional to the reciprocal square of the side length or diameter. Does this...
  4. P

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

    Plugging in the supposed ##G## into the delta function equation ##\nabla^2 G = -\frac{1}{4 \pi} \frac{1}{r^2} \frac{\partial}{\partial r} \left(\frac{r^2 \left(ikr e^{ikr} - e^{ikr} \right)}{r^2} \right)## ##= -\frac{1}{4 \pi} \frac{1}{r^2} \left[ike^{ikr} - rk^2 e^{ikr} - ike^{ikr} \right]##...
  5. Matt Chu

    Proving a complex wave satisfies Helmholtz equation

    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...
  6. Matt Chu

    Proving a wave satisfies the Helmholtz equation

    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...
  7. DrClaude

    Green's function for the Helmholtz equation

    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

    Hello Everyone, Helmholtz equations derives from the wave equation by using separation of variables and assuming that the solution is indeed separable ##g(x,y,z,t) = f(x,y,z) T(t)##. The solutions to Helmholtz equations are functions of space, like f(x,y,z), and do not depend on time t. the...
  9. T

    A Observations Helmholtz equation is unable to explain

    Can someone explain to me some of the key observations Helmholtz equation cannot explain and why that is so? Thanks!
  10. Vajhe

    Fourier transform of the Helmholtz equation

    Hi guys, I have been trying to solve the Helmholtz equation with no luck at all; I'm following the procedure found in "Engineering Optics with MATLAB" by Poon and Kim, it goes something like this: Homework Statement Homework Equations Let's start with Helmholtz eq. for the complex amplitude ##...
  11. S

    Engineering Eigenfunctions of the vector Helmholtz equation

    Hi everyone, I'm looking for a reference book that treats the theory behind the eigenfunctions solution of the so called vector Helmholtz equation and its Neumann and Dirichlet problems. I've already found a theory inside the last chapter of Morse & Feshbach's Methods of theoretical physics...
  12. J

    When can I use Helmholtz equation for electromagnetics

    The complete Maxwell wave equation for electromagnetic field using the double curl operator "∇×∇×". Only when the transverse condition is hold, this operator can equal to the Laplace operator and form the helmholtz. My question is what's the condition can we use the helmoltz equation instead of...
  13. A

    How Do You Solve the Vector Modal Equation for Step-Index Fiber?

    I need help solving the vector modal equation for a step index fiber having a constant refractive index in the core and the cladding. (Under the conditions of zero dispersion and absorption.
  14. A

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

    I only took one class of PDE and even though I do remember the relationship between Laplace and Poisson I really do not recall Helmholtz at all. Anyways, I am trying to figure out if my software (a software I found online, FISKPACK) that solves Helmholtz equation can be used to solve Poisson...
  15. E

    Helmholtz equation and Multislice approach

    Hi there! I have a problem about the proof of an equation in microscopy. I think this is the right section because it is about solving the Helmholtz equation. I'm looking to Wikipedia page http://en.wikipedia.org/wiki/Multislice where they try to solve the Schrödinger equation for an electron...
  16. G

    Pertubations To Helmholtz Equation

    Consider the Helmholtz Equations with a perturbation p(r) [gradient^2 + p(r) + omega^2/c(r)^2 ]u(r,w) = 0 Does anyone know where I can find resources to the solutions/discussion of this equation? I can find many things such that p(r) = 0 , but the RHS = forcing function, but that is not...
  17. Y

    Question on using solution from Helmholtz equation in Poisson equation

    Helmholtz equation stated that \nabla^2 u(r,\theta,\phi) =-ku(r,\theta,\phi) = f(r,\theta,\phi) This is being used for Poisson equation with zero boundary: \nabla^2 u(r,\theta,\phi) = f(r,\theta,\phi) and u(a,\theta,\phi)=0 I just don't see how this can work as k=m^2 is a number only...
  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

    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 = (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. Homework Equations Helmholtz...
  20. Y

    Is Helmholtz equation a Poisson Equation?

    Helmholtz equation:##\nabla^2 u=-ku## is the same form of ##\nabla^2 u=f##. So is helmholtz equation a form of Poisson Equation?
  21. M

    NH Modified Helmholtz Equation with Robin Boundary Condition

    Hi, I am working on a quite difficult, though seemingly simple, non-homogeneous differential equation in cylindrical coordinates. The main equation is the non homogeneous modified Helmholtz Equation \nabla^{2}\psi - k^{2}\psi =...
  22. W

    The best method to solve Helmholtz equation for a irregular boundary

    i have an almost square region. By 'almost' i mean the edges are curvy, not completely straight. i now need to solve the Helmholtz equation with Dirichlet boundary condition what is the best numerical method? how is Finite element, though i do not know what Finite element is
  23. T

    Scaling Helmholtz equation

    Hi, I have tried to solve eigenvalue problem of the Helmholtz equation ∇×1/μ∇×E-k2E=0 in 2D, where k2=k20-β2 is the eigenvalue and k0=2*π/λ0 is the wavenumber in vacuum. Also β=neff*2*π/λ0 where λ0 is the wavelength in vacuum. Because constants ε=εrε0 and μ=μrμ0 are not very convinient...
  24. B

    Solving the Helmholtz Equation for a Point Source

    Homework Statement By integrating (2-55), over a small volume containing the origin, substituting ψ = Ce-jβr/r, and letting r approach zero, show that C = 1/4π, thus proving (2-58). Homework Equations (2-55): ∇2ψ + β2ψ = -δ(x)δ(y)δ(z) (2-58): ψ = e-jβr/(4πr) The Attempt at a...
  25. B

    Solving the Helmholtz Equation for a Point Source

    Homework Statement By integrating (2-55), over a small volume containing the origin, substituting ψ = Ce-jβr/r, and letting r approach zero, show that C = 1/4π, thus proving (2-58). Homework Equations (2-55): ∇2ψ + β2ψ = -δ(x)δ(y)δ(z) (2-58): ψ = e-jβr/(4πr) The Attempt at a...
  26. J

    Sturm-Liouville theory applied to solve Helmholtz equation

    Homework Statement Find the eigenfunctions of the Helmholtz equation: \frac{d^2y}{dx^2}+k^2y = 0 with boundary conditions: y(0)=0 y'(L)=0 Homework Equations General Solution: y = Asin(kx) + Bcos(kx) The Attempt at a Solution I found that at y(0) that B=0 and that...
  27. Q

    A strange but reasonable solution for Helmholtz equation

    Hi guys, I have a question when solving 3D Helmholtz equation derived from Maxwell equations. Normally I will get a plane wave solution. But when I used the method of separation variables for the three components Ex Ey and Ez, I found that the vector k in these three can be different as long...
  28. R

    Bessel Function / Helmholtz equation

    Homework Statement I'm interested in the solution of an equation given below. (It's not a homework/coursework question, but can be stated in a similar style, so I thought it best to post here.) Homework Equations A \nabla^2 f(x)-Bf(x)+C \exp(-2x^2/D^2)=0 where A,B,C,D are...
  29. A

    Inhomogeneous Helmholtz equation

    Homework Statement it is stated in wikipedia http://en.wikipedia.org/wiki/Helmholtz_equation that "Here, G is the Green's function of this equation, that is, the solution to the inhomogeneous Helmholtz equation with ƒ equaling the Dirac delta function, so G satisfies [del^2 +...
  30. Demon117

    Green's function for Helmholtz Equation

    Homework Statement Arfken & Weber 9.7.2 - Show that \frac{exp(ik|r_{1}-r_{2}|)}{4\pi |r_{1}-r_{2}|} satisfies the two appropriate criteria and therefore is a Green's function for the Helmholtz Equation. Homework Equations The Helmholtz operator is given by \nabla ^{2}A+k^{2}A...
  31. S

    PDE : Can not solve Helmholtz equation

    PDE : Can not solve Helmholtz equation (This is not a homework. I doing my research on numerical boundary integral. I need the analytical solution to compare the results with my computer program. I try to solve this equation, but it not success. I need urgent help.) I working on anti-plane...
  32. W

    Showing that an equation satisfied the helmholtz equation

    Homework Statement Show that \epsilon(r)=\frac{A}{r}e^{ikr} is a solution to \nabla^{2}\epsilon(r)+k^{2}\epsilon(r)=0 Homework Equations The Attempt at a Solution Is \nabla^{2} in this case equal to \frac{\partial^2}{\partial r^2} or \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial...
  33. H

    Helmholtz Equation with non-homogeneous b.c.

    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

    Helmholtz equation and the Maximum Principle

    hi everyone, I am dealing with a numerical method to solve the Helmholtz equation. As far as I know it is a second order elliptic PDE. I would like to know, if the Maximum Principle (aka Boundary Maximum Principle) holds for the Helmholtz eq., or where can I find explicit literature about it...
  35. Y

    Solution of Helmholtz equation

    hi guys.. I'm trying desperately to solve the following Helmholtz equation: *(all parameters are known) \frac{\partial^2 E_z}{\partial x^2}+ \frac{\partial^2 E_z}{\partial y^2} +j\omega\sigma E_z=0 (Ez is a scalar of course) in the boundaries -inf<x<inf, -inf<y<0 with the...
  36. X

    Helmholtz equation in N dimensions

    Hello! I'm in a search for information on the topic to devise a strategy of solving H. equation in N dimensions with Dirichlet - von Neumann type of boundary conditions. I'm assured that problem can be solved in closed form. As far as I can see, boundary conditions in any number of...
  37. H

    Helmholtz equation Neumann and divergence

    "Helmholtz equation" Neumann and divergence Hello, I'm trying to solve the following elliptic problem : S = B - \mu\nabla^2 B Where S(x,y) and B(x,y) are 3 component vectors. I have \nabla\cdot S = 0 and I want B such that \nabla\cdot B = 0 everywhere. I'm using finite differences on a...
  38. L

    Solving the 3D Helmholtz Equation Directly

    Homework Statement Given 3D Helmholtz eqn. u_xx + U-yy + U_zz + Lamda*u = 0 ,Lamda > 0. We are asked to "Calculate the fundamental solution directly (without using the Bessel identity for J_1/2 given)" where: Bessel identity given is w(r)=C_n*r^-(n-2)/2*J-(n-2)/2*(Lamda^1/2*r)...
  39. S

    How Do Magnetic Coils Affect Bird Navigation?

    Possible Helmholtz equation? Homework Statement Heres my problem: In recent studies of migrating birds using the Earth's magnetic field for navigation, birds have been fitted with coils as "caps" and "collars". a. If the identical coils have radii of 1.2cm and are 2.2cm apart, with 50...
  40. Repetit

    Solving a partial differential equation (Helmholtz equation)

    Hey! I am trying to solve this quite nasty (as least I think so : - ) partial differential equation (the Helmholtz equation): \frac{1}{r}\frac{\partial}{\partial r} \left( r \frac{\partial\Psi}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 \Psi}{\partial \phi^2} + \frac{\partial^2...
  41. M

    Classical and nonclassical symmetries for Helmholtz Equation

    " Classical and nonclassical symmetries for Helmholtz Equation " solitions help. Thank you.
  42. M

    Can Quantum Chaos Link to the Riemann Hypothesis Through Helmholtz Equations?

    dear friends :) "Classical and noncllasical symetries for helmholtz equation" help help.
  43. J

    Solving Helmholtz Equation in a Hollow Cylinder

    Im trying to solve Helmholtz equation \nabla ^2u(r,\phi,z) + k^2u(r,\phi,z) = 0 in a hollow cylinder with length L and a < r < b and the boundary conditions: u(a,\phi,z) = F(\phi,z) u(b,\phi,z) = G(\phi,z) u(r,\phi,0) = P(\phi,z) u(r,\phi,L) = Q(\phi,z)...
  44. U

    Helmholtz Equation in a Square

    I'm solving a Helmholtz equation uxx+uyy+lambda*u=0 in a rectangle: 0<=x<=L, 0<=y<=H with the following boundary conditions: u(x,0)=u(x,H)=0 and ux(0,y)=ux(L,y)=0 I found the eigenvalues to be: lambda(nm)=(n Pi/L)^2+(m Pi/H)^2 and the eigenfunctions to be: u(nm)=Cos(n Pi x/L)*Sin(m Pi...