In mathematics, the Helmholtz equation, named for Hermann von Helmholtz, is the partial differential equation ∇ 2 A + k 2 A = 0 , displaystyle nabla ^ 2 A+k^ 2 A=0, where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude. Contents 1 Motivation and uses
2 Solving the
2.1 Vibrating membrane 2.2 Three-dimensional solutions 3 Paraxial approximation 4 Inhomogeneous Helmholtz equation 5 See also 6 Notes 7 References 8 External links Motivation and uses[edit]
The
( ∇ 2 − 1 c 2 ∂ 2 ∂ t 2 ) u ( r , t ) = 0. displaystyle left(nabla ^ 2 - frac 1 c^ 2 frac partial ^ 2 partial t^ 2 right)u(mathbf r ,t)=0.
u ( r , t ) = A ( r ) T ( t ) . displaystyle u(mathbf r ,t)=A(mathbf r )T(t). Substituting this form into the wave equation and then simplifying, we obtain the following equation: ∇ 2 A A = 1 c 2 T d 2 T d t 2 . displaystyle frac nabla ^ 2 A A = frac 1 c^ 2 T frac d^ 2 T dt^ 2 . Notice that the expression on the left side depends only on r, whereas the right expression depends only on t. As a result, this equation is valid in the general case if and only if both sides of the equation are equal to a constant value. From this observation, we obtain two equations, one for A(r), the other for T(t): ∇ 2 A A = − k 2 displaystyle frac nabla ^ 2 A A =-k^ 2 and 1 c 2 T d 2 T d t 2 = − k 2 , displaystyle frac 1 c^ 2 T frac d^ 2 T dt^ 2 =-k^ 2 , where we have chosen, without loss of generality, the expression −k2 for the value of the constant. (It is equally valid to use any constant k as the separation constant; −k2 is chosen only for convenience in the resulting solutions.) Rearranging the first equation, we obtain the Helmholtz equation: ∇ 2 A + k 2 A = ( ∇ 2 + k 2 ) A = 0. displaystyle nabla ^ 2 A+k^ 2 A=(nabla ^ 2 +k^ 2 )A=0. Likewise, after making the substitution ω = d e f k c , displaystyle omega stackrel mathrm def = kc, the second equation becomes d 2 T d t 2 + ω 2 T = ( d 2 d t 2 + ω 2 ) T = 0 , displaystyle frac d^ 2 T dt^ 2 +omega ^ 2 T=left( frac d^ 2 dt^ 2 +omega ^ 2 right)T=0, where k is the wave vector, and ω is the angular frequency.
We now have Helmholtz's equation for the spatial variable r and a
second-order ordinary differential equation in time. The solution in
time will be a linear combination of sine and cosine functions, with
angular frequency of ω, while the form of the solution in space will
depend on the boundary conditions. Alternatively, integral transforms,
such as the Laplace or Fourier transform, are often used to transform
a hyperbolic PDE into a form of the Helmholtz equation.
Because of its relationship to the wave equation, the Helmholtz
equation arises in problems in such areas of physics as the study of
electromagnetic radiation, seismology, and acoustics.
Solving the
( ∇ 2 + k 2 ) A = 0 displaystyle (nabla ^ 2 +k^ 2 )A=0 can be obtained for simple geometries using separation of variables.
Vibrating membrane[edit]
The two-dimensional analogue of the vibrating string is the vibrating
membrane, with the edges clamped to be motionless. The Helmholtz
equation was solved for many basic shapes in the 19th century: the
rectangular membrane by
A r r + 1 r A r + 1 r 2 A θ θ + k 2 A = 0. displaystyle A_ rr + frac 1 r A_ r + frac 1 r^ 2 A_ theta theta +k^ 2 A=0. We may impose the boundary condition that A vanish if r = a; thus A ( a , θ ) = 0. displaystyle A(a,theta )=0., The method of separation of variables leads to trial solutions of the form A ( r , θ ) = R ( r ) Θ ( θ ) , displaystyle A(r,theta )=R(r)Theta (theta ),, where Θ must be periodic of period 2π. This leads to Θ ″ + n 2 Θ = 0 , displaystyle Theta ''+n^ 2 Theta =0,, and r 2 R ″ + r R ′ + r 2 k 2 R − n 2 R = 0. displaystyle r^ 2 R''+rR'+r^ 2 k^ 2 R-n^ 2 R=0., It follows from the periodicity condition that Θ = α cos n θ + β sin n θ , displaystyle Theta =alpha cos ntheta +beta sin ntheta ,, and that n must be an integer. The radial component R has the form R ( r ) = γ J n ( ρ ) , displaystyle R(r)=gamma J_ n (rho ),, where the
ρ 2 J n ″ + ρ J n ′ + ( ρ 2 − n 2 ) J n = 0 , displaystyle rho ^ 2 J_ n ''+rho J_ n '+(rho ^ 2 -n^ 2 )J_ n =0, and ρ = kr. The radial function Jn has infinitely many roots for each value of n, denoted by ρm,n. The boundary condition that A vanishes where r = a will be satisfied if the corresponding wavenumbers are given by k m , n = 1 a ρ m , n . displaystyle k_ m,n = frac 1 a rho _ m,n ., The general solution A then takes the form of a doubly infinite sum of terms involving products of sin ( n θ ) or cos ( n θ ) , and J n ( k m , n r ) . displaystyle sin(ntheta ) text or cos(ntheta ), text and J_ n (k_ m,n r). These solutions are the modes of vibration of a circular drumhead. Three-dimensional solutions[edit] In spherical coordinates, the solution is: A ( r , θ , φ ) = ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ ( a ℓ m j ℓ ( k r ) + b ℓ m y ℓ ( k r ) ) Y ℓ m ( θ , φ ) . displaystyle A(r,theta ,varphi )=sum _ ell =0 ^ infty sum _ m=-ell ^ ell left(a_ ell m j_ ell (kr)+b_ ell m y_ ell (kr)right)Y_ ell ^ m (theta ,varphi ). This solution arises from the spatial solution of the wave equation and diffusion equation. Here j ℓ ( k r ) displaystyle j_ ell (kr) and y ℓ ( k r ) displaystyle y_ ell (kr) are the spherical Bessel functions, and Y ℓ m ( θ , φ ) displaystyle Y_ ell ^ m (theta ,varphi ) are the spherical harmonics (Abramowitz and Stegun, 1964). Note that these forms are general solutions, and require boundary conditions to be specified to be used in any specific case. For infinite exterior domains, a radiation condition may also be required (Sommerfeld, 1949). Writing r 0 = ( x , y , z ) displaystyle mathbf r_ 0 =(x,y,z) function A ( r 0 ) displaystyle A(r_ 0 ) has asymptotics A ( r 0 ) = e i k r 0 r 0 f ( r 0 r 0 , k , u 0 ) + o ( 1 r 0 ) as r 0 → ∞ displaystyle A(r_ 0 )= frac e^ ikr_ 0 r_ 0 fleft( frac mathbf r _ 0 r_ 0 ,k,u_ 0 right)+oleft( frac 1 r_ 0 right) text as r_ 0 to infty where function f is called scattering amplitude and u 0 ( r 0 ) displaystyle u_ 0 (r_ 0 ) is the value of A at each boundary point r 0 displaystyle r_ 0 . Paraxial approximation[edit] Further information: Slowly varying envelope approximation In the paraxial approximation of the Helmholtz equation,[1] the complex amplitude A is expressed as A ( r ) = u ( r ) e i k z displaystyle A(mathbf r )=u(mathbf r )e^ ikz where u represents the complex-valued amplitude which modulates the sinusoidal plane wave represented by the exponential factor. Then under a suitable assumption, u approximately solves ∇ ⊥ 2 u + 2 i k ∂ u ∂ z = 0 , displaystyle nabla _ perp ^ 2 u+2ik frac partial u partial z =0, where ∇ ⊥ 2 = d e f ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 displaystyle textstyle nabla _ perp ^ 2 stackrel mathrm def = frac partial ^ 2 partial x^ 2 + frac partial ^ 2 partial y^ 2 is the transverse part of the Laplacian. This equation has important applications in the science of optics, where it provides solutions that describe the propagation of electromagnetic waves (light) in the form of either paraboloidal waves or Gaussian beams. Most lasers emit beams that take this form. The assumption under which the paraxial approximation is valid is that the z derivative of the amplitude function u is a slowly-varying function of z:
∂ 2 u ∂ z 2
≪
k ∂ u ∂ z
. displaystyle bigg partial ^ 2 u over partial z^ 2 bigg ll bigg k partial u over partial z bigg . This condition is equivalent to saying that the angle θ between the wave vector k and the optical axis z is small: θ ≪ 1 displaystyle theta ll 1 .
The paraxial form of the
∇ 2 ( u ( x , y , z ) e i k z ) + k 2 u ( x , y , z ) e i k z = 0. displaystyle nabla ^ 2 (uleft(x,y,zright)e^ ikz )+k^ 2 uleft(x,y,zright)e^ ikz =0. Expansion and cancellation yields the following: ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 ) u ( x , y , z ) e i k z + ( ∂ 2 ∂ z 2 u ( x , y , z ) ) e i k z + 2 ( ∂ ∂ z u ( x , y , z ) ) i k e i k z = 0. displaystyle left( frac partial ^ 2 partial x^ 2 + frac partial ^ 2 partial y^ 2 right)u(x,y,z)e^ ikz +left( frac partial ^ 2 partial z^ 2 u(x,y,z)right)e^ ikz +2left( frac partial partial z u(x,y,z)right)ik e^ ikz =0. Because of the paraxial inequalitiy stated above, the ∂2u/∂z2 term is neglected in comparison with the k·∂u/∂z term. This yields the paraxial Helmholtz equation. Substituting u ( r ) = A ( r ) e − i k z displaystyle u(mathbf r )=A(mathbf r )e^ -ikz then gives the paraxial equation for the original complex amplitude A: ∇ ⊥ 2 A + 2 i k ∂ A ∂ z + 2 k 2 A = 0. displaystyle nabla _ perp ^ 2 A+2ik frac partial A partial z +2k^ 2 A=0. The
∇ 2 A ( x ) + k 2 A ( x ) = − f ( x ) in R n displaystyle nabla ^ 2 A(x)+k^ 2 A(x)=-f(x) text in mathbb R ^ n where ƒ : Rn → C is a given function with compact support, and n = 1, 2, 3. This equation is very similar to the screened Poisson equation, and would be identical if the plus sign (in front of the k term) is switched to a minus sign. In order to solve this equation uniquely, one needs to specify a boundary condition at infinity, which is typically the Sommerfeld radiation condition lim r → ∞ r n − 1 2 ( ∂ ∂ r − i k ) A ( r x ^ ) = 0 displaystyle lim _ rto infty r^ frac n-1 2 left( frac partial partial r -ikright)A(r hat x )=0 uniformly in x ^ displaystyle hat x with
x ^
= 1 displaystyle hat x =1 , where the vertical bars denote the Euclidean norm. With this condition, the solution to the inhomogeneous Helmholtz equation is the convolution A ( x ) = ( G ∗ f ) ( x ) = ∫ R n G ( x − y ) f ( y ) d y displaystyle A(x)=(G*f)(x)=int limits _ mathbb R ^ n !G(x-y)f(y),dy (notice this integral is actually over a finite region, since f displaystyle f has compact support). Here, G displaystyle G is the
∇ 2 G ( x ) + k 2 G ( x ) = − δ ( x ) in R n . displaystyle nabla ^ 2 G(x)+k^ 2 G(x)=-delta (x) text in mathbb R ^ n ., The expression for the
n displaystyle n of the space. One has G ( x ) = i e i k
x
2 k displaystyle G(x)= frac ie^ ikx 2k for n = 1, G ( x ) = i 4 H 0 ( 1 ) ( k
x
) displaystyle G(x)= frac i 4 H_ 0 ^ (1) (kx) for n = 2,[6] where H 0 ( 1 ) displaystyle H_ 0 ^ (1) is a Hankel function, and G ( x ) = e i k
x
4 π
x
displaystyle G(x)= frac e^ ikx 4pi x for n = 3. Note that we have chosen the boundary condition that the
x
→ ∞ displaystyle xto infty . See also[edit]
Notes[edit] ^ J. W. Goodman. Introduction to Fourier
References[edit] Abramowitz, Milton; Stegun, Irene, eds. (1964). Handbook of Mathematical functions with Formulas, Graphs and Mathematical Tables. New York: Dover Publications. ISBN 0-486-61272-4. Riley, K. F.; Hobson, M. P.; Bence, S. J. (2002). "Chapter 19". Mathematical methods for physics and engineering. New York: Cambridge University Press. ISBN 0-521-89067-5. Riley, K. F. (2002). "Chapter 16". Mathematical Methods for Scientists and Engineers. Sausalito, California: University Science Books. ISBN 1-891389-24-6. Saleh, Bahaa E. A.; Teich, Malvin Carl (1991). "Chapter 3". Fundamentals of Photonics. Wiley Series in Pure and Applied Optics. New York: John Wiley & Sons. pp. 80–107. ISBN 0-471-83965-5. Sommerfeld, Arnold (1949). "Chapter 16". Partial Differential Equations in Physics. New York: Academic Press. ISBN 0126546568. Howe, M. S. (1998).
External links[edit] Helmholtz Equation at EqWorld: The World of Mathematical Equations. Hazewinkel, Michiel, ed. (2001) [1994], "Helmholtz equation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 Vibrating Circular Membrane by Sam Blake, The Wolfram Demonstrations Project. Green's functions for the wave, Helmholtz and Poisson equations in a two-dimensiona |