Helmholtz Equation, Boundary-Value Problems, and Eigenfunction Expansions

Helmholtz Equation, Boundary-Value Problems, and Eigenfunction Expansions

The time-dependent scalar wave equation is

(1)

If the motion is harmonic in time, the time dependence can be separated from the spatial dependence.

Write

(2)

Then

(3)

Substitution into the wave equation gives

(4)

Define

(5)

The spatial field therefore satisfies

(6)

This is the Helmholtz equation.

The central idea of this article is that the differential equation alone is not enough. The allowed solutions are determined by the differential equation together with the boundary conditions.

The combination is a boundary value problem.

1 From the wave equation to the Helmholtz equation

Start from

(7)

Use a complex harmonic representation,

(8)

The second time derivative produces

(9)

Thus the complex spatial amplitude satisfies

(10)

where

(11)

The physical field is the real part of the complex representation.

2 Why the Helmholtz equation matters

The Helmholtz equation appears whenever a linear wave system is examined at one angular frequency.

Examples include:

• vibrating strings,
• vibrating membranes,
• acoustic cavities,
• electromagnetic cavities,
• scalar diffraction models,
• quantum stationary-wave analogues.

The geometry enters through the Laplacian and the boundary conditions.

The allowed values of

(12)

become discrete when the domain is bounded and the boundary conditions confine the field.

3 Boundary value problem

A boundary value problem consists of:

1. a differential equation inside a domain,
2. boundary conditions specified on the boundary of that domain.

For the Helmholtz equation:

(13)

inside a region

(14)

the boundary is denoted

(15)

Different physical boundaries correspond to different mathematical conditions.

4 Dirichlet boundary condition

A Dirichlet condition specifies the value of the field on the boundary:

(16)

The homogeneous Dirichlet condition is

(17)

Examples include:

• a string fixed at an endpoint,
• a membrane fixed at its rim,
• a scalar field constrained to vanish on a wall.

M06-17 and M06-18 both used homogeneous Dirichlet conditions.

5 Neumann boundary condition

A Neumann condition specifies the Normal derivative:

(18)

Here

(19)

means derivative in the outward normal direction.

The homogeneous Neumann condition is

(20)

Depending on the physical system, this can represent a free boundary, a rigid acoustic wall, or a no-flux condition.

6 Mixed and Robin boundary conditions

Different parts of the boundary may obey different conditions.

For example:

(21)

on one part and

(22)

on another.

This is a mixed boundary value problem.

A Robin condition combines field value and normal derivative:

(23)

Robin conditions often model finite boundary compliance or impedance-like effects.

7 The Laplacian eigenvalue problem

For homogeneous boundary conditions, the Helmholtz equation can be written

(24)

where

(25)

This is an eigenvalue problem.

The operator is

(26)

The eigenvalue equation is

(27)

The boundary conditions determine which eigenfunctions

(28)

and eigenvalues

(29)

are allowed.

For wave problems,

(30)

8 One-dimensional fixed interval

Consider

(31)

with

(32)

The Helmholtz equation is

(33)

Its general solution is

(34)

The condition at

(35)

gives

(36)

The condition at

(37)

gives

(38)

Thus

(39)

and

(40)

The eigenvalues are

(41)

9 One-dimensional Neumann interval

Now impose

(42)

The allowed functions become

(43)

with

(44)

The

(45)

mode is a constant function.

Its eigenvalue is

(46)

This zero mode is allowed by Neumann conditions but not by homogeneous Dirichlet conditions.

10 Rectangular domain

For a rectangle

(47)

with

(48)

on all four edges, separation gives

(49)

and

(50)

Thus

(51)

These are exactly the membrane modes derived in M06-17.

11 Circular domain

For a circular domain of radius

(52)

with

(53)

the separated radial equation becomes Bessel’s equation.

Regularity at the center gives radial factors

(54)

The fixed boundary requires

(55)

Therefore

(56)

and

(57)

These are the circular-membrane eigenvalues from M06-18.

12 Green’s first identity

Orthogonality of Laplacian eigenfunctions can be derived from Green’s first identity:

(58)

Interchanging

(59)

and

(60)

and subtracting gives Green’s second identity:

∫ V dV = ∫ ∂V dS.

(61)

For homogeneous Dirichlet or homogeneous Neumann boundary conditions, the surface term vanishes for pairs of eigenfunctions.

13 Orthogonality of distinct eigenfunctions

Let

(62)

and

(63)

Use Green’s second identity with

(64)

If the boundary term vanishes,

(65)

Therefore

(66)

If

(67)

then

(68)

Distinct eigenvalues therefore correspond to orthogonal eigenfunctions.

14 Self-adjointness

An operator

(69)

is self-adjoint under an inner product if

(70)

for functions satisfying the relevant boundary conditions.

For the Laplacian problem,

(71)

Green’s identity shows that the difference between the two inner products becomes a boundary term.

Homogeneous Dirichlet, homogeneous Neumann, and many suitable Robin conditions make that term vanish.

This is the structural reason the eigenvalues are real and the eigenfunctions can be chosen orthogonal.

15 Positivity of Dirichlet eigenvalues

For a homogeneous Dirichlet eigenfunction,

(72)

Multiply by

(73)

and integrate:

(74)

Using Green’s first identity and the zero boundary value,

(75)

Both integrals are nonnegative.

For a nonzero Dirichlet eigenfunction,

(76)

Thus

(77)

16 Neumann zero mode

For homogeneous Neumann conditions, a constant function satisfies

(78)

and

(79)

Therefore

(80)

is allowed.

This is why Neumann spectra commonly contain a zero mode.

17 Degeneracy

Two linearly independent eigenfunctions can share the same eigenvalue.

This is degeneracy.

For a square membrane,

(81)

Thus

(82)

Any linear combination

(83)

is also an eigenfunction with that same eigenvalue.

Orthogonality of distinct eigenvalues is automatic.

Within a degenerate eigenspace, one chooses an orthogonal basis.

18 Eigenfunction normalization

An eigenfunction may be multiplied by any nonzero constant without changing its eigenvalue.

It is often convenient to normalize:

(84)

Then

(85)

For non-unit normalization,

(86)

where

(87)

is the normalization constant.

19 Eigenfunction expansion

For many regular bounded domains and standard self-adjoint boundary conditions, the eigenfunctions form a complete basis for functions satisfying the appropriate conditions.

Then a field

(88)

can be expanded as

(89)

For an orthogonal but not necessarily normalized basis,

(90)

This is the multidimensional generalization of Fourier coefficients.

20 Example 1: sine expansion on a fixed interval

On

(91)

with homogeneous Dirichlet conditions,

(92)

The normalization is

(93)

Therefore

(94)

This is the ordinary Fourier sine series.

21 Example 2: one exact eigenfunction

Suppose

(95)

Orthogonality gives

(96)

and

(97)

A field already equal to one eigenfunction has exactly one nonzero modal coefficient.

22 Example 3: rectangular eigenfunction expansion

For the rectangle of M06-17,

(98)

The normalization is

(99)

Thus

(100)

This is the two-dimensional Fourier sine expansion.

23 Example 4: circular eigenfunction expansion

For the circular membrane, the basis functions contain

(101)

times sine or cosine angular factors.

The area element is

(102)

Thus the radial inner product carries the Weight

(103)

For fixed

(104)

the radial orthogonality relation is

(105)

for

(106)

This is the Fourier-Bessel analogue of sine-series orthogonality.

24 Recovering time-dependent wave motion

Once the spatial eigenfunctions are known, the full wave field is expanded as

(107)

Substitute into the wave equation:

(108)

Using

(109)

orthogonality separates the modal equations:

(110)

Thus

(111)

Every spatial eigenfunction becomes one independent normal coordinate.

25 Initial displacement and initial velocity

Let

(112)

and

(113)

Write

(114)

Then

(115)

so

(116)

Also

(117)

so

(118)

26 Modal reconstruction

The complete solution is

ψ(r,t)

= ∑ nΦn(r) .

(119)

The problem has therefore been decomposed into:

• a spatial eigenvalue problem,
• projection of the initial data,
• independent harmonic evolution of each mode.

27 Boundary conditions change the spectrum

The differential operator may stay the same while the spectrum changes because the boundary conditions change.

For example, on

(120)

Dirichlet conditions give

(121)

with

(122)

Neumann conditions give

(123)

with

(124)

Thus the boundary is part of the physics, not an afterthought.

28 Boundary geometry changes the basis

Rectangular geometry produces products of sine and cosine functions.

Circular geometry produces Bessel functions and angular trigonometric factors.

Spherical geometry produces spherical Bessel functions and spherical harmonics.

The underlying operator may still be the Laplacian.

The geometry determines the natural coordinate system and the corresponding eigenfunctions.

29 Completeness and approximation

In practical calculations, an infinite eigenfunction expansion is truncated:

(125)

As more modes are included, the approximation generally improves for sufficiently regular functions.

Sharp spatial features require higher spatial-frequency modes.

This is the multidimensional analogue of Fourier-series convergence.

30 Parseval-type relation

For an orthonormal eigenbasis,

(126)

the squared norm is

(127)

For non-unit normalization,

(128)

This is the eigenfunction version of Parseval’s relation.

31 Why eigenfunction expansions are powerful

A partial differential equation can couple infinitely many spatial points.

The eigenfunction basis reorganizes the same problem into independent modal coordinates.

For the homogeneous wave equation,

(129)

For forced wave equations, each mode becomes a driven oscillator.

That is the next step developed in M06-20.

32 Common mistakes

1. Treating the Helmholtz equation as time dependent.
2. Forgetting that k = ω∕c.
3. Solving the differential equation without imposing boundary conditions.
4. Assuming the same boundary conditions produce the same spectrum in every geometry.
5. Confusing Dirichlet field values with Neumann normal derivatives.
6. Forgetting the outward normal derivative in a Neumann condition.
7. Assuming homogeneous Dirichlet and homogeneous Neumann spectra have the same lowest mode.
8. Forgetting the Neumann zero mode.
9. Writing the eigenvalue equation with inconsistent signs.
10. Assuming any two eigenfunctions are orthogonal even when they belong to a degenerate eigenvalue and have not been orthogonalized.
11. Forgetting that boundary terms determine whether the Laplacian is self-adjoint.
12. Ignoring the geometry-dependent integration measure in the inner product.
13. Forgetting the radial weight r in polar coordinates.
14. Using expansion coefficients without dividing by the eigenfunction norm.
15. Treating completeness as meaning a finite number of modes reproduces every function exactly.

33 Practice exercises

1. Derive the Helmholtz equation from the wave equation using harmonic time dependence.
2. Show that k = ω∕c has units of inverse length.
3. State homogeneous Dirichlet, Neumann, and Robin boundary conditions.
4. Solve the one-dimensional Dirichlet Helmholtz problem on 0 < x < L.
5. Solve the one-dimensional Neumann Helmholtz problem and identify the zero mode.
6. Re-derive the rectangular eigenvalues of M06-17 from the Laplacian eigenvalue problem.
7. Re-derive the circular eigenvalues of M06-18 from J_m(ka) = 0.
8. Use Green’s second identity to prove orthogonality for distinct eigenvalues.
9. Show that homogeneous Dirichlet Laplacian eigenvalues are positive.
10. Explain why a homogeneous Neumann problem may contain λ = 0.
11. Normalize the one-dimensional Dirichlet eigenfunctions.
12. Derive the general projection coefficient for a non-unit orthogonal eigenbasis.
13. Derive the modal equations for the full time-dependent wave field.
14. Derive the initial-displacement and initial-velocity coefficients.
15. Explain why a change in boundary conditions changes the eigenvalue spectrum even if the differential operator is unchanged.

34 Summary

Harmonic time dependence converts the wave equation into

(130)

Equivalently,

(131)

The boundary conditions select the allowed eigenfunctions and eigenvalues.

For wave motion,

(132)

For distinct eigenvalues under suitable self-adjoint boundary conditions,

(133)

A general field can then be expanded as

(134)

with

(135)

The time-dependent wave field is reconstructed from independent modal oscillators:

(136)

The Helmholtz eigenvalue problem therefore supplies the spatial foundation for normal modes, Resonances, cavity waves, and later forced-wave calculations.

References

References