Method of Images and Eigenfunction Expansions of Green's Functions

The two representations are still the working tools of anyone solving a field equation in a bounded region. In room acoustics, the image-source method of Allen and Berkley (1979) models a room's early impulse response as a lattice of mirror sources reflected in the walls — the direct descendant of the interval derivation above — and it is the engine inside spatial-audio and virtual-reality rendering, where a few dozen images reproduce the first reflections while the diffuse late tail is left to a statistical modal model. In biomolecular simulation, image charges enforce the electrostatic boundary condition at a metal electrode or a dielectric membrane without meshing it. And across device physics the eigenfunction resolvent has become the non-equilibrium Green's function (NEGF) formalism: the poles this article calls resonances are the transmission peaks of a nanoscale transistor, and the trace of $G$ is the density of states experiment measures. Continued to complex energy, the same object gives the semiclassical trace formulas of quantum chaos, where long classical orbits — descendants of multiply reflected images — reappear as oscillations in the level density. Choosing the ray sum or the mode sum remains the judgement Green's essay first made possible.

The Green's function is the impulse response of a differential operator. Strike the domain with a unit point source at $\mathbf{r}'$ and $G(\mathbf{r},\mathbf{r}')$ is the field that rings out at $\mathbf{r}$; because $\mathcal{L}$ is linear, the response to any source $f$ is the superposition $\int G f$. Everything here is a way to compute that one response for a bounded region. The method of images replaces a conductor by a ghost. A grounded plane holds the potential at zero; a negative charge reflected behind the plane achieves the same zero on the surface, so in the physical half-space the true field is just the real charge plus its mirror. The conductor has vanished, and with it the hard boundary-value problem — replaced by two point charges in free space. The first figure shows the trick and the force it predicts.

The eigenfunction picture is the opposite temperament: a decomposition into standing waves. A point source excites every normal mode at once, each weighted by its amplitude at source and observer, $\phi_n(\mathbf{r}')\phi_n(\mathbf{r})$, and by how far its eigenvalue $\lambda_n$ sits from the operating value $\lambda$. Drive the system near a resonance, $\lambda\to\lambda_m$, and that one denominator $\lambda_m-\lambda$ collapses: a single mode dominates and $G$ diverges — exactly what "resonance" means. The two representations are one function summed differently, and the second figure makes the equivalence literal on the simplest domain, an interval.

Which series to trust is a question of regime. Near the source the singularity is universal — every method reproduces the same $1/r$ blow-up — and the ray picture gets there in one or two terms. Far away, or right on a resonance, the standing-wave sum is dominated by a handful of modes and wins. Choosing between them is not choosing different physics; it is choosing a fast-converging expansion of one and the same $G$.

Fix the convention up front: $\mathcal{L}$ is a positive self-adjoint operator such as $-\nabla^2$ or $-\partial_x^2$, so $-\nabla^2 G=\delta$ gives the everyday Coulomb-like potential with a positive free-space kernel. The defining equation pins $G$ down as the response to a unit point source inside the domain.

where: $\mathcal{L}$ — linear self-adjoint operator (e.g. $-\nabla^2$ or $-\nabla^2-\lambda$); $\Omega$ — bounded or half-space domain with boundary $\partial\Omega$; $\delta(\mathbf{r}-\mathbf{r}')$ — Dirac delta, a unit point source at $\mathbf{r}'$. The homogeneous boundary condition (Dirichlet $G=0$ or Neumann $\partial_n G=0$) is part of the definition — without it $G$ is not unique. Knowing $G$ solves the inhomogeneous problem for any source by a single integral.

where: $f$ — the source of $\mathcal{L}u=f$; $u$ — the sought field; boundary terms carry inhomogeneous data via Green's second identity. Read it as superposition: each element $f(\mathbf{r}')\,\mathrm{d}^n r'$ radiates its response $G$, and the field is their sum. For the Laplacian the free-space kernel is elementary — the seed every image construction grows from.

where: $G_3,\,G_2$ — solutions of $-\nabla^2 G=\delta$ in three and two dimensions; $|\mathbf{r}-\mathbf{r}'|$ — source-to-observer distance. They follow from $\nabla^2(1/r)=-4\pi\delta$ (3D) and $\nabla^2\ln r=2\pi\delta$ (2D), the sign fixed by the positive-operator convention. The half-space Dirichlet problem is the archetype of the method of images: one mirror charge does the whole job.

where: $\mathbf{r}'_{\!*}$ — mirror of $\mathbf{r}'$ across $z=0$, at $z=-z'$ outside the physical region. On the plane the two distances are equal, so the bracket vanishes and $G=0$ holds identically; inside $\Omega$ only the real singularity survives. For a sphere the mirror becomes an inversion: Kelvin's image lives on the same ray, scaled inward, with a reduced charge.

where: $q'$ — image charge magnitude; $\mathbf{r}'_{\!*}$ — image position along $\hat{\mathbf{r}}'$, inside a grounded sphere of radius $a$; $r'$ — source distance from the centre. This pair is the unique choice that zeroes the potential on $|\mathbf{r}|=a$, derived below. The eigenfunction expansion is the universal alternative — no special geometry, only a complete self-adjoint spectrum.

where: $\phi_n$ — orthonormal eigenfunctions with $\mathcal{L}\phi_n=\lambda_n\phi_n$ satisfying the boundary conditions; $\lambda_n$ — real eigenvalues (self-adjointness); $\lambda$ — spectral parameter, so $(\mathcal{L}-\lambda)^{-1}$ is the resolvent. Each pole at $\lambda=\lambda_n$ is a resonance. Self-adjointness leaves a measurable fingerprint on $G$ worth naming on its own.

where: $G(\mathbf{r},\mathbf{r}')$ — symmetric under exchange of source and observer. A source at $A$ seen at $B$ reads identically to a source at $B$ seen at $A$; this is manifest term-by-term in the spectral sum (6). Once $G$ is known, observables such as the induced density $\sigma_{\text{ind}}=-\varepsilon_0\,\partial_n\Phi|_{\partial\Omega}$ and the image force follow by differentiation, as Example 1 shows.

Which representation is efficient, and whether the image method works at all, depends on the geometry and on how close the spectral parameter sits to the spectrum.

1 · Charge above a grounded plane — force and induced charge (fully worked)

A point charge $q=+1\ \mathrm{nC}$ is held at height $d=3\ \mathrm{mm}$ above a large grounded conducting plane. Find the force on it and the total charge it induces on the plane.

Step 1. Replace the plane by its image: a charge $q'=-1\ \mathrm{nC}$ at depth $d$ below the surface, separation $2d=6\ \mathrm{mm}$. The half-space field is just these two charges, so the force on the real charge is its Coulomb attraction to the image, $F=\dfrac{1}{4\pi\varepsilon_0}\dfrac{q^2}{(2d)^2}=\dfrac{q^2}{16\pi\varepsilon_0 d^2}$, directed toward the plane.

Step 2. Insert numbers with $1/4\pi\varepsilon_0=8.988\times10^9\ \mathrm{N\,m^2/C^2}$:

This gives F = 2.5×10⁻⁴ N (0.25 mN), attractive. The force falls as $1/d^2$, so halving the gap quadruples the pull — the origin of the diverging adhesion a charge feels approaching a metal surface.

Step 3. The induced density $\sigma(\rho)=-\dfrac{q\,d}{2\pi(\rho^2+d^2)^{3/2}}$ peaks under the charge over a patch of size $\sim d$. Integrating over the plane,

The plane draws up exactly the image charge, which is why the far field looks like a dipole. F = 2.5×10⁻⁴ N attractive, Q_ind = −1 nC.

2 · Kelvin image for a grounded sphere

A charge $q$ sits at distance $r'=2a$ from the centre of a grounded sphere of radius $a$. Locate and size the image, and find the force on $q$.

Step 1. Equation (5) gives the image charge $q'=-\dfrac{a}{r'}q=-\dfrac{a}{2a}q=\mathbf{-q/2}$, placed at $\mathbf{r}'_{\!*}=\dfrac{a^2}{r'^2}\mathbf{r}'$, i.e. at radial distance $\dfrac{a^2}{r'}=\dfrac{a^2}{2a}=\mathbf{a/2}$ from the centre, inside the sphere. The image is weaker than the source and sits at the inversion point $a^2/r'$.

Step 2. The force on $q$ is its attraction to the image, separated by $r'-a^2/r'=2a-a/2=3a/2$:

So the charge is pulled toward the sphere with F = (2/9) q²/4πε₀a². Unlike the plane's pure $1/d^2$ law, the sphere's force interpolates between the far-field $1/r'^{4}$ behaviour (an induced-dipole attraction) and the near-surface $1/(\text{gap})^2$ divergence as $r'\to a$.

Step 3. As a check, let the sphere grow, $a\to\infty$ with the gap $\ell=r'-a$ held fixed: $q'\to-q$ and the image distance below the surface tends to $\ell$, recovering the flat-plane result of Example 1. The sphere image reduces to the plane mirror in the large-radius limit.

3 · A rectangular cavity and its resonances

For $(-\nabla^2-k^2)$ on a Dirichlet box $[0,a]\times[0,b]$, write the Green's function and find where it blows up.

Step 1. The separable modes $\phi_{mn}=\dfrac{2}{\sqrt{ab}}\sin\dfrac{m\pi x}{a}\sin\dfrac{n\pi y}{b}$ have eigenvalues $\lambda_{mn}=(m\pi/a)^2+(n\pi/b)^2$, so the spectral formula (6) with $\lambda=k^2$ gives

Step 2. The denominator vanishes as $k^2\to\lambda_{mn}$: the cavity rings at the resonant wavenumbers $k_{mn}=\sqrt{(m\pi/a)^2+(n\pi/b)^2}$ (lowest $k_{11}=\pi\sqrt2/a$ for a square box). A small loss $k^2\to k^2+\mathrm{i}\epsilon$ lifts each pole off the axis; summing one index in closed form converts the slow double sum into a fast ray-like series. Resonances at $k_{mn}=\sqrt{(m\pi/a)^2+(n\pi/b)^2}$ are the poles of the cavity Green's function.

The whole apparatus of this article — the "function," the identities that manipulate it, the idea of solving a field problem by superposing point-source responses — descends from one self-taught Nottingham miller who published a single, almost unread pamphlet.

Texts. Stone & Goldbart, Mathematics for Physics (2009), Ch. 11 (Green functions) · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., Ch. 10 · Riley, Hobson & Bence, Mathematical Methods for Physics and Engineering, 3rd ed., Ch. 21 · Jackson, Classical Electrodynamics, 3rd ed., Ch. 2–3 (images and expansions) · Morse & Feshbach, Methods of Theoretical Physics, Vol. I, Ch. 7. Key results to keep in mind: the half-space image satisfies $G|_{\partial\Omega}=0$ and gives the plane force $F=q^2/16\pi\varepsilon_0 d^2$; the Kelvin image is $q'=-a/r'$ at $a^2/r'$, recovering the flat-plane result in the limit $a\to\infty$; the interval kernel is $x_<(L-x_>)/L$, with peak $2L/9$ at $x'=L/3$, reproduced by both the sine series and the jump-matching construction; the box eigenvalues are $\lambda_{mn}=(m\pi/a)^2+(n\pi/b)^2$ with resonance poles at $k_{mn}=\sqrt{\lambda_{mn}}$; and the $-\nabla^2$ free-space kernels are $+1/4\pi r$ (3D) and $-(1/2\pi)\ln r$ (2D). Numerical spot-checks: plane force $2.5\times10^{-4}\ \mathrm{N}$ for $q=1\ \mathrm{nC}$, $d=3\ \mathrm{mm}$; induced charge $-q$; sphere force $\tfrac29\,q^2/4\pi\varepsilon_0 a^2$ at $r'=2a$.

Builds on: Sturm–Liouville theory and self-adjoint operators · Distributions and the Dirac delta · Fourier and Bessel series · Electrostatics of conductors. Leads to: WKB Approximation · Method of Steepest Descent and Saddle Points · Bohr–Sommerfeld Quantization.

Ask the tutor: Ask me to work a specific geometry end-to-end — a charge near a grounded sphere, a slab's image train, or a cavity's resonances — and I'll build its Green's function both ways and show you which series converges faster.