6.6
What is sound?
§6 · Sources and radiation

6.6 The Green’s function — every source is a sum of point sources

The monopole, dipole, and piston of the previous lessons are particular sources with particular fields. A real source — a voice, an engine, a loudspeaker driven by music — has an arbitrary shape and an arbitrary time history. We need a method that produces the radiated field of any source without starting from scratch each time. That method is the Green’s function, and it rests on a single idea: solve the problem for one point impulse, then build every other source by superposition.

The forced wave equation

So far the wave equation has had nothing on its right-hand side; it described the field once a disturbance was already present. A source is precisely a term that injects disturbance. Writing the d’Alembertian operator

□  ≡  1c2∂2∂t2−∇2,\Box \;\equiv\; \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2,

a region that is creating sound — injecting volume, like a small pulsating sphere — obeys the inhomogeneous wave equation

□ p′(r,t)  =  s(r,t),\Box\, p'(\mathbf{r}, t) \;=\; s(\mathbf{r}, t),

where ss is the source density: nonzero where sound is being generated, zero in the surrounding air. The homogeneous case s=0s = 0 is everything from chapter 4 onward; the new content is the right-hand side.

The impulse response

The Green’s function GG is the field produced by a unit point impulse — a source concentrated at one place r′\mathbf{r}' and one instant t′t':

□ G(r,t; r′,t′)  =  δ3(r−r′) δ(t−t′).\Box\, G(\mathbf{r}, t;\, \mathbf{r}', t') \;=\; \delta^3(\mathbf{r} - \mathbf{r}')\,\delta(t - t').

Once GG is known, the field of any source follows by superposition, with no further solving:

    p′(r,t)  =  ∫ ⁣ ⁣∫G(r,t; r′,t′) s(r′,t′)  d3r′ dt′.    \boxed{\;\; p'(\mathbf{r}, t) \;=\; \int\!\!\int G(\mathbf{r}, t;\, \mathbf{r}', t')\, s(\mathbf{r}', t')\; d^3r'\, dt'. \;\;}

The field is the source convolved with the Green’s function (refresher: convolution →). GG is exactly the impulse response of the medium regarded as a linear, time-invariant system: feed it an impulse and it answers with GG; feed it anything else and it answers with the corresponding sum of delayed, scaled impulse-responses.

▶ Why the impulse response is the whole answer Derivation

The Dirac delta has the sifting property: any source distribution is its own superposition of point impulses,

s(r,t)  =  ∫ ⁣ ⁣∫s(r′,t′) δ3(r−r′) δ(t−t′)  d3r′ dt′.s(\mathbf{r}, t) \;=\; \int\!\!\int s(\mathbf{r}', t')\, \delta^3(\mathbf{r} - \mathbf{r}')\,\delta(t - t')\; d^3r'\, dt'.

The wave operator □\Box is linear, and the medium does not change in time, so the response to a sum of impulses is the sum of the responses to each. The response to the single impulse at (r′,t′)(\mathbf{r}', t') is G(r,t;r′,t′)G(\mathbf{r}, t; \mathbf{r}', t') by definition; weighting by s(r′,t′)s(\mathbf{r}', t') and summing gives the boxed formula. Linearity plus the sifting property is the entire argument — no special structure of the wave equation is used, which is why the same construction works for the heat equation, electromagnetism, and any other linear field theory.

The free-space Green’s function

In unbounded three-dimensional space the Green’s function is

G(r,t; r′,t′)  =  δ ⁣(t−t′−∣r−r′∣c)4π ∣r−r′∣.G(\mathbf{r}, t;\, \mathbf{r}', t') \;=\; \frac{\delta\!\left(t - t' - \dfrac{|\mathbf{r} - \mathbf{r}'|}{c}\right)}{4\pi\,|\mathbf{r} - \mathbf{r}'|}.

Read it directly: a flash at r′\mathbf{r}' at time t′t' produces an infinitely thin spherical shell that expands at speed cc, reaching radius R=∣r−r′∣R = |\mathbf{r} - \mathbf{r}'| at the retarded time t=t′+R/ct = t' + R/c, with amplitude falling as 1/R1/R. This is the monopole field of 6.1–6.2 reduced to its essence — the 1/r1/r spherical wave is the free-space Green’s function, and the 4π4\pi is the surface area of the unit sphere over which a point’s output is shared.

For a single frequency e−iωte^{-i\omega t} the time delay becomes a phase and the Green’s function solves the inhomogeneous Helmholtz equation (∇2+k2) G^=−δ3(r−r′)(\nabla^2 + k^2)\,\hat G = -\delta^3(\mathbf{r} - \mathbf{r}'):

G^(r,r′)  =  eik∣r−r′∣4π ∣r−r′∣.\hat G(\mathbf{r}, \mathbf{r}') \;=\; \frac{e^{ik|\mathbf{r} - \mathbf{r}'|}}{4\pi\,|\mathbf{r} - \mathbf{r}'|}.
▶ That $1/4\pi r$ solves the equation, and where the $4\pi$ comes from Derivation

Away from the source (r>0r > 0, with r=∣r−r′∣r = |\mathbf{r} - \mathbf{r}'|), the steady-state G^=A eikr/r\hat G = A\,e^{ikr}/r must satisfy ∇2G^+k2G^=0\nabla^2 \hat G + k^2 \hat G = 0. Using the radial Laplacian ∇2u=1rd2dr2(ru)\nabla^2 u = \frac{1}{r}\frac{d^2}{dr^2}(r u):

∇2 ⁣(eikrr)=1rd2dr2 ⁣(eikr)=−k2eikrr,\nabla^2\!\left(\frac{e^{ikr}}{r}\right) = \frac{1}{r}\frac{d^2}{dr^2}\!\left(e^{ikr}\right) = \frac{-k^2 e^{ikr}}{r},

so ∇2G^=−k2G^\nabla^2 \hat G = -k^2 \hat G — the homogeneous Helmholtz equation holds everywhere except the origin.

The strength of the singularity at the origin fixes AA. Integrate ∇2G^=−δ3\nabla^2 \hat G = -\delta^3 over a small ball of radius ε\varepsilon around the source. The right side gives −1-1. The left side, by the divergence theorem, is the flux of ∇G^\nabla \hat G through the enclosing sphere. As ε→0\varepsilon \to 0 the near field behaves like the static A/rA/r, whose gradient is −A/r2 r^-A/r^2 \,\hat{\mathbf r}; the flux through the sphere of area 4πε24\pi\varepsilon^2 is −A/ε2⋅4πε2=−4πA-A/\varepsilon^2 \cdot 4\pi\varepsilon^2 = -4\pi A. Setting −4πA=−1-4\pi A = -1 gives A=1/4πA = 1/4\pi. The 4π4\pi is the area of the unit sphere — the same geometric factor behind the inverse-square law.

Retarded time: the field as a sum of delayed echoes

Substituting the free-space GG into the superposition integral and doing the t′t' integral against the delta collapses the time history to its retarded value:

p′(r,t)  =  14π∫s ⁣(r′, t−∣r−r′∣/c)∣r−r′∣  d3r′.p'(\mathbf{r}, t) \;=\; \frac{1}{4\pi}\int \frac{s\!\left(\mathbf{r}',\, t - |\mathbf{r} - \mathbf{r}'|/c\right)}{|\mathbf{r} - \mathbf{r}'|}\; d^3r'.

Every patch of the source contributes what it was doing earlier, by exactly the travel time ∣r−r′∣/c|\mathbf{r} - \mathbf{r}'|/c, attenuated by 1/∣r−r′∣1/|\mathbf{r} - \mathbf{r}'|. The field at the listener is the sum of these delayed, geometrically-weakened echoes. The convolution structure — output equals input slid against an impulse response — is the same one that governs every linear time-invariant system; the interactive below shows it in the cleanest one-dimensional setting.

f(τ) (blue) and g(t − τ) (orange, shifted by t = 0.00)(f ∗ g)(t) — convolution
f(t):
g(t):

Everything in this chapter is a special case

The Green’s function is not a new kind of source; it is the atom of which the others are built:

Stated as one of the load-bearing facts of the book: Green’s functions are the elementary answers; every other solution is a weighted sum of them.

What it buys

Any radiation or scattering problem now has a recipe: find the Green’s function for the geometry, then integrate it against the source. Boundaries do not change the principle — they change GG. A rigid wall, for instance, is handled by the method of images: add a mirror-image source so that GG automatically satisfies the boundary condition, exactly the construction that turns a room into a sum of image-source impulse responses (7.9). The hard part of acoustics is rarely the superposition; it is finding GG for a complicated boundary — after which the field of any source is one integral away.

⏳ The history — A miller's essay, and the function named for it

The Green’s function is named for George Green (1793–1841), a self-taught miller from Nottingham who published, at his own expense in 1828, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (Green 1828). It introduced both the idea of a potential’s response to a point source and the integral identities — now “Green’s theorem” and “Green’s identities” — that turn a volume problem into a surface one. The essay was almost unknown until William Thomson (Lord Kelvin) rediscovered and republicised it in 1846. Green himself had by then been dead five years, having finally entered Cambridge as an undergraduate at the age of forty.

The acoustic application matured later: Gustav Kirchhoff in 1882 wrote the diffraction field as a surface integral of the free-space Green’s function — the Helmholtz–Kirchhoff integral of 7.4 — and Rayleigh systematised the method for radiation and scattering in The Theory of Sound (Rayleigh 1894).