History

In Principia (1687), Newton computed the speed of sound assuming isothermal compression — i.e., that the temperature of the gas stays fixed during a sound wave's compressions and rarefactions (Newton 1687). His formula $c = \sqrt{p_0 / \rho_0}$ gives about 280 m/s for air, which was already known by then to be about 15% low (Mersenne and others had timed the round-trip of cannon-fire echoes).

The discrepancy stood for 130 years. In 1816 Laplace pointed out that the compressions in a sound wave are too fast for heat to flow between adjacent regions — they are essentially adiabatic. The right formula is then $c = \sqrt{\gamma p_0 / \rho_0}$, and for diatomic air $\gamma = 7/5$, recovering $c \approx 343$ m/s (Laplace 1816).

The factor $\gamma$ — the ratio of specific heats $c_p / c_v$ — is the same $\gamma$ that distinguishes adiabatic from isothermal in thermodynamics, and it counts the active molecular degrees of freedom. For a monatomic gas (helium) $\gamma = 5/3$; for a diatomic gas like air at room temperature $\gamma = 7/5$ (translation + rotation); for a polyatomic gas with active vibrational modes, $\gamma$ approaches 1 from above. Laplace's correction connects acoustics to thermodynamics to kinetic theory in a single step.

In 1747 Jean le Rond d'Alembert, then 29 years old, published a paper Recherches sur la courbe que forme une corde tendue mise en vibration in the Berlin Academy's proceedings (d'Alembert 1747). It contained the general solution $y(x, t) = F(x - ct) + G(x + ct)$ to the 1-D wave equation he had derived for a vibrating string. The result is the same formula we use today.

What followed was one of the great mathematical controversies of the 18th century. Euler argued that $F$ and $G$ could be any functions — including those with corners (e.g., the initial shape of a plucked string, which has a sharp peak). D'Alembert insisted they had to be analytic, drawn from the class of well-behaved functions Newton and Leibniz had developed calculus for. Daniel Bernoulli proposed yet a third view: any vibration is a sum of sinusoidal modes — what we now call a Fourier series.

The dispute lasted decades. It was only resolved in 1822 by Fourier (Fourier 1822), whose work on heat flow showed that arbitrary functions could be expanded as trigonometric series, vindicating Bernoulli and forcing a redefinition of what "function" even meant. The controversy is the origin of modern analysis.

Ernst Florens Friedrich Chladni demonstrated in 1787 that a metal plate, bowed at its edge and dusted with fine sand, reveals its mode shapes as the sand collects along the nodal lines where the plate does not vibrate. The resulting "Chladni figures" were the first visualisation of two-dimensional standing-wave patterns. Chladni toured Europe with the demonstration, including a performance for Napoleon in 1809 that led to a prize offered by the French Academy for a mathematical theory of plate vibration — eventually won by Sophie Germain in 1816.

Chladni's figures predate the Fourier methods and eigenvalue theory that would later explain them. The nodal patterns are the zero sets of the plate's eigenfunctions, and the frequencies at which each pattern appears are the eigenvalues of the biharmonic operator on the plate domain. The demonstration remains one of the most effective ways to make modal structure visible; modern versions use loudspeaker-driven plates and appear in physics classrooms worldwide.

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).

Christian Doppler presented his prediction of the frequency shift for moving sources in 1842, in a paper titled "On the coloured light of double stars and certain other stars of the heavens." His original application was to explain the colours of binary stars — an incorrect application, since the effect on light frequency from stellar velocities is far too small to account for observed star colours. The acoustic version of the prediction, however, was confirmed experimentally by Christoph Buys Ballot in 1845 using a locomotive and a group of trumpet players.

Doppler's paper was criticised on both physical and mathematical grounds during his lifetime. The correct relativistic treatment for light came only with Einstein's special relativity in 1905. For sound, the nonrelativistic formula Doppler derived is exact and remains the foundation of every application from radar speed guns to medical ultrasound.

Hermann von Helmholtz, in his 1863 Die Lehre von den Tonempfindungen, demonstrated that complex musical tones could be analysed into their Fourier components using a set of precisely tuned acoustic resonators — hollow brass spheres, each with a narrow opening, that amplified a single frequency from the ambient sound field. By holding different resonators to his ear, Helmholtz could identify the individual harmonics present in a sung vowel or a bowed string. The experiments provided the first empirical confirmation that Fourier's mathematics described the physical reality of sound.

The resonators also let Helmholtz demonstrate that timbre — the quality distinguishing a violin from a flute playing the same note — is determined by the relative amplitudes and phases of the harmonics, not by the fundamental frequency alone. This insight connects the physics of sound (this chapter) to the neuroscience of hearing: the cochlea performs the same Fourier-like decomposition that Helmholtz did with his brass spheres, but continuously and in real time.

Hermann von Helmholtz built sets of tuned brass spheres in the 1860s, each with a small neck to insert in the ear and a nipple opposite. Held to the ear, a sphere would ring loudly only when the surrounding sound contained its resonant frequency — a mechanical spectrum analyser. With them Helmholtz picked apart the overtones ("partials") of vowels and instruments one by one, establishing that timbre is the pattern of a tone's harmonics. The resonators, described in On the Sensations of Tone (1863), gave the frequency his name is now attached to, and the experiments seeded the whole idea — central to What is hearing? — that the ear performs a running Fourier analysis of sound. The cochlea, it turned out, is a bank of exactly such resonators, graded along its length.

John William Strutt, third Baron Rayleigh, published The Theory of Sound in two volumes (1877, 1894) — the first comprehensive mathematical treatment of acoustics. The work covered vibrating strings, membranes, and plates; the propagation of sound in tubes and in the open air; diffraction, scattering, and radiation from sources of various geometries. Rayleigh's formulation of the monopole, dipole, and quadrupole radiation patterns — the subject of this chapter — remains standard.

Rayleigh wrote the book while managing his family estate at Terling Place in Essex, before taking up the Cavendish Professorship at Cambridge. The Theory of Sound was unusual for its time in treating acoustics as a branch of mathematical physics rather than of music or physiology. It set the vocabulary and the methods for the field for the next century; nearly every derivation in the Sound book traces a lineage to Rayleigh's two volumes.

By the 1870s the pieces of this reading existed in scattered form — d'Alembert's travelling waves, Bernoulli's and Fourier's modal sums, George Green's impulse responses, Helmholtz's resonators. John William Strutt, third Baron Rayleigh, assembled them into a single coherent theory in The Theory of Sound (two volumes, 1877–78; Rayleigh 1894). His organising move was the one this lesson takes: treat the wave equation as a linear operator, expand every problem in its normal modes, and read energy, radiation, and resonance off the expansion. Much of the vocabulary of acoustics — "normal mode," the Rayleigh quotient for estimating frequencies, the Rayleigh integral for radiation — descends directly from it.

Modern architectural acoustics began in 1895 at Harvard. Wallace Clement Sabine — a 26-year-old assistant professor of physics — was asked to fix the Fogg Art Museum's new lecture hall, where speech was unintelligible because reverberation lasted nearly six seconds. Sabine had no acoustic training; he taught himself by experiment.

His protocol: at night, after the building had emptied, he carried seat cushions from a neighbouring lecture theatre into the Fogg's lecture room, played a tone on an organ pipe, and timed (with a stopwatch and a sensitive ear) how long the sound was audible after the pipe stopped. He repeated this with different numbers of cushions — that is, different amounts of absorbing surface area — and looked for a pattern. After thousands of measurements over five years, he saw the relation $T \cdot A =$ constant times $V$, and published the result in 1900 (Sabine 1900).

The constant 0.161 in $T_{60} = 0.161\,V/A$ (in SI units) traces back to Sabine's stopwatch measurements at Harvard. The Fogg lecture room, once fixed, became the prototype for acoustic design of every concert hall built since. Sabine went on to consult on Boston's Symphony Hall (opened 1900), which remains one of the finest-sounding concert halls in the world — a direct application of the formula he had derived with seat cushions and patience.

For two centuries, conservation of energy and momentum were empirical facts, patched into mechanics by hand. In 1918 Emmy Noether, working in Göttingen at Hilbert and Klein's invitation — and barred from a salaried professorship because she was a woman — proved that they are consequences of symmetry: every continuous symmetry of a system's action implies a conserved quantity, and vice versa. Time-translation symmetry $\Rightarrow$ energy; space-translation symmetry $\Rightarrow$ momentum; rotation symmetry $\Rightarrow$ angular momentum. Her theorem reframed conservation laws as statements about the sameness of the laws of physics from moment to moment and place to place, and it became the organising principle of all modern field theory, from electromagnetism to the Standard Model. The humble acoustic energy density above is one of its smallest and most concrete instances.

The electrical analogy grew out of the telephone industry's need to design earpieces, microphones, and horns quantitatively. Arthur Kennelly and Kurô Nukiyama measured the "motional impedance" of telephone receivers around 1919, treating the acoustic load as a circuit element seen through the coil. Through the 1920s and 30s, A. G. Webster, Warren Mason, and Harry Olson at Bell Labs and RCA turned the correspondence into a full discipline: Olson's Dynamical Analogies (1943) tabulated the mechanical, acoustical, and electrical duals and let engineers draw a loudspeaker or a resonator as a schematic and compute its response. The framework is why a modern hearing aid or cochlear-implant microphone can be designed on a circuit simulator before any air moves (Beranek 1954).

Claude Shannon's 1948 paper "A Mathematical Theory of Communication" established information theory and, along the way, provided the definitive statement of the sampling theorem: a band-limited signal with maximum frequency $B$ is completely determined by samples taken at rate $2B$ or higher. The result had precursors — Harry Nyquist stated a version in 1928, and the Soviet mathematician Vladimir Kotelnikov proved a similar theorem independently in 1933 — but Shannon's formulation embedded sampling in a complete theory of communication and gave it the form used today.

The sampling theorem is the bridge between continuous acoustics and digital signal processing. Every digital audio recording, every cochlear-implant stimulation strategy, every FFT computation on a finite data record rests on it. The theorem also establishes the fundamental tradeoff: higher sample rates preserve more bandwidth but require more data. CD audio's 44.1 kHz sample rate captures frequencies up to 22.05 kHz — just above the nominal upper limit of human hearing.

Michael James Lighthill published "On Sound Generated Aerodynamically" in 1952, reformulating the Navier-Stokes equations as a wave equation with a quadrupole source term — the Lighthill stress tensor $T_{ij}$. The move was conceptual rather than computational: by rearranging the exact equations into a form where the left side is the simple wave operator and the right side is a known source, Lighthill showed that turbulent flow generates sound as if it were an assembly of quadrupole sources, and that the radiated acoustic power scales as $U^8/c^5$ for flow speed $U$ — the famous eighth-power law.

The paper founded the field of aeroacoustics and earned Lighthill a knighthood and the Royal Medal. The eighth-power law explained why jet engines are so loud and why reducing jet velocity by even a modest factor produces dramatic noise reduction. Modern computational aeroacoustics still uses Lighthill's analogy (and its extensions by Curle and Ffowcs Williams-Hawkings) as its conceptual and sometimes computational framework.

By the early 1960s the electrical analogy had matured enough to model the human middle ear as a specific circuit. Jozef Zwislocki built lumped acoustic-analog networks of the ear canal, eardrum, ossicles, and cochlear load whose measured input impedance matched real ears, and used them to design the acoustic couplers and artificial ears still used to calibrate audiometers. The tympanometer on every audiology bench is a direct descendant: it assumes the ear is the circuit this chapter builds, and reports the value of one of its elements. The physics that began with a slab of air and a variational principle ends, four chapters of the sound book and one lumped-element chapter later, as a number on a clinical readout.