Lord Rayleigh

How to use this course

Rayleigh’s name is attached to more physics than almost anyone else’s: Rayleigh scattering, the Rayleigh criterion, Rayleigh waves, the Rayleigh number, the Rayleigh–Taylor instability, the Rayleigh–Jeans law, the Rayleigh quotient, the Rayleigh distribution. Behind the list sits a single method. He took a physical problem, reduced it to small oscillations about an equilibrium, and asked which modes grow, which decay and how fast. This course follows that method through acoustics, optics, fluids, elasticity and radiation, and ends with the experimental precision that won him the Nobel Prize.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The order is roughly chronological: vibration theory (1870s), scattering (1871), optics (1879), fluid stability (1878–1916), elastic surface waves (1885), radiation (1900–1905) and gas densities (1892–1895).

Timeline

A fragile heir, 1842–1861

John William Strutt was born on 12 November 1842 at Langford Grove, near Maldon in Essex, the eldest son of John James Strutt, second Baron Rayleigh, and Clara Elizabeth La Touche. The family owned the Terling estate, several thousand acres of Essex farmland. As heir to a barony, he was expected to run the land, sit in the House of Lords and perhaps serve in the army. No one expected a scientist.

He was a frail child, and illness broke up every stage of his schooling. He went to Eton at about ten and spent much of his time there in the sanatorium. He then spent three years at a private school in Wimbledon and a short spell at Harrow, which also ended in ill health. In 1857 he was sent to the Rev. George Townsend Warner, who took a few pupils at Torquay. He stayed four years. The small, quiet setting suited him, and he discovered there that mathematics came easily.

The pattern of these years recurs through his life. He worked best alone or with a few people, at his own pace, in a place he controlled. His greatest asset as a physicist would be a laboratory in his own house.

Trinity and the Tripos, 1861–1871

In October 1861 Strutt entered Trinity College, Cambridge, to read mathematics. The Cambridge Mathematical Tripos was then a ferocious competitive examination, and success depended on private coaches. His coach was Edward Routh, the most successful of them all, who drilled pupils in rigid-body dynamics, oscillations and the problems that would fill Rayleigh’s later work. Strutt also attended the lectures of George Gabriel Stokes, the Lucasian Professor, whose blend of mathematical analysis and careful experiment on light and fluids became his model.

In 1865 he was Senior Wrangler, first in the Tripos, and also first Smith’s Prizeman. The economist Alfred Marshall was second wrangler that year. In 1866 Trinity elected him a Fellow.

Choosing science

Experimental physics hardly existed as a subject at Cambridge in the 1860s, and there was no teaching laboratory. Strutt bought his own apparatus and began experimenting. His first papers, from the late 1860s, dealt with electromagnetism and with colour, and he read Maxwell’s new electromagnetic theory closely. Friends of his family thought a peer’s heir had no business doing science for a living. He did it anyway.

In 1871 he married Evelyn Balfour, sister of Arthur Balfour, the future prime minister, and of the Cambridge embryologist Francis Maitland Balfour. Marriage meant giving up his Trinity fellowship. The same year he published the paper on the colour of the sky that is the subject of Lesson 05.

The Nile, Terling and the book, 1872–1879

Soon after his marriage Strutt fell seriously ill with rheumatic fever. To recover, he and Evelyn spent the winter of 1872–73 in Egypt and Greece, sailing up the Nile on a houseboat. He took books and paper, and on the river he began writing The Theory of Sound.

His father died in 1873, shortly after the couple returned, and he became the third Baron Rayleigh. He moved to Terling Place and turned part of the house into a laboratory. In the same year he was elected a Fellow of the Royal Society. Running a large estate took time he wanted for physics, and in 1876 he handed its management to his younger brother Edward. Terling remained his scientific home for the rest of his life; much of the work in Part II was done there, often with improvised apparatus.

The Theory of Sound

The two volumes appeared in 1877 and 1878. They are much more than a book on acoustics. The first volume develops a general theory of vibrating systems with finitely many degrees of freedom, then strings, bars, membranes and plates. The second treats sound in air and other fluids: pipes, resonators, reflection, diffraction and the stability of jets and flames. Hermann von Helmholtz had written on the physiology of tone; Rayleigh gave the physics its mathematical framework. The book is still in print, and many results in it remain the standard ones.

From 1876 to 1878 he was president of the London Mathematical Society, where he published many of his stability papers. In 1879 Maxwell died, and Cambridge needed a new Cavendish Professor. That story continues in Lesson 11.

The Theory of Sound and Rayleigh’s principle

For the general reader. A guitar string, a drum or a bridge vibrates at certain natural frequencies. Computing them exactly is often impossible. Rayleigh found that a rough guess of the shape of the vibration gives a surprisingly accurate frequency, and that the guess always errs on the high side. Engineers still use his idea every day, in the finite element method.

Small oscillations

Let a conservative system have generalized coordinates \(q\in\mathbb R^n\), measured from a stable equilibrium. To second order the kinetic and potential energies are

with \(M\) symmetric positive definite and \(K\) symmetric positive semidefinite. Lagrange’s equations give \(M\ddot q+Kq=0\). Solutions \(q=u\,e^{i\omega t}\) require \(Ku=\omega^2Mu\). The eigenvalues \(\omega_1^2\le\omega_2^2\le\cdots\le\omega_n^2\) are the squared normal frequencies, and the eigenvectors can be chosen \(M\)-orthonormal. In normal coordinates the system splits into independent oscillators. Rayleigh called these principal modes and used them throughout The Theory of Sound.

Proof

Expand \(u=\sum_j c_ju_j\) in the \(M\)-orthonormal eigenvectors. Then \(u^{\mathsf T}Mu=\sum c_j^2\) and \(u^{\mathsf T}Ku=\sum\omega_j^2c_j^2\), so \(R(u)\) is a weighted average of the \(\omega_j^2\) with weights \(c_j^2/\sum c_k^2\). It is at least \(\omega_1^2\), with equality iff only modes with \(\omega_j=\omega_1\) appear. With \(c_1=1\) and the other \(c_j=O(\varepsilon)\), the average differs from \(\omega_1^2\) by \(\sum_{j\ge2}(\omega_j^2-\omega_1^2)c_j^2/\sum c_k^2=O(\varepsilon^2)\).

For continuous systems the sums become integrals. For a string of tension \(\tau\), linear density \(\rho\) and length \(L\), fixed at both ends, with transverse displacement \(y(x)\,e^{i\omega t}\),

Rayleigh used this to estimate the pitch of strings and bars of variable density, and the correction for the open end of an organ pipe. In 1908–1909 Walther Ritz turned it into a systematic method: minimize \(R\) over a finite-dimensional space of trial functions. The resulting Rayleigh–Ritz method is the variational core of the finite element method.

Dissipation and reciprocity

In an 1873 paper, Some general theorems relating to vibrations, Rayleigh added friction proportional to velocity through a quadratic dissipation function \(F=\tfrac12\dot q^{\mathsf T}C\dot q\), so that the equations of motion become

where \(Q_i\) are external forces; \(2F\) is the rate at which energy is dissipated. The same paper proves a reciprocity theorem: in a linear system with symmetric \(M,C,K\), the response at point B to a harmonic force at A equals the response at A to the same force at B. Acousticians and antenna engineers use it constantly.

Why the sky is blue

For the general reader. Sunlight contains all colours. When it passes through air, tiny particles and the air molecules themselves scatter a little of it sideways. Rayleigh showed in 1871 that very small scatterers deflect short wavelengths much more strongly than long ones, in proportion to the inverse fourth power of the wavelength. Blue light is scattered across the sky; the sun at sunset, seen through a long path of air, looks red.

The dimensional argument (1871)

In On the light from the sky, its polarization and colour (Philosophical Magazine, 1871), Rayleigh considered a particle of volume \(V\), much smaller than the wavelength \(\lambda\), with optical properties different from its surroundings. The ratio of scattered to incident amplitude at distance \(r\) is dimensionless, proportional to \(V\) (each part of the particle scatters in phase) and inversely proportional to \(r\) (spherical spreading). The only other length available is \(\lambda\). So the amplitude ratio must be proportional to \(V/(r\lambda^2)\), and the intensity ratio to \(V^2/(r^2\lambda^4)\). John Tyndall had shown experimentally that fine suspensions scatter blue light and polarize it; Rayleigh supplied the law.

The electromagnetic derivation

A small sphere of radius \(a\ll\lambda\) and refractive index \(n\), in a field \(E_0\cos\omega t\), acquires a dipole moment \(p=\alpha E_0\cos\omega t\) with polarizability \(\alpha=4\pi\varepsilon_0a^3\,\frac{n^2-1}{n^2+2}\). An oscillating dipole radiates the average power \(P=\frac{\alpha^2E_0^2\omega^4}{12\pi\varepsilon_0c^3}\). Dividing by the incident intensity \(I=\tfrac12\varepsilon_0cE_0^2\) gives the cross-section.

In later papers Rayleigh worked out the electromagnetic version himself, and in 1899 he argued that the air molecules alone, without dust, scatter enough light to make the sky blue. Since the scattering depends on the number of molecules per unit volume, measurements of sky brightness give an estimate of Avogadro’s number. The violet end of the spectrum is scattered even more than blue, but sunlight contains less violet and the eye is less sensitive to it, so the sky looks blue.

Resolving power

For the general reader. No lens or mirror can focus a star to a perfect point. Diffraction spreads the light into a small disk with faint rings. When two stars are very close, their disks overlap and blur into one. In 1879 Rayleigh proposed a simple rule for when two images can just be told apart, and the same reasoning gave the resolving power of prisms and gratings.

The criterion

A circular aperture of diameter \(D\) illuminated by a distant point source of wavelength \(\lambda\) produces the Airy pattern found by George Biddell Airy in 1835:

where \(J_1\) is the Bessel function of order one. The first zero of \(J_1\) is at \(x\approx3.8317\).

Rayleigh chose the criterion as a convention, and he said so. Its value lies in a fixed, reproducible standard. Other definitions, such as Sparrow’s (the separation at which the central dip just disappears), give slightly smaller numbers.

Gratings and prisms

A grating of \(N\) slits with spacing \(d\) has principal maxima at \(d\sin\theta=m\lambda\). The angular half-width of each maximum, to its first zero, is \(\Delta\theta=\lambda/(Nd\cos\theta)\). Applying the same criterion to two wavelengths \(\lambda\) and \(\lambda+\Delta\lambda\), with dispersion \(d\theta/d\lambda=m/(d\cos\theta)\), gives

For a prism with base length \(t\) filled by the beam, Rayleigh found \(\mathcal R=t\,|dn/d\lambda|\). Both formulas say that resolving power depends on the largest optical path difference across the beam, not on the shape of the instrument.

The quarter-wave rule

In the same 1879 investigations Rayleigh asked how perfect an optical surface must be. He showed that if the wavefront deviates from a perfect sphere by less than a quarter of a wavelength, the image is barely degraded. This tolerance is still the starting point for specifying telescope mirrors.

Instabilities of fluids

For the general reader. A thin stream from a tap breaks into drops. Water held above air in an upturned glass falls out, even though air pressure could support it. A pan of water heated from below begins to churn only when the heating is strong enough. Rayleigh explained each case by asking whether a small ripple grows or dies, and which ripple grows fastest.

The breakup of jets (1878)

Joseph Plateau had shown from surface energy that a cylinder of liquid of radius \(a\) is unstable to varicose disturbances of wavelength longer than its circumference, \(\lambda>2\pi a\). In On the instability of jets (1878) Rayleigh added the dynamics. For an inviscid jet of density \(\rho\) and surface tension \(\gamma\), a disturbance \(\propto e^{\sigma t+ikz}\) grows at the rate

where \(I_0,I_1\) are modified Bessel functions. Growth requires \(x<1\), Plateau’s condition, and is fastest at \(x\approx0.697\), that is, at wavelength \(\lambda\approx9.01\,a\). The jet breaks into drops spaced at about this wavelength.

Heavy fluid over light (1883)

Derivation. In each layer the flow is irrotational with potential \(\phi_j\) satisfying Laplace’s equation; decaying solutions are \(\phi_1=B_1e^{kz}e^{ikx+\sigma t}\) below and \(\phi_2=B_2e^{-kz}e^{ikx+\sigma t}\) above. The interface \(z=\eta=\eta_0e^{ikx+\sigma t}\) moves with the fluid: \(\sigma\eta_0=kB_1=-kB_2\). The linearized Bernoulli equation \(p_j=-\rho_j(\partial_t\phi_j+gz)\) and continuity of pressure at \(z=\eta\) give \(\rho_1(\sigma B_1+g\eta_0)=\rho_2(\sigma B_2+g\eta_0)\). Substituting \(B_1=\sigma\eta_0/k\) and \(B_2=-\sigma\eta_0/k\) gives \(\sigma^2(\rho_1+\rho_2)=gk(\rho_2-\rho_1)\). Rayleigh found this in 1883 for continuously stratified fluids; Geoffrey Ingram Taylor showed in 1950 that acceleration plays the role of gravity, which is why the instability matters in supernovae and inertial-confinement fusion.

Convection (1916)

Henri Bénard had photographed hexagonal convection cells in thin heated layers in 1900. In his 1916 paper on convection in a layer heated from below, Rayleigh analysed a layer of depth \(d\) with temperature difference \(\Delta T\), kinematic viscosity \(\nu\), thermal diffusivity \(\kappa\) and expansion coefficient \(\beta\). Convection begins when the dimensionless number

exceeds a critical value. For free-slip boundaries, the marginal disturbances with horizontal wavenumber \(k\) satisfy \(\mathrm{Ra}=(\pi^2+a^2)^3/a^2\) with \(a=kd\). The number is now called the Rayleigh number. Bénard’s own cells, it turned out much later, were driven mainly by surface tension rather than buoyancy.

Shear flows (1880)

Proof sketch. Divide by \(U-c\), multiply by \(\bar\phi\) and integrate by parts: \(\int(|\phi'|^2+k^2|\phi|^2)\,dy+\int\frac{U''}{U-c}|\phi|^2dy=0\). The imaginary part is \(c_i\int\frac{U''|\phi|^2}{|U-c|^2}dy=0\). If \(c_i\ne0\), the integral must vanish, so \(U''\) takes both signs.

Waves on surfaces

For the general reader. An earthquake sends several kinds of wave through the Earth. The ones that arrive last at a distant station, and shake the ground hardest, travel along the surface. Rayleigh predicted them in 1885 from the theory of elasticity, years before seismographs could record them. He also worked out the shape of the solitary water wave that had puzzled engineers since the 1830s.

Rayleigh waves (1885)

In an isotropic elastic solid with Lamé constants \(\lambda,\mu\) and density \(\rho\), body waves travel at \(c_p=\sqrt{(\lambda+2\mu)/\rho}\) (compressional) and \(c_s=\sqrt{\mu/\rho}\) (shear). In On waves propagated along the plane surface of an elastic solid (1885), Rayleigh looked for waves travelling along the free surface \(z=0\) of the half-space \(z<0\) and decaying with depth. Write the displacement through potentials, \(\mathbf u=\nabla\varphi+\nabla\times(\psi\,\hat{\mathbf y})\), with

where \(c=\omega/k\). Requiring both tractions \(\sigma_{zz}\) and \(\sigma_{xz}\) to vanish at \(z=0\) gives two homogeneous equations for \(A,B\). A non-trivial solution exists only if \(c\) satisfies the following.

For typical rocks \(c_R\approx0.92\,c_s\). The same waves, at megahertz frequencies on piezoelectric crystals, are the surface acoustic wave filters inside mobile phones.

The solitary wave

John Scott Russell had observed a single heap of water travelling unchanged along a canal in 1834. George Airy and Stokes doubted that such a wave could exist. In On waves (1876) Rayleigh, working independently of Joseph Boussinesq, showed that a long wave of height \(a\) on water of depth \(h\) can keep a permanent \(\operatorname{sech}^2\) profile and travel at speed \(c=\sqrt{g(h+a)}\). Korteweg and de Vries wrote the governing equation in 1895; the soliton theory of the 1960s grew from it.

The Rayleigh–Jeans law

For the general reader. A hot oven glows with a colour that depends only on its temperature. Around 1900 physicists tried to explain the spectrum of that glow. Rayleigh applied the most trusted rule of classical physics, the equal sharing of energy among all modes of motion. His formula matched experiments at long wavelengths and predicted absurdly infinite energy at short ones. The failure helped force the quantum theory.

Counting modes

In June 1900, in Remarks upon the law of complete radiation (Philosophical Magazine), Rayleigh treated radiation in a cavity as a system of standing waves, exactly as in The Theory of Sound. In a cube of side \(\ell\) with conducting walls, the allowed wavevectors are \(\mathbf k=\frac{\pi}{\ell}(n_1,n_2,n_3)\) with positive integers \(n_i\). The number of modes with \(|\mathbf k|<K\) is one octant of a sphere in \(n\)-space, times two polarizations:

In 1900 Rayleigh gave only the frequency dependence. When he supplied the constant in Nature in 1905, he counted all sign combinations of the \(n_i\) and was too large by a factor of 8; James Jeans pointed out the error shortly afterwards. Albert Einstein reached the same law independently in 1905.

The law agreed with the measurements of Heinrich Rubens and Ferdinand Kurlbaum at long wavelengths in 1900. But \(\int_0^\infty u\,d\nu\) diverges: classical physics gives a cavity infinite energy. Paul Ehrenfest named this the ultraviolet catastrophe in 1911. Max Planck had already found, in late 1900, the formula

which replaces \(k_BT\) by the mean energy of a quantized oscillator. Rayleigh, who disliked the quantum hypothesis, had nonetheless given the classical baseline against which it had to be judged.

Precision: the ohm and argon

For the general reader. Rayleigh believed that the next great discovery often hides in the last decimal place. He spent years measuring electrical units and the densities of gases more accurately than anyone before. A tiny, stubborn disagreement between two samples of nitrogen led him and William Ramsay to an unknown gas that makes up nearly one percent of the air.

Electrical standards

At the Cavendish Laboratory in the early 1880s, Rayleigh redetermined the ohm in absolute units, working with Arthur Schuster and with his sister-in-law Eleanor Sidgwick. They used rotating-coil methods and checked every source of error. With Sidgwick he also measured the electrochemical equivalent of silver, which later defined the international ampere. These measurements helped fix the practical electrical units adopted internationally in the 1880s and 1890s.

The nitrogen anomaly

In the early 1890s Rayleigh set out to test Prout’s hypothesis by measuring gas densities precisely. Nitrogen gave trouble. Nitrogen prepared from air, by removing oxygen, water and carbon dioxide, was consistently about half a percent denser than nitrogen made from chemical compounds such as ammonia. In September 1892 he wrote to Nature asking chemists for an explanation.

William Ramsay, at University College London, took up the problem. Rayleigh repeated Henry Cavendish’s experiment of 1785, sparking air with oxygen over alkali until no more nitrogen was absorbed; Cavendish had noticed a small bubble that refused to react. Ramsay absorbed the nitrogen with hot magnesium. Both were left with a residue whose spectrum showed lines belonging to no known element. They announced the new gas at the British Association meeting in Oxford in August 1894 and published a full account in 1895. They named it argon, from the Greek for idle, because it would not combine with anything.

Argon did not fit in the periodic table as then known. Ramsay went on to find helium, neon, krypton and xenon, a whole new column. In 1904 Rayleigh received the Nobel Prize in Physics for his investigations of the densities of the most important gases and his discovery of argon, and Ramsay the Nobel Prize in Chemistry.

Cavendish, Royal Institution, public science, 1879–1908

Maxwell died in November 1879, and Rayleigh was elected to succeed him as Cavendish Professor of Experimental Physics at Cambridge. He accepted partly because the agricultural depression had cut the income from his estates. In five years he turned Maxwell’s new laboratory into a working teaching institution. With Richard Glazebrook and Napier Shaw as demonstrators, he organized systematic practical classes in physics, open to women students as well as men. The research programme centred on the absolute electrical units of Lesson 10.

He resigned in 1884, as he had always intended, and returned to Terling. The chair passed to the young J. J. Thomson, who had worked in the laboratory under him. Among the students who attended Rayleigh’s Cambridge teaching was the Indian physicist Jagadish Chandra Bose.

From 1887 to 1905 he was Professor of Natural Philosophy at the Royal Institution in London, succeeding John Tyndall. He gave public lectures there and had the use of its laboratory. He was Secretary of the Royal Society from 1885 to 1896 and its President from 1905 to 1908. He served as Lord Lieutenant of Essex (1892–1901), as scientific adviser to Trinity House on lighthouses and fog signals (1896–1919), and as President of the government’s Advisory Committee for Aeronautics from its creation in 1909. He took a leading part in establishing the National Physical Laboratory in 1900.

His curiosity was wide. In 1883 he explained in Nature how albatrosses can soar without flapping by exploiting the change of wind speed with height, now called dynamic soaring. In 1907 he gave the duplex theory of sound localization: we locate low tones by the time difference between the two ears, high tones by the intensity difference caused by the shadow of the head. In 1887 he analysed waves in media with periodic structure and found the bands of strong reflection that underlie today’s photonic crystals.

Last years and legacy, 1908–1919

In 1908 Rayleigh became Chancellor of the University of Cambridge, a post he held until his death. He kept publishing to the end. His collected Scientific Papers fill six volumes and reprint 446 papers; the convection paper of 1916 came when he was 73. In 1919 he served as President of the Society for Psychical Research, a field that had interested him, as it did his Balfour and Sidgwick in-laws, for decades. He died at Terling Place on 30 June 1919 and was buried in the churchyard of All Saints’ Church, Terling.

His eldest son, Robert John Strutt, fourth Baron Rayleigh, was also a physicist, professor at Imperial College London, known for work on radioactivity and on the glow of the night sky. He wrote his father’s biography.

Honours

Royal Medal (1882), De Morgan Medal (1890), Copley Medal (1899), original member of the Order of Merit (1902), Nobel Prize in Physics (1904), Privy Councillor (1905), Albert Medal (1905), Rumford Medal (1914), among many others. Craters on the Moon and on Mars bear his name.

Legacy

• Method: linear stability analysis, normal modes and variational estimates of frequencies are the working tools of every branch of continuum physics. Rayleigh used them everywhere first.
• Acoustics: The Theory of Sound founded the modern subject.
• Optics and atmospheres: Rayleigh scattering underlies remote sensing, lidar and the study of planetary atmospheres; his criterion is the standard measure of resolution.
• Fluids and Earth science: the Rayleigh number governs convection in pans, oceans, the Earth’s mantle and stars; Rayleigh waves are basic seismology.
• Quantum theory: the Rayleigh–Jeans law is the classical limit that every account of Planck’s discovery starts from.
• Chemistry: argon opened the family of noble gases.

Connections

People linked to Lord Rayleigh across this collection. A name without a link has no page here yet. See Lord Rayleigh in the network of all 337 lives →

Teachers and advisors

• Edward Routh 1861-1865

Students

• J. J. Thomson
• Jagadish Chandra Bose

Influenced by

• George Biddell Airy
• George Gabriel Stokes
• Henry Cavendish
• Hermann von Helmholtz
• James Clerk Maxwell
• John Tyndall

Influenced

• Albert Einstein
• G. I. Taylor
• Max Planck

Colleagues and collaborators

• Albert A. Michelson
• Dmitri Mendeleev
• James Clerk Maxwell
• James Jeans 1905
• Osborne Reynolds 1863-1867
• William Ramsay 1894-1895
• William Thomson, Lord Kelvin
• Arthur Schuster 1881
• Richard Glazebrook 1880-1884

Family

• Eleanor Sidgwick
• Robert John Strutt, 4th Baron Rayleigh

Also mentioned in this life

• Amedeo Avogadro
• Friedrich Wilhelm Bessel
• Subrahmanyan Chandrasekhar
• Augustus De Morgan
• Paul Ehrenfest
• Leonhard Euler
• Joseph Fourier
• Joseph-Louis Lagrange
• Pierre-Simon Laplace
• John Napier
• Siméon Denis Poisson
• Count Rumford
• Bertrand Russell

Fields

• acoustics 9 lives
• optics 38 lives
• fluid dynamics 6 lives
• elasticity 11 lives
• experimental physics 16 lives

On CoursesHub

• Optics & Photonics
• Fluid Mechanics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, John William Strutt, Lord Rayleigh, MacTutor History of Mathematics.
2. Lord Rayleigh: biographical, NobelPrize.org.
3. The Nobel Prize in Physics 1904, NobelPrize.org.
4. John William Strutt, 3rd Baron Rayleigh, Wikipedia.
5. R. J. Strutt (fourth Baron Rayleigh), John William Strutt, Third Baron Rayleigh, Edward Arnold, 1924.

Physics

1. Lord Rayleigh, The Theory of Sound, 2 vols., Macmillan, 1877–78; 2nd ed. 1894–96, reprinted by Dover.
2. Lord Rayleigh, Scientific Papers, 6 vols., Cambridge University Press, 1899–1920.
3. J. W. Strutt, On the light from the sky, its polarization and colour, Phil. Mag. 41 (1871).
4. Lord Rayleigh, Some general theorems relating to vibrations, Proc. London Math. Soc. 4 (1873).
5. Lord Rayleigh, On the instability of jets, Proc. London Math. Soc. 10 (1878).
6. Lord Rayleigh, Investigations in optics, with special reference to the spectroscope, Phil. Mag. 8 (1879).
7. Lord Rayleigh, On waves propagated along the plane surface of an elastic solid, Proc. London Math. Soc. 17 (1885).
8. Lord Rayleigh, Remarks upon the law of complete radiation, Phil. Mag. 49 (1900).
9. Lord Rayleigh and W. Ramsay, Argon, a new constituent of the atmosphere, Phil. Trans. R. Soc. A 186 (1895).
10. Lord Rayleigh, On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side, Phil. Mag. 32 (1916).
11. L. Rayleigh, The density of gases in the air and the discovery of argon, Nobel Lecture, 1904.
12. A concise history of the black-body radiation problem, arXiv:2208.06470.
13. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford, 1961 (Rayleigh–Bénard and Rayleigh–Taylor problems).
14. P. G. Drazin and W. H. Reid, Hydrodynamic Stability, Cambridge, 2nd ed. 2004.