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 sea. He and A. Oberbeck showed that when the waves on the sea attain lengths of from 16 to 33 feet, the air waves must attain lengths of from 10 to 20 miles, and proportional depths. Superposed strata would thus mix more thoroughly, and their energy would be partly dissipated. From hydrodynamical equations of rotation Helmholtz established the reason why the observed velocity from equatorial regions is much less in a latitude of, say, 20° or 30°, than it would be were the movements unchecked.

About 1860 acoustics began to be studied with renewed zeal. The mathematical theory of pipes and vibrating strings had been elaborated in the eighteenth century by Daniel Bernoulli, D'Alembert, Euler, and Lagrange. In the first part of the present century Laplace corrected Newton's theory on the velocity of sound in gases, Poisson gave a mathematical discussion of torsional vibrations; Poisson, Sophie Germain, and Wheatstone studied Chladni's figures; Thomas Young and the brothers Weber developed the wave-theory of sound. Sir J. F. W. Herschel wrote on the mathematical theory of sound for the Encyclopædia Metropolitana, 1845. Epoch-making were Helmholtz's experimental and mathematical researches. In his hands and Rayleigh's, Fourier's series received due attention. Helmholtz gave the mathematical theory of beats, difference tones, and summation tones. Lord Rayleigh (John William Strutt) of Cambridge (born 1842) made extensive mathematical researches in acoustics as a part of the theory of vibration in general. Particular mention may be made of his discussion of the disturbance produced by a spherical obstacle on the waves of sound, and of phenomena, such as sensitive flames, connected with the instability of jets of fluid. In 1877 and 1878 he published in two volumes a treatise on The Theory of Sound. Other mathematical researches on this subject have been made in England by Donkin and Stokes.