The Spirited Horse, the Engineer, and the Mathematician: Water Waves in Nineteenth-Century Hydrodynamics

Once upon a time, a spirited horse was dragging a boat along a peaceful canal of Northern Scotland. Suddenly, a ferocious dog charged from the bushes. The horse took fright and ran off, drawing the boat after him. To the horse's surprise, the boat offered almost no resistance at this high speed. A clever naval engineer, who chanced to witness the scene, pursued the matter through numerous experiments and confirmed the horse's discovery. Not knowing his limits, the engineer ventured to propose a mechanical explanation of this paradox based on the existence of a wonderful, solitary wave that carried the boat with it. A renowned mathematician, who excelled in the learned calculus of water motion, mocked this amateurish attempt. There was nothing in the engineer's equations to justify his reasoning, and much to condemn it. Yet for a few years Scottish canal travelers enjoyed the commercial exploitation of the paradox. Half a century later, the aged mathematician resumed his calculations. Thanks to his long experience, he now saw new meanings in his old symbols. From this enriched analysis the solitary wave and the possibility of vanishing ship resistance emerged as if by magic. At last, the wise man rejoiced, mathematics could do as well as a galloping horse. This fable is an imaginary simplification of a real story of which the engineer John Scott Russell, and the mathematicians (in a broad sense) George Biddell Airy, George Gabriel Stokes, Joseph Boussinesq, and Lord Kelvin were the main actors (besides the spirited horse). It is intended to indicate a major nineteenth-century transformation of the mathematical physicist's tool kit through which practically important solutions of long-known equations became much more easily accessible. The main purpose of the present article is to analyze the nature and the water-wave circumstances of this transformation. Waves on the surface of water were an obvious field of application of the new hy-drodynamics of Jean le Rond d'Alembert, Leonhard Euler, and Joseph Louis Lagrange. The latter mathematician himself wrote the basic equations of water waves, and solved them in the simplest case of small waves on shallow water. Most of what is today known on water waves was found in the nineteenth century: the celerity of small, plane, mono-chromatic waves on water of constant depth, the pattern of waves created by a local action on the water surface, the shape of oscillatory or solitary waves of finite size, the …