Introduction

Waves occur throughout Nature in an astonishing diversity of physical, chemical and biological systems. During the late nineteenth and the early twentieth century, the linear theory of wave motion was developed to a high degree of sophistication, particularly in acoustics, elasticity and hydrodynamics. Much of this ‘classical’ theory is expounded in the famous treatises of Rayleigh (1896), Love (1927) and Lamb (1932).

The classical theory concerns situations which, under suitable simplifying assumptions, reduce to linear partial differential equations, usually the wave equation or Laplace's equation, together with linear boundary conditions. Then, the principle of superposition of solutions permits fruitful employment of Fourier-series and integral-transform techniques; also, for Laplace's equation, the added power of complex-variable methods is available.

Since the governing equations and boundary conditions of mechanical systems are rarely strictly linear and those of fluid mechanics and elasticity almost never so, the linearized approximation restricts attention to sufficiently small displacements from some known state of equilibrium or steady motion. Precisely how small these displacements must be depends on circumstances. Gravity waves in deep water need only have wave-slopes small compared with unity; but shallow-water waves and waves in shear flows must meet other, more stringent, requirements. Violation of these requirements forces abandonment of the powerful and attractive mathematical machinery of linear analysis, which has reaped such rich harvests. Yet, even during the nineteenth century, considerable progress was made in understanding aspects of weakly-nonlinear wave propagation, the most notable theoretical accomplishments being those of Rayleigh in acoustics and Stokes for water waves.