On three-dimensional packets of surface waves

Abstract In this note we use the method of multiple scales to derive the two coupled nonlinear partial differential equations which describe the evolution of a three-dimensional wave-packet of wavenumber k on water of finite depth. The equations are used to study the stability of the uniform Stokes wavetrain to small disturbances whose length scale is large compared with 2π/k. The stability criterion obtained is identical with that derived by Hayes under the more restrictive requirement that the disturbances are oblique plane waves in which the amplitude variation is much smaller than the phase variation.