Finite amplitude surface waves in a liquid layer

An analysis is presented for the interaction of capillary and gravity waves in a liquid layer of finite depth. The method of multiple scales is used to obtain a third-order expansion uniformly valid for all times. Although this expansion is valid for a wide range of wave-numbers, it breaks down at two critical wave-numbers if the liquid depth is larger than √3/ k c , k c = (ρ g / T ) ½ , where g is the gravitational acceleration, and ρ and T are the liquid density and surface tension, respectively. For a deep liquid, the singularities are at k c /√2 and k c /√3 respectively, as found by Wilton (1915), and Pierson & Fife (1961). A second-order expansion valid for wave-numbers near the first critical value (corresponding to a wavelength of 2·44 cm in deep water) is obtained. This expansion shows that two different wave profiles could exist at or near the first critical wave-number. One of these profiles is gravity-like while the other is capillary-like.