Two-dimensional periodic permanent waves in shallow water

When two periodic permanent wave trains on shallow water intersect obliquely, the regions of intersection are two-dimensional waves of permanent shape. This shape varies from a nearly linear superposition of the two wave trains at large angles of intersection between the wave normals, to a structure predominantly transverse to the direction of propagation at small angles of intersection. The latter shape is found to be governed by the two-dimensional Korteweg–de Vries (Kadomtsev–Petviashvili) equation. The two-dimensional permanent waves are stable to periodic disturbances parallel to their direction of propagation, but are unstable to certain oblique periodic disturbances.