A theoretical formulation for wave propagations over uneven bottom

A first-order solution is obtained with a mild-slope parameter within the framework of the linear wave theory using the classical multiple scale method. This solution satisfies (1) the governing equations and free surface condition; (2) the coastal bottom condition in terms of multiple scale with its form analytical. As to two-dimensional wave propagating towards a slope, its solution reduces to the famous Biesel [1952. Study of wave progression in water of gradually varying depth. Gravity Waves, vol. 521. US National Bureau of Standards Circular, pp. 243–253] solution.