Nonlinear Long Wave in Shallow Water

Contents  Two Important Parameters For Waves In Shallow Water  Nondimensional Variables  Nondimensional Governing Equation & Boundary Conditions  Perturbation Series of Potential  Depth Average Horizontal Velocity & Wave Elevation  Ariy's Approximation  Boussenesq's Approximation  Steady Kdv Equation  Solitary Wave  Cnoidal Wave

Two Important Nonlinear Parameters for Waves in Shallow Water

Non-dimensional Variables

Nondimensional Governing Equation & Boundary Conditions

Perturbation Series for Potential :

Depth Average Horizontal Velocity & Wave Elevation

Ariy's Theory For Very Long & Very Steep Waves

Boussinesq's Approximation

Steady KdV (Korteweg-de Vries) Equation

Solitary Waves

Figure 1. The sketch of a non-dimensional solitary wave train.

Cnoidal Wave

Computation Procedure : When water depth h, wave length L (or T) and wave height H are given, we may use a try and error (or iterative) method to determine m based on Equation (50) (or (53)). Once m is obtained, the characteristics of a Cnoidal wave train can be computed using Equations (47), (48), (49), (51) & (52).