Diagram Wave Profiles by Cnoidal Wave Equation

The displayed wave profile is calculated using the cnoidal wave equation.
The lower part of the page describes the complete procedure for determining the calculated parameters and the displayed wave profile.

The range of possible inputs for modulus \( m \), wave height \( H \) and water depth \( d \) is not restricted.
Thereby, a verification, whether physically reasonable inputs were entered, does not take place.

Lernplattform des Leichtweiß-Institut für Wasserbau

Modulus m:

Wave height H [m]:

Water depth d [m]:

Wave period T = 0 s

d/gT² = 0

H/gT² = 0

Wave length L = 0 m

Ursell-Parameter U_R = 0

*Illustration taken from: EAK 2002, Empfehlungen für Küstenschutzwerke, Korrigierte Ausgabe 2007
des Kuratoriums für Forschung im Küsteningenieurwesen (erschienen in: Die Küste, Heft 65, Jahr 2002)

Procedure for determining the calculated parameters and the displayed wave profile:

Firstly using the entered (elliptical) modulus \( m \) the complete elliptic integral of the first order \( K(m) \), in the following simply called \( K \), is generated.
With the equation \( m \cdot K^2 = 2 \pi^2 \cdot \frac{3 \cdot a}{4 \cdot k^2 \cdot d^3} \) as well as with the aid of \( a = \frac{H}{2} \) and by the simplifying use of the linear wave number \( k = \frac{2 \pi}{L} \) follows the wave length \( L = \sqrt{\frac{16}{3} \cdot K^2 \cdot m \cdot d^3 / H} \).
The wave velocity is determined as follows \( c = \sqrt{\frac{g \cdot d}{1 + \frac{H}{d} \cdot \left( \frac{1}{m} - 2 \right)}} \), whereby the acceleration due to gravity is \( g = 9.81 \frac{m}{s^2} \).
This results in the wave period \( T = \frac{L}{c} \).
The Ursell parameter is \( Ur = \frac{H \cdot L^2}{d^3} \).
In the next step the different points of the wave profile are calculated by using the cnoidal wave equation \( \eta_i = H \cdot cn^2 \left\{ \frac{K}{\pi} \cdot \left( \frac{2 \pi}{L} \cdot x_i \right) ; m \right\} \), whereby \( cn \) represents the Jacobian elliptic function cosine amplitudinis \( cn \left\{ z ; m \right\} \).
In order to generate an average still water level of \( \eta = 0 \) in the displayed wave profile \( \eta \), the mean value of all \( \eta_i \) is subtracted from the values \( \eta_i \) determined with the help of the cnoidal wave equation.

The field of Wave Theories is part of the module Basic Coastal Engineering within the specialization Coastal and Ocean Engineering of the master programmes Civil and Environmental Engineering.