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Elliptic Integral Modules

Links : ±
CALCULATOR MODULE : Maths Special Function   ±

Calculate special function values.

Elliptic integrals are calculated using Carlsons forms. Jacobi elliptic functions are calculated using Landens transformation. The Gamma function is calculated using the Lanczos approximation.

Reference : Numerical Recipes, The Art Of Scientific Computing, Press, Teukolsky, Vetterling, Flannery, Cambridge University Press

Change Module :

CALCULATOR MODULE : Maths Elliptic Integral   ±

Calculate the complete and incomplete elliptic integrals of the first, second and third kind from the elliptic k modulus.

Elliptic integrals are calculated for an ellipse of the form

`x^2 + (y / b)^2 = 1 `
`k = √(1 - 1 / b^2) `

where :

k = the elliptic k modulus

For a circle k = 0. k tends to 1 as b tends to infinity. Use the Result Plot option to plot the integrals versus the k modulus.

Reference : Numerical Recipes, The Art Of Scientific Computing, Press, Teukolsky, Vetterling, Flannery, Cambridge University Press

Change Module :

CALCULATOR MODULE : Cnoidal Fifth Order Wave   ±

Calculate Cnoidal wave velocity, acceleration and surface profile using Fentons 1999 fifth order wave method.

The Cnoidal wave is defined by the elliptic modulus m, the wave trough depth w, and the wave alpha parameter α. The Cnoidal wave model is a truncated series and is only valid within certain ranges. The Cnoidal wave theory is not recommended where the wavelength over water depth ratio (Lod) is less than 8. The recommended wave type is displayed below the calc bar.

Note : The cnoidal wave theory uses a truncated infinite series. The truncated series is only valid for conditions where the series converges (m > 0.8). For deep water waves with small m, the series does not converge (use the Stokes wave instead).

Check that the convergence is close to or equal to one. The wave period should be measured at zero current velocity to avoid Doppler effects.

Reference : J D Fenton, The Cnoidal Theory Of Water Waves, Developments in Offshore Engineering, Gulf, Houston, chapter 2, 1999

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