Determining Nome and Complementary Nome in First‐Order Cnoidal Theory

Abstract

Evaluation of the complete elliptic integrals and Jacobian elliptic functions can be made using power series in terms of the modulus, k, of the elliptic integrals, or in terms of the name, q, or complementary nome, q′, of the Theta functions. For the ranges of these parameters encountered in cnoidal wave theory, computations are much more efficient using q or f′. An approximate, explicit expression for q in first-order cnoidal wave theory is derived. Based on accuracy and computational efficiency, the expression is recommended for applications where the Ursell parameter is less than approximately 55. Isobe's approximate, explicit expression for q′ is recommended for applications where the Ursell parameter is greater than approximately 55. One or the other of these expressions is accurate over the entire range of Ursell parameter for which cnoidal wave theory is applicable. The explicit expressions for q and q′ are particularly useful for desktop calculations and for computationally intensive applications such as numerical models of the refraction and shoaling of cnoidal waves.