Semi-analytical Karhunen–Loeve representation of irregular waves based on the prolate spheroidal wave functions

Highlights: • A semi-analytical Karhunen–Loeve (K–L) representation of random data is presented. • Analytical prolate spheroidal wave functions are used in the K–L representation. • Present method requires less memory than purely numerical K–L representation. • Present method requires less computation time than other semi-analytical methods. A new semi-analytical approach is presented to solving the matrix eigenvalue problem or the integral equation in Karhunen–Loeve (K–L) representation of random data such as irregular ocean waves. Instead of direct numerical approach to this matrix eigenvalue problem, which may suffer from the computational inaccuracy for big data, a pair of integral and differential equations are considered, which are related to the so-called prolate spheroidal wave functions (PSWF). First, the PSWF is expressed as a summation of a small number of the analytical Legendre functions. After substituting them into the PSWF differential equation, a much smaller size matrix eigenvalue problem is obtained than the direct numerical K–L matrix eigenvalue problem. By solving this with a minimal numerical effort, the PSWF and the associated eigenvalue of the PSWF differential equation are obtained. Then, the eigenvalue of the PSWF integral equation is analytically expressed by the functional values of the PSWF and the eigenvalues obtained in the PSWF differential equation. Finally, the analytically expressed PSWFs and the eigenvalues in the PWSF integral equation are used to form the kernel matrix in the K–L integral equation for the representation of exemplary wave data such as ordinary irregular waves. It is found that, with the same accuracy, the required memory size of the present method is smaller than that of the direct numerical K–L representation and the computation time of the present method is shorter than that of the semi-analytical method based on the sinusoidal functions.