Abstract
Various kinds of approximate and explicit solutions(AESs) to the wave length in the Airy wave theory have been made because of analytical insolvability of the basic dispersion relationship. In this paper, 25 kinds of 1 step-AESs and their 2 step-AESs based on Newton's method using each AES for an initial estimate are respectively classified into 7 groups according to the maximum relative errors ε(1)max associated with the 1 step-AESs. Then a range of relative error ε and behavior of ε with relative water depth are investigated as to each of grouped AESs as well as a Combined Piecewise AES (CPAES) for separate usage in shallow water and deeper water. Main findings are (1)1 step-AES-based ε(1)max widely distributes from 0.0012 % to 5.2 %, (2)2 step-AES yields an ε(2)max smaller than 1 step-AES with a ratio from 10-2 to less than 10-5, (3)the CPAES gives higher accuracy than the best of 1-step AESs, and (4)selection of the AES or CPAES properly meeting the requested conditions may be possible from the candidates by taking εmax and computation load into account.
