There are many analytical solutions to the mild-slope equation that employ Hunt’s direct solution of linear dispersion  to solve wave-scattering problems involving varying quiescent depths. The advantage of Hunt’s direct solution of linear dispersion is that it can extend the range of applicability of the mild-slope equation from long-wave to deep-wave conditions. However, because the bottom curvature and slope-squared terms are neglected, the mild-slope equation cannot preserve mass conservation. In this investigation, we derived an analytic solution of the modified mild-slope equation for a conical island by adopting Hunt’s direct solution of linear dispersion to be applicable to intermediate water depth waves. We studied three differently slopes of conical islands in this paper. The relative difference between the present solution, including extended terms, and the conventional mild-slope equation proposed by Liu and Lin  was also estimated. The relative difference is insignificant in the long waves cases of the conical island; the maximal difference is just 0.4%. In contrast with the long waves conditions, the relative difference in the case of intermediate water waves is as high as 8.7% in T = 120 sec with a bottom slope of 1:3. Our analytical procedure shows that the 3rd Hunt’s approximation can perform as well as the conventional mild-slope equation proposed by Liu and Lin , which involves a 5th approximation. Finally, the relative difference between the present solution and the conventional mild-slope equation proposed by Liu and Lin  increases as the azimuth of the conical island decreases.
Hsiao, Sung-Shan; Chang, Chun-Ming; and Wen, Chih-Chung
"AN ANALYTICAL SOLUTION TO THE MODIFIED MILD-SLOPE EQUATION FOR WAVES PROPAGATING AROUND A CIRCULAR CONICAL ISLAND,"
Journal of Marine Science and Technology: Vol. 18:
4, Article 6.
Available at: https://jmstt.ntou.edu.tw/journal/vol18/iss4/6