The Kelvin-Havelock-Peters farfield approximation to ship waves

The classical asymptotic approximations to farfield ship waves given by Kelvin, Peters and Havelock for points located inside, outside or at the cusps of the Kelvin wake, and the uniform approximation given by Ursell, are considered. A simple approximation that is nearly equivalent to the Kelvin and Peters approximations inside or outside the cusps of the Kelvin wake, but is finite and agrees with Havelock's approximation at the cusps, is given. This approximation only involves elementary functions, whereas Ursell's approximation involves the Airy function and its derivative. The simple approximation given here, based on a modification of Ursell's uniform approximation, combines the approximations given by Kelvin, Havelock and Peters and is then called the Kelvin-Havelock-Peters (KHP) approximation. This approximation is less accurate than Ursell's approximation. However, the KHP approximation is more realistic than, yet as simple as, Kelvin's and Peters' approximations, which are singular at the cusps.