MACH REFLECTION OF A LARGE-AMPLITUDE SOLITARY WAVE

Reflection of an obliquely incident solitary wave by a vertical wall is studied numerically by applying the 'high-order spectral method' developed by Dommermuth & Yue (1987). According to the analysis by Miles (1977a, b) which is valid when a(i) much less than 1, the regular type of reflection gives way to 'Mach reflection' when psi(i)/(3a(i))1/2 less-than-or-equal-to 1, where a(i) is the amplitude of the incident wave divided by the quiescent water depth d and psi(i) is the angle of incidence. ln Mach reflection, the apex of the incident and the reflected waves moves away from the wall at a constant angle (psi*, say), and is joined to the wall by a third solitary wave called 'Mach stem'. Miles model predicts that the amplitude of Mach stem, and so the run-up at the wall, is 4a(i) when psi(i) = (3a(i))1/2. Our numerical results shows, however, that the effect of large amplitude tends to prevent the Mach reflection to occur. Even when the Mach reflection occurs, it is 'contaminated' by regular reflection in the sense that all the important quantities that characterize the reflection pattern, such as the stem angle psi*, the angle of reflection psi(r), and the amplitude of the reflected wave a(r), are all shifted from the values predicted by Miles' theory toward those corresponding to the regular reflection, i.e. psi* = 0, psi(r) = psi(i), and a(r) = a(i). According to our calculations for a(i) = 0.3, the changeover from Mach reflection to regular reflection happens at psi(i) almost-equal-to 37.8-degrees, which is much smaller than (3a(i))1/2 = 54.4-degrees, and the highest Mach stem is observed for psi(i) = 35-degrees (psi(i)/(3a(i))1/2 = 0.644). Although the 'four-fold amplification' is not observed for any value of psi(i), considered here, it is found that the Mach stem can become higher than the highest two-dimensional steady solitary wave for the prescribed water depth. The numerical result is also compared with the analysis by Johnson (1982) for the oblique interaction between one large and one small solitary wave, which shows much better agreement with the numerical result than the Miles' analysis does when psi(i) is sufficiently small and the Mach reflection occurs.