Wave transformation by two-dimensional bathymetric anomalies with sloped transitions

Abstract

The reflection and transmission of normally incident waves by two-dimensional trenches and shoals of finite width with sloped transitions between the depth changes are studied. Three methods are developed using linearized potential theory. The step method is valid in arbitrary water depth and is an extension of the solution of Kirby and Dalrymple [J. Fluid Mech. 133 (1983) 47] for asymmetric trenches that allows for sloped transitions to be approximated by a series of steps. The slope method is an extension of Dean [J. Waterw. Harb. Div., ASCE 90 (1964) 1] that allows trenches and shoals with a linear transition between changes in depth to be modeled in the shallow water limit. A numerical method is also developed using a backward space-stepping routine commencing from the downwave side of the trench or shoal to model the wave field for arbitrary bathymetry in the shallow water limit. The reflection and transmission coefficients are compared for both symmetric and asymmetric trenches and shoals with abrupt transitions and sloped transitions. The sloped transitions cause a reduction in the reflection coefficient; a reduction that increases as the waves progress from long waves to shorter period waves. For symmetric bathymetric anomalies complete transmission is found for certain dimensionless wavelengths, a result not found for asymmetric trenches. The wave transformation by transitions with Gaussian forms is also investigated with results indicating minimal reflection for waves beyond the shallow water limit. Comparison between the three methods indicates good agreement in the shallow water range for the cases studied. A comparison of the step method to the numerical model FUNWAVE 1.0 [1-D] indicates satisfactory agreement between the models for linear incident waves. Several new results were established during the study. The wave field modification is shown to be independent of the incident wave direction for asymmetric changes in depth, a result shown by Kreisel [Q. Appl. Math. 7 (1949) 21] for a single step. For asymmetrical bathymetric anomalies with the same depth upwave and downwave of the anomaly, a zero reflection coefficient occurs only at k 1 h 1=0. For asymmetrical bathymetric anomalies with unequal depths upwave and downwave of the anomaly, the only k 1 h 1 value at which the reflection coefficient equals zero is that approached asymptotically at deep-water conditions.