Swash flow due to obliquely incident bores

We present a new solution to the nonlinear shallow water equations and show that it accurately predicts the swash flow due to obliquely approaching bores in large-scale wave basin experiments. The solution is based on an application of Snell's law of refraction in settings where the bore approach angle $\theta$ is small. We use the weakly two-dimensional nonlinear shallow water equations [Ryrie (J. Fluid Mech., vol. 129, 1983, p. 193)], where the cross-shore dynamics are independent of, and act as a forcing to, the alongshore dynamics. Using a known solution to the cross-shore dynamics [Antuono (J. Fluid Mech., vol. 658, 2010, p. 166)], we solve for the alongshore flow using the method of characteristics and show that it differs from previous solutions. Since the cross-shore solution assumes a constant forward-moving characteristic variable, $\alpha$, we term our solution the `small-$\theta$, constant-$\alpha$' solution. We test our solution in large-scale experiments with data from fifteen wave cases, including normally incident waves and obliquely incident waves generated using the wall reflection method. We measure water depths and fluid velocities using in situ sensors within the surf and swash zones and track shoreline motion using quantitative imaging. The data show that the basic assumptions of the theory (Snell's law of refraction and constant-$\alpha$) are satisfied and that our solution accurately predicts the swash flow. In particular, the data agrees well with our expression for the time-averaged alongshore velocity, which is expected to improve predictions of alongshore transport at coastlines.