  We show higher interior regularity for the Westervelt equation with strong
nonlinear damping term of the $q$-Laplace type. Secondly, we investigate an
interface coupling problem for these models, which arise, e.g., in the context
of medical applications of high intensity focused ultrasound in the treatment
of kidney stones. We show that the solution to the coupled problem exhibits
piecewise $H^2$ regularity in space, provided that the gradient of the acoustic
pressure is essentially bounded in space and time on the whole domain. This
result is of importance in numerical approximations of the present problem, as
well as in gradient based algorithms for finding the optimal shape of the
focusing acoustic lens in lithotripsy.
