I am trying to find a mathematical proof of a statement contained in MIT lecture notes on shallow water gravity waves. On the 13th page the author says in reference to a disturbance caused by a projectile:
If the source is very localized(eg. a stone dropped into water), the wavefronts will be circular.
However, if I look for plane wave solutions to the differential equations of the form provided by the author:
\begin{equation} \frac{\partial^2 X}{\partial t^2} -c_0^2 \nabla^2X=0 \tag{*} \end{equation}
Then it isn't clear that the wave fronts will always form a closed curve in the plane. If I could guarantee that the wavefronts always form a closed curve then I can prove that asymptotically these curves resemble the circle.
In fact, assuming negligible friction/viscosity all the waves will travel in a direction normal to the boundary of the projectile. However, let's suppose that the boundary of my projectile is not smooth then I can't guarantee that the wave fronts will be closed curves in the plane.
In fact, in the case that my projectile is a triangle-based pyramid dropped so that its apex hits the surface of an initially motionless body of water, then I can guarantee that the wavefront isn't closed due to the discontinuous change in the unit normal around the base of the pyramid. It also follows that over time the shape of the wavefront deviates from that of a circle.
My hunch is that the author should have said smooth and convex projectile. Otherwise, I think my counterexample should suffice.
Note 1: I'd appreciate any clarification if there happens to be a flaw in my arguments.
Note 2: If the link doesn't work try this https://ocw.mit.edu/courses/earth-atmospheric-and-planetary-sciences/12-333-atmospheric-and-ocean-circulations-spring-2004/lecture-notes/ch1.pdf