Friction forces when calculating the height of water waves when throwing a stone into the water

I recently watched people throwing stones into a lake and I was wondering about the profile of the waves. How does the height decline over time?

I believe it’s clear that the energy of the wave will (at least for stones which are not too large) be proportional to the gravitational energy of the stone. There a many different waves generated which move at different speed, but the slow waves don’t differ too much, so I can assume that the effective wavelength of the large and slow waves is approximately constant over its (short) lifetime. Therefore the height of the wave should be proportional to its energy.

Since the waves travel in circles, the energy dissipates as $1/r$ with the radius of the circle, i.e. the height of the wave should, too.

But what about dissipative forces such as friction? I guess it’s safe to assume that the waves travel mostly at the surface, so surface tension might play a role? Is friction negligible until the wave becomes very small?

Is there a simple heuristic for the effect of friction forces on the height of the wave? And can we also determine the effect size relative to the $1/r$ energy dissipation due to circular waves?

In other words: what other terms would I encounter if I try to write down an equation for the height of the wave as a function of the radius from its origin? Can I just neglect them?