The assumption of deep water I made earlier requires only that the water depth be greater than half a wavelength, or 5.5 cm. Their wavelength is this times the period, or g T / 2π. Waves in deep water travel at g T 2 / 2π. The period of our waves is T = 0.0676 seconds. This seems to be a little quick (like maybe by a factor of 2), but I'll go with it. It takes 0.0169 seconds, for a total oscillation period of 0.0676 seconds. If someone wants to finish solving this analytically, that would be awesome. Here's a matlab script I wrote so I didn't have to do math. I'll now proceed with a specific example: A granite rock with a 2 cm radius. So the answer to OP's actual question is that " the wavelength of the ripples is dependent on the rock's radius and density". This is a question I can answer!īut not analytically, because that's more calculus than I feel like doing right now. We're now down to "how long does something starting with zero speed take to travel from x = r1 to x = 0, with instantaneous acceleration 2 x 9.81 x r(ρ/750 + 2) / r1". Don't worry, most of this is about to cancel out. This approximation seems consistent with my observations. As it fills in, the force decreases, but not as quickly as the volume decreases, which results in an extremely high acceleration just before the void closes. I'm going to make another leap and say that it acts on the exact volume of water required to fill the void. The volume this force is acting on is difficult. r1 is the current radius of the cylindrical void which I'm keeping separate from the radius of the rock for reasons which will become apparent. Skipping the boring math, the total force comes out to 1000 x 9.81 x r 2 x (ρ/750 + 2) 2 x 2πr1. Pressure is 1000 x 9.81 x d, where d is the depth below the mean surface. The total hydrostatic force around the edge of this void is equal to the integral of pressure times area. So now we have a cylindrical void of radius r and depth r(ρ/750 + 2), and we want to know how long this will take to fill in. The waves generated will be at least a few cm in wavelength, so gravity is the only restoring forceĪ sphere of radius r meters has an equivalent Newton impactor length of 4/3 r meters. Transients caused by the rock impact are higher frequency than the oscillation The system oscillates at this period for some amount of time, then decreases in period as energy is removed from the system The amount of time taken to fill the void is 1/4 of the system's period of oscillation The water to fill this void is driven in from the sides by hydrostatic pressure difference The rock displaces a cylindrical column of water as predicted by Newton's impact depth approximation plus the rock diameter (in other words, it punches a hole, then gets out of the way). So if you look at this expanding circle, those ripples should have a wavelength of around 1.7 cm. That means that if you look around the where the pebble entered, you will see a clear circle around the pebble, expanding at the speed of these slowest moving waves. The critical wavelength - which is about 1.7 cm for water in a lake on Earth - is the wavelength that moves out most slowly. Thus, the long and short wavelength components move away from the stone faster. Because of the way surface tension and gravity contribute to the behavior of the waves, it turns out there is a critical wavelength such that waves of that critical wavelength travel more slowly than waves of any other wavelength. When the rock drops into the water, it will excite waves of a variety of wavelengths. In general, both effects matter, and this leads to an interesting effect. For long wavelengths, gravity dominates, and we get gravity waves. The behavior also depends on the depth of the water and the densities of the air and water also matter, but for the case at hand, except at the edges of the lake (and as long as we're not dealing with a puddle), we can treat the depth of the lake as effectively infinite, and the density of air is so much less than the density of water that it has minimal effect.įor short wavelengths, the surface tension term dominates, and we get capillary waves. Two effects dominate: surface tension and gravity. This is not a simple question, but there's one feature of these waves that is relatively simple.
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