The calculation of the speed of surface waves on a fluid is a little lengthy to do in full generality for all fluid heights and all wavelengths. When you say that you "dropped it" do you mean you dropped the whole tray full of water on the floor, or did you drop something into the tray to make some waves? I would set up something with a card or a board on one end, shaken side to side so as to make waves of a desired wavelength -- dropping something in the tray results in a complicated mixture of waves of different wavelengths, which travel at different speeds.
I found a full expression for the speed of a wave in an incompressible fluid with no viscosity (there are no such fluids, but water is pretty close to this approximation for small trayfuls at normal temperatures and pressures). The speed of surface waves is:
speed = sqrt( ((g*lambda)/(2*pi))*tanh(2*pi*h/lambda) )
from Fetter and Walecka, "Theoretical Mechanics of Particles and Continua", McGraw-Hill, 1980. Here, g is the acceleration due to gravity, 9.81 m/s**2, lambda is the wavelength of the waves under study, and h is the depth of the fluid. For very shallow fluids (compared to the wavelength), the speed increases proportionally to the square root of the depth, and for very deep fluids, the speed increases with the square root of the wavelength.
In words why this is the case -- for a shallow fluid, the motion of the fluid is mostly side-to-side, and in order to accumulate more fluid in one place (to make the crest of the wave), each little bit of fluid must travel a little farther than it would have to in deeper water. When a wave passes, the bits of fluid, if you could watch one at a time, travel in ellipses. For shallow water, the ellipses are stretched out horizontally, and in very deep water, they are very nearly circular. So for a wave of the same height (top to bottom of the ellipse), the bits of water must travel farther in the shallow tray than the deep tray. Because the waves of the same height in shallow and deep water exert the same pressure differences due to gravity to get the water moving (although the motion is different due to the fact that the bottom is there), similar forces push and pull on the water. To get the water moving farther and faster with the same force takes a longer time for each push, and hence a slower speed for the wave, in the shallow water.
Note: this speed formula assumes that the waves are small -- for waves whose heights are comparable to the depth of the tray, you will get even more complicated behavior -- the most spectacular of these is the formation of "breakers" where the waves will curl over and crash as they do on beaches, making for good surfing.
Tom
(published on 10/22/2007)
