Is there a relation between the depth of the water and the speed of the wave?
No, however… one must answer this question carefully. First of all the speed an acoustic wave propagation depends on density of the medium trough which it propagates. The denser the medium the faster it will move. Since water is uncompressible the height of water should not have any affect on propagation. If you are referring to surface wave it is a different story. The energy of the wave must be preserved. An example is a tsunami wave. When it travels across the ocean it travels very fast but as it comes closer to the shore it rises increasing its potential energy and decreasing its kinetic energy. With decrease of kinetic energy its speed falls dramatically.
