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I'm doing a experiment that revolves around dropping an object into a container with water.

When I increase the mass of the object, will the velocity of water waves increase when the object is dropped into the container?

Will the kinetic energy coming from the object transfer its energy to the water wave, thus increase the velocity of the water wave? If it does, then what is the relationship between kinetic energy and water wave velocity?

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No, the velocity of the particles of water and the propagation velocity are two different things. By exciting the wave with higher energy you increase the amplitude of the wave and this translate in higher displacements and higher particle velocities (this is the velocity of water oscillating up and down). The propagation velocity depends only on the properties of water and maybe depth of the tank (depends on the type of waves).

Of course, this is true for the linear regime, which is what you probably have. For very large amplitudes the waves may become nonlinear and propagation velocity may depend on amplitude and frequency.

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Classically, the velocity of a wave is only dependent on the properties of the medium. The velocity is given by

$v = \sqrt{(T/\mu )}$

where T is the restoring force returning the system to equilibrium (e.g. tension in a rope) and $\mu$ is the inertia resisting that change (e.g. the mass of a rope). When considering small perturbations in deep water where there is no horizontal motion of particles, this law applies without added corrections.

By dropping a more massive object into the water, the depth it penetrates should increase and therefore it should change the amplitude of the resulting waves.

If however the object creates enough disturbance and vortexes as it enters the water, it may affect the wave velocity locally.

Really the best way to find out is to actually try it out.

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