Question

How would we model/compute the circular waves in a pool of water (wavelength and amplitude) from a mass falling into it from a given height, and from a fountain of water falling into it continuously?

Answer 1

To model and compute the circular waves in a pool of water generated by a mass falling into it or a continuous fountain, we can use the principles of wave propagation and fluid dynamics. The resulting waves will exhibit certain characteristics such as wavelength and amplitude.

1. Mass falling into the water: When a mass falls into the water, it creates a disturbance that generates waves spreading outward from the impact point. The waves will have a certain wavelength and amplitude determined by the initial conditions and the properties of the water. The exact calculation of these parameters can be complex and depend on factors like the mass of the object, its velocity upon impact, the depth of the water, and the surface tension of the water.

To estimate the wavelength, we can use a simplified approach based on linear wave theory. The wavelength (λ) can be approximated using the following formula:

λ = 2πh

where h is the water depth. This approximation assumes that the wavelength is much larger than the depth of the water.

The amplitude of the waves generated by the falling mass will depend on various factors, including the energy transferred to the water upon impact and the distance from the impact point. It can be challenging to determine the exact amplitude without specific measurements or further information.

2. Continuous fountain of water: A continuous fountain falling into the pool will also create circular waves that propagate outward. In this case, the waves are generated continuously, resulting in a steady-state wave pattern. The wavelength and amplitude of the waves will depend on the flow rate and the height from which the water falls.

Again, estimating the exact wavelength and amplitude can be challenging without more specific information about the flow rate, the shape and size of the fountain, and other factors. However, similar to the mass falling into the water, we can approximate the wavelength using the formula mentioned above, assuming the water depth is much smaller than the wavelength.

It's important to note that the actual behavior of the waves in both scenarios can be influenced by various factors such as wave reflections, interference, and dissipation, which may complicate the analysis.