Throwing a small stone on the surface of a calm pond of water creates concentric ripples on the water surface starting from the point of impact and travelling outwards like transverse waves.

At the same time the momentum of the sinking stone creates longitudinal underwater pressure waves that propagate outwards with the speed of sound in water.

I am interested for an equation that describes the propagation speed relation of these two different types of waves created.

Will the surface transverse waves have the same propagation speed with the underwater longitudinal pressure waves and if not how can the propagation speed of each type of wave calculated for water?


1 Answer 1


Since nobody seems to be interested in answering this question here is an answer.

It is a combination of transverse surface waves and longitudinal pressure waves (i.e. sound waves):

waves on water

Kraaiennest, CC BY-SA 4.0, via Wikimedia Commons

Water molecules move with circular motion.

The closer to the surface the more the transverse up-down motion component of water molecules perpendicular to the energy poynting vector dominates. Circular motion radius becomes smaller but with the same angular velocity the deeper we go underwater where the longitudinal wave component dominates (i.e. motion of molecules towards the direction of the energy poynting vector).

The longitudinal motion component of these combinatoric waves is the speed of sound in the particular gas or liquid medium:

$$ c=\sqrt{\frac{K}{\rho_0}} \text { with } K=\frac{1}{\kappa} \text { the compressibility (bulk) modulus }\left(c=343 \mathrm{~m} / \mathrm{s} \text { at } 20^{\circ} \mathrm{C}\right. \text { ). } $$

And $ρ_{0}$ being the mass density of a homogeneous medium.

For pure water $K=2 \times 10^9 \mathrm{~N} / \mathrm{m}^2$ and $\rho_0=1000 \mathrm{~kg} / \mathrm{m}^3$, so $c=1414 \mathrm{~m} / \mathrm{s}$.

Concerning now the transverse wave's motion component and surface waves poynting vector propagation speed. This speed is much smaller than the longitudinal sound component and for shallow water surfaces can be calculated using the more complicated Korteweg–De Vries (KdV) equation.

A more easy empirical calculation can be also used:

$$ \text { Speed }=\text { Wavelength } x \text { Wave Frequency } $$

By measuring the separation distance between waves (wavelength) and the frequency by which they pass a point in space (frequency) $c=λf$.

So, for example when waves are $10 m$ apart from each other and one wave is passing every $5 s$ then the measured propagation speed of the surface waves is $2m/s$.


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