If an Earth sized planet had 1.5-5x more or less gravity than that of the Earth, how would it affect the shape of waves in water bodies?

The best way to study water waves of many sizes is to use Airy wave theory (see also here), a mathematical model using several simplifications that nonetheless produces reasonable results. The linear theory works best when the amplitude ($a$) is small in comparison to the water depth ($h$) and the wavelength ($L$). Using the continuity equation for fluids, the Navier-Stokes equations, Bernoulli's principle, Laplace's equation, appropriate boundary conditions, and the assumption of linearity, we reach the surprisingly simple approximation of $$\begin{array}{}\text{(1)}& \eta (x,t)=a\cos (\omega t-kx)\end{array}$$ where $\eta$ is the elevation of a particle above the plane of water at rest, $\omega$ is the angular frequency, and $k$ is the wavenumber. We also get a nice dispersion relation and a somewhat complicated expression for the phase velocity of the wave, $c$: $$\begin{array}{}\text{(2)}& {\omega }^2=gk\tanh (kh)\end{array}$$ $$\begin{array}{}\text{(3)}& c=\sqrt{\frac{gL}{2\pi }\tanh \left(\frac{2\pi h}{L}\right)}\end{array}$$ where $g$ is of course the acceleration due to gravity; we get this from the equation $k=2\pi /L$. Additionally, as Green pointed out, the mean kinetic energy $\overline{E}$ is given by¹ $$\begin{array}{}\text{(4)}& \overline{E}=\frac{1}{16}\rho g{H}^2\end{array}$$ where $\rho$ is the density of the water. If an energy $E_w$ is imparted to a wave, then we have the relation $$H=\sqrt{\frac{16E_w}{\rho g}}\to H\propto \frac{1}{\sqrt{g}}$$ as JDÅ‚ugosz said. Stronger gravity means smaller waves.

Holding all other variables constant, $$\omega \propto \sqrt{g},c\propto \sqrt{g},H\propto \frac{1}{\sqrt{g}}$$ Therefore, on a planet with higher gravity, you'll see . . .

• Smaller waves
• Faster waves

and on a planet with lower gravity, you'll see . . .

• Larger waves
• Slower waves

This is all very straightforward, but what happens when waves reach the shore? At this point, linear wave theory breaks down (pun intended), and numerical modeling is often your best shot (see here). The shallow water equations are useful here, especially when discussing tsunamis. Let's look at one analytical case where we do get results. We look at the Carrier-Greenspan criterion for wave-breaking. When approaching a beach where the seafloor slopes up at a rate $s=\left|\frac{dH}{dx}\right|$, a wave will break if $$\begin{array}{}\text{(5a)}& s^2<\frac{{\omega }^2{a}_{\text{shore}}}{g}\end{array}$$ where $a_{\text{shore}}$ is the amplitude of the shore. Substituting in the dispersion relation for ${\omega }^2$ shows that this is entirely independent of $g$! A more rigorous model gives us a better relation (still without influence from $g$): $$\begin{array}{}\text{(5b)}& s^{5/2}<\sqrt{2\pi }ka\end{array}$$ This is elegant. While gravity will, of course, influence how specifically a wave moves near the shoreline, it will not affect whether it breaks or reflects.

Further reading:

• Tsunami - Mathematical classification on Earth Science Stack Exchange
• Aalborg University lecture notes
• University of San Diego lecture notes
• Lecture notes on other types of waves (Kelvin waves, Rossby waves, etc.)

posted about 8 years ago

HDE 226868‭

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