We will ignore some of the more obvious issues with the movie and assume all other things are consistent to have fun with some of these questions.
Simple [hopefully] Pre-questions:
1) If the water is only a meter or so deep, then how can there be enough water to produce waves hundreds of meters in amplitude?
2) Are we to assume the source of the waves are tidal forces from the black hole nearby? If so, wouldn't this significantly alter the local gravity experienced by the crew?
3) Would the waves be more appropriately defined as gravity waves or shallow water waves?
In the case of shallow water waves, the phase speed is, assuming a wavelength ($\lambda$) much larger than the water depth ($h$), given by:
$$
\frac{\omega}{k} = \sqrt{g h}
$$
We are told in the movie that $g_{w} = 1.3 \ g_{E}$, or ~12.71-12.78 $m \ s^{-2}$. If we assume a water depth of 1 meter, then the phase speed should have been ~3.6 $m \ s^{-1}$ (roughly 8 mph).
If we ignore surface tension for the moment and assume the waves were gravity waves, then their phase speed is given by:
$$
\frac{\omega}{k} = \sqrt{\frac{g}{k}} \sim \sqrt{\frac{g \ \lambda}{2 \ \pi}}
$$
From my limited memory, I would estimate that the wavelength of these waves was ~100-1000 meters (let's make the numbers easy to deal with) and we already know the gravity, so we have phase speeds of ~14-45 $m \ s^{-1}$ (roughly 32-100 mph).
It's difficult to estimate speeds from a movie, but I am not sure if these results seem reasonable or not. The speeds are certainly more reasonable (i.e., they seem close to the actual movie speeds, I think) than I thought they would be prior to calculation, but the results bother me.
Intuitive Issue [and main question]
The soliton-like pulse of the waves in the movie makes me doubt both the movie and my estimates. The reason is that the phase speed of solitons depends upon their amplitude and FWHM. My intuition says that the amplitude of the waves alone should have resulted in much higher phase speeds than my estimates and the speeds shown in the movie.
Updates
I am not so much worried about the black hole or any direct general relativistic effect it might have on the planet. I am only interested in the waves on the planet.
Questions
- Can anyone suggest a possible explanation that might alleviate my concerns?
- The water is very shallow, as shown by the characters walking through it. So how can there be several hundred meter waves?
- Is it that all the planet's water is coalesced into these wave-like distortions (i.e., Are these just extreme tides?)?
- Are the waves actual a distortion of the planet's surface and the water is still only a meter or so deep?
- [Just to be nit-picky] If the previous question is true, then how would such a world not have significant volcanic activity (e.g., see Jupiter's moon Io)?
- If the waves are entirely water-based (i.e., they are effectively extreme tides) then their amplitude is orders of magnitude larger than the water depth or $\delta \eta/\eta_{o} \gg 1$. Is this a wave or just an extreme tide?
- If a wave, then:
- would the propagation speed of such a wave(?) be dominated by tidal effects?
- would it act like a soliton-like pulse once formed?
- If a tide, then:
- would there not be (extreme?) weather changes near these mounds of water?
- If a wave, then:
Best Answer
The following interpretations are taken from Thorne [2014].
Chapter 17, entitled Miller's Planet, discusses the issue of the large waves on the water planet in the movie Interstellar. There Kip mentions that the waves are due to tidal bore waves with height of ~1.2 km. In the appendix entitled Some Technical Notes, Kip estimates the density of Miller's planet to be $\sim 10^{4} \ kg \ m^{-3}$. For comparison, Earth's density is $\sim 5.514 \times 10^{3} \ kg \ m^{-3}$. We are also told that the planet itself has ~130% of the gravitational acceleration of Earth. From this we can estimate the mass and radius of Miller's planet (ignoring tidal distortions to make things easy): $$ \begin{align} r_{M} & = \frac{3 g_{M}}{4 \pi \ G \ \rho_{M}} \tag{1a} \\ & = \frac{3.9 g_{E}}{4 \pi \ G \ \rho_{M}} \tag{1b} \\ r_{M} & \sim 4546-4572 \ km \tag{1c} \\ M_{M} & = \frac{ 9 g_{M}^{3} }{ \left( 4 \pi \ \rho_{M} \right)^{2} \ G^{3} \ \rho_{M}} \tag{2a} \\ & = \frac{ 19.773 g_{E}^{3} }{ \left( 4 \pi \ \rho_{M} \right)^{2} \ G^{3} \ \rho_{M}} \tag{2b} \\ M_{M} & \sim 3.936 \times 10^{24} - 4.002 \times 10^{24} \ kg \tag{2c} \end{align} $$
For reference, the Earth's mean equatorial radius is $\sim 6.3781366 \times 10^{3} \ km$ and the Earth's mass is $\sim 5.9722 \times 10^{24} \ kg$.
Unfortunately, the answer is extremely boring. The planet is tidally locked with the nearby black hole and nearly all of the surface water of the planet is locked into two regions on opposite sides of the planet. The planet itself is shaped much like an American football rather than an oblate spheroid.
There is a slight problem with this interpretation, though. In the movie, the Ranger appears to float. Though I do not doubt that the vehicle is well sealed, I am curious if it could displace more water than its weight allowing it to float on the massive waves.
Just an extreme tide, and according to the wiki on this planet, they do not actually propagate, the planet rotates beneath you due to a slight difference in the planet's rotation rate and its orbital motion (i.e., the planet "rocks" back-and-forth during its orbit about the black hole).
I would be very surprised if such large mounds of water were not surrounded by or at least affecting the nearby weather, much the same as mountains on Earth. However, this is starting to split hairs in an already speculative subject I guess.
Updated Thoughts
I updated the following computations for fun merely because I found them more interesting than the tidal bores.
Gravity Waves
If we assume that the wave height were the same as the wavelength and we assume these were gravity waves, then their phase speed would be ~49 m/s.
Shallow Water Waves
If we assume the wavelength is $\sim r_{M} \gg h$ (i.e., from Equation 1c), where we now assume $h$ ~ 1.2 km, then the phase speed would go to ~124 m/s.
References
Typos and/or mistakes in book
I only found a few typos/mistakes in the book, which are listed below:
I consider these fairly minor and honest mistakes, but worth taking note of...