# Explanation of the waves on the water planet in the movie Interstellar?

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.

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

1. Can anyone suggest a possible explanation that might alleviate my concerns?
2. 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)?
3. 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?
• Can we start a tag for Interstellar-related questions? I would have so many ;) – SuperCiocia Jan 4 '15 at 18:13
• I would have, but I do not think I have enough reputation to do so. I also have several other questions, but this one seemed like an approachable one (I hope). – honeste_vivere Jan 4 '15 at 18:16
• I was joking, it wouldn't be very professional... the next tag would be Inception... ;) – SuperCiocia Jan 4 '15 at 18:23
• I don't think the planet/bh scenario is nearly as interesting as people think. For a stellar size bh the Roche limit of the planet is deep inside classical gravity territory. That's probably pretty much equivalent to a tidal locked planet around an ordinary star. I think the much more interesting cases are that of 1) Early moon, where a small moon is near its Roche limit orbiting the planet 2) The Melancholia scenario, where two planet size bodies nearly collide. That is when the crust really hits the fan! The effects shown in "Melancholia" were pretty close IMHO. – CuriousOne Jan 4 '15 at 18:23
• Who says that tides aren't waves? See Does Earth really have two high-tide bulges on opposite sides? for more details, but even if, on Earth, you (i) took away the land, and (ii) made all oceans as deep as the deepest trench, you would still have significant wave effects coming from the fact that the water simply cannot respond fast enough to the changes in gravity caused by the Moon's tidal force. Moreover, once you get to amplitudes as high as the depth itself, there are strong nonlinear effects to be considered. – Emilio Pisanty Oct 9 '15 at 12:50

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 $\sim 5.9722 \times 10^{24} \ kg$.

The water is very shallow, as shown by the characters walking through it. So how can there be several hundred meter waves?

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.

Is this a wave or just an extreme tide?

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).

would there not be (extreme?) weather changes near these mounds of water?

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

• Thorne, K. "The Science of Interstellar," W.W. Norton & Company, New York, NY, ISBN:978-0-393-35137-8, 2014.

# Typos and/or mistakes in book

I only found a few typos/mistakes in the book, which are listed below:

• Chapter 2
• He confuses the north and south magnetic poles (i.e., the north magnetic pole is located near the south geographic pole, not the north).
• He assigns the source of the aurora to protons. However, the aurora are due to energetic electrons exciting oxygen and nitrogen.
• Chapter 7
• He states that the Cassini spacecraft used "...Saturn's moon Io..." for a gravitational slingshot. However, Io is one of the four Galilean moons of Jupiter and Saturn is the planet to which Cassini was headed.

I consider these fairly minor and honest mistakes, but worth taking note of...

The issue with massive waves on a 1 meter deep ocean is that the waves cannot propagate fast enough on a planetary size object. We get fast moving shallow tsunami waves in the open ocean over a thousand meters deep. The tsunami piles up when the wave slows down due to contact with a shallow shoreline. Hundred meter high waves could never propagate fast enough in a one meter deep ocean to pile up. The same for a tidal bulge, water could not flow fast enough around a planetary distance to pile up in a one meter deep ocean.

What might be possible is that tidal effects on the land would be large enough to slosh waves of considerable height, especially if there were a sypathetic frequency. On Earth the land tides are a few inches, close to a black hole there might be land tides of over a meter. Of course, this creates problems of its own - since the energy absorbed from such huge land motions would remelt the planet.