As I watched a muskrat swimming a bit offshore, I wished I had a better grasp of naval architecture (watercraft design) for it looked as if the muskrat were travelling at approximately its hull speed. Its concave body seemed to be draped between two wave crests of its own making.
An object (boat, muskrat) travelling across the surface of water creates a bow wave that necessarily travels at the same speed as the object. However, a bow wave is a wave so its subsequent troughs and crests will extend back alongside the hull. The wavelength depends on the wave speed and thus the boat speed. A slowly moving boat will produce a slowly moving, and thus a short wave. If the boat is long, this leads to many distinct waves along the hull. With greater and greater boat speed, and thus wave speed, there comes a time when the wavelength of the water wave has grown to equal the length of the hull. That speed is called the hull speed and the boat now sits neatly between the wave crests it created, just as did the muskrat I was watching.
For an animal moving across the surface, its hull speed may well constitute a practical speed limit. If it attempts to swim faster, the wavelength increases and the animal must now struggle to continuously travel uphill between the trough and crest of its own bow wave. In a boat, when sufficient power is applied, the bow will first tip up as that hill is climbed, but soon, with even more power, the boat is up and planing. About the only animals that seem to be (temporarily) capable of planing are birds landing on water. Apparently a muskrat swims at a speed just below its hull speed. (Fish, 1993). But what is that speed?
The relationship between wave speed and wavelength in a deep–water wave is,
c = √(g λ/2π), where c is wave velocity, λ is wavelength, and g is gravity.
If we put metric (MKS) numbers in this we get c = 1.25√λ and if λ = 0.3m, the average length, ℓ, of a muskrat body, we get a velocity of 0.7 m/s or about 2.5 kph.
Of course, estimating the actual speed of an object moving in the distance is difficult, so I will play a trick. Again set λ = ℓ, the muskrat’s body length, but now divide through by ℓ so the velocity, V, is expressed as the number of hull lengths travelled per second. We now get,
V = √(g/2πℓ).
Numerically, this is V = 1.25/√ℓ, and with ℓ = 0.3 m, we get that the muskrat travels at about two body lengths each second.
Indeed, that is what the muskrat appeared to be doing; travelling at its hull speed of about two body lengths per second. Incidentally, when travelling underwater, an animal will not encounter this problem with waves and so can travel faster than it can on the surface. Now, if only it didn’t have to return to the surface to breath, it could move really quickly.
The concave body of a muskrat is suspended between the crest of two waves of its own making as it travels at about its hull speed of two body lengths each second.