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In this video I go over the water wave velocity equation and approximate it when the waves are deep and when the waves are shallow. For deep waves, the hyperbolic tan or tanh(x) term in the water wave velocity equation approaches 1, which greatly simplifies the equation. For shallow waves, we can use the Maclaurin series approximation for tanh(x) and show that 3rd order or higher terms approach zero for shallow waves. I also calculate the error of this shallow wave speed approximation by using the alternating series estimation theorem.
Timestamps:
Exercise 4: Water waves speed: 0:00
Solution to (a): Deep water wave speed approximation: 1:43
Approximation using hyperbolic tanh(x) = 1: 3:38
Deep wave speed approximation formula: 5:29
Solution to (b): Shallow water wave speed approximation: 6:04
Maclaurin series approximation for tanh(x): 7:05
Determining derivatives of tanh(x) at a = 0: 8:44
Writing out our Maclaurin series approximation of tanh(x): 17:18
3rd order terms approach zero when water is shallow: 20:07
Shallow water speed approximation: 21:43
Shallow water speed approximation formula: 23:22
Solution to (c): Alternating Series Estimation Theorem for the error: 24:25
Estimation for tanh(x) approximation: 26:53
tanh(x) remainder approximation if L is greater than 10d: 30:00
Estimating accuracy of shallow water wave speed: 32:33
Error is less than 0.0132gL: 34:02
Exercise 4: Approximating the Velocity of a Shallow Water Wave using Maclaurin Series