If a water wave with length L moves with velocity v across a...

If a water wave with length $L$ moves with velocity $v$ across a body of water with depth $d,$ as in the figure, then $$v^{2}=\frac{g L}{2 \pi} \tanh \frac{2 \pi d}{L}$$ (a) If the water is deep, show that $v \approx \sqrt{g L /(2 \pi)}$ . (b) If the water is shallow, use the Maclaurin series for tanh to show that $v \approx \sqrt{g d}$ . (Thus in shallow water the velocity of a wave tends to be independent of the length of the wave.) (c) Use the Alternating Series Estimation Theorem to show that if $L>10 d$ , then the estimate $v^{2} \approx g d$ is accurate to within 0.014$g L .$

If a water wave with length $L$ moves with velocity $v$ across a body of water with depth $d,$ as in the figure, then
$$v^{2}=\frac{g L}{2 \pi} \tanh \frac{2 \pi d}{L}$$
(a) If the water is deep, show that $v \approx \sqrt{g L /(2 \pi)}$ .
(b) If the water is shallow, use the Maclaurin series for tanh to show that $v \approx \sqrt{g d}$ . (Thus in shallow water the velocity of a wave tends to be independent of the length of the wave.)
(c) Use the Alternating Series Estimation Theorem to show that if $L>10 d$ , then the estimate $v^{2} \approx g d$ is accurate to within 0.014$g L .$

Key Concepts

Dispersion Relation for Water Waves

This concept describes the relationship between the wave velocity, wavelength, and water depth. In the study of water waves, the dispersion relation reveals how the speed at which waves propagate is influenced by both gravitational acceleration and the physical parameters of the fluid, such as depth and wavelength. It provides the foundation for understanding different regimes of wave behavior in fluids.

Deep Water Approximation

When the water depth is much greater than the wavelength, the behavior of the wave simplifies. Under deep water conditions, certain terms in the dispersion relation become negligible, leading to an approximation where the wave velocity depends primarily on the wavelength and gravitational acceleration. This approximation helps in analyzing wave properties when the influence of the bottom is minimal.

Shallow Water Approximation

In contrast, when the water is shallow relative to the wavelength, the wave dynamics are significantly influenced by the bottom of the water body. Under shallow water conditions, the dispersion relation simplifies such that the wave velocity becomes independent of the wavelength and instead depends mainly on the water depth and gravitational acceleration. This approximation is fundamental for understanding coastal and near-shore wave phenomena.

Maclaurin Series Expansion

The Maclaurin series is a representation of a function as an infinite sum of terms calculated from the derivatives of the function at zero. In the context of water waves, it is used to approximate hyperbolic functions, such as the tanh function, when the argument is small. This allows for simpler analytical expressions and facilitates the derivation of approximations in shallow water scenarios.

Alternating Series Estimation Theorem

This theorem provides a method for estimating the error when truncating an alternating series after a finite number of terms. It is particularly useful in ensuring that the approximations derived from a series expansion, like the Maclaurin series for the tanh function, are accurate to a specific error bound. In wave analysis, it helps quantify the accuracy of approximations made in calculating wave velocity in different regimes.

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