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 on page $496,$ 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 on page $496,$ 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

Wave Dispersion Relation

This concept refers to the mathematical relationship that connects the velocity of a wave to its wavelength, the depth of the medium (in this case, water), and physical constants such as gravitational acceleration. Understanding the dispersion relation is essential for predicting how waves propagate in different physical contexts and for determining how various parameters influence wave behavior.

Deep and Shallow Water Approximations

These approximations simplify wave analysis by assuming the water depth is either much larger or much smaller than the wavelength. In deep water, the approximation leads to a simplification of functions like the hyperbolic tangent, while in shallow water, a series expansion of these functions results in a velocity that depends mainly on the depth. This approach allows for easier computations and conceptual understanding in different physical regimes.

Maclaurin Series Expansion

The Maclaurin series is an expansion of a function into an infinite sum of terms calculated from the derivatives at a single point (usually zero), which is especially useful for approximating functions when their arguments are small. In wave analysis, expanding functions like tanh using a Maclaurin series provides a simpler polynomial expression that captures the leading-order behavior in shallow water conditions.

Alternating Series Estimation Theorem

This theorem is used to provide an estimate of the error incurred when an alternating series is truncated after a finite number of terms. It asserts that the magnitude of the error is no greater than the absolute value of the first omitted term. This is particularly important in assessing the accuracy of approximations derived from series expansions in wave speed calculations.

Transcript

00:01 So let's get right at it.

00:01 And we are given this equation and asked to show this if the water is deep.

00:07 Ok, so what we're looking for is when d is very large.

00:16 So when d is very large, let's take a look at this.

00:20 So the limit of d to infinity, v squared, equals the limit of gl over 2 pi d to infinity times the tangent of h to pi d over l.

00:44 This will equal g over l 2 pi in the limit of d to infinity tangent h over l.

01:06 And that will equal gl over 2 pi times 1 equals gl over 2 pi.

01:16 So we can approximate the tangent h when d is very large.

01:25 When d is very large, this approaches.

01:36 Let me look at this.

01:39 This approaches one.

01:42 So this section right here approaches one.

01:47 Thus, d will be equal to the square root of gl over 2 pi.

01:55 High part two says where is my part two um table oopsie gotta go over here everything's taking a long time here arrange surface area temperature directions these guidelines use guidelines what what is this one? i am on 826.

02:46 If water.

02:47 Ok, for b, b says if water is shallow, then we are asked to use the maclaurin series for the tangent h to show that v is approximately equal to the square root of g d.

03:30 So thus, in shallow water, the velocity of the wave tends to be independent of the length of the wave.

04:02 Okay, let's do this one...

Please give Ace some feedback