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Chapter 6: Problem 71

a. Confirm that the linear approximation to \(f(x)=\tanh x\) at \(a=0\) is \(L(x)=x.\) b. Recall that the velocity of a surface wave on the ocean is \(v=\sqrt{\frac{g \lambda}{2 \pi} \tanh \frac{2 \pi d}{\lambda}} .\) In fluid dynamics, shallow water refers to water where the depth-to-wavelength ratio \(d / \lambda<0.05 .\) Use your answer to part (a) to explain why the shallow water velocity equation is \(v=\sqrt{g d}.\) c. Use the shallow-water velocity equation to explain why waves tend to slow down as they approach the shore.

Short Answer

Expert verified
To summarize: a. We confirmed that the linear approximation of \(f(x)=\tanh x\) at \(a = 0\) is indeed \(L(x) = x\). b. We used the linear approximation of \(\tanh x\) to show that the shallow water velocity equation becomes \(v=\sqrt{gd}\) in shallow water where \(d / \lambda < 0.05\). c. We explained that waves slow down as they approach the shore because the water depth decreases, and the wave velocity is directly proportional to the square root of the water depth according to the shallow-water velocity equation, \(v=\sqrt{gd}\).

Step by step solution

01

a. Linear approximation of \(\tanh x\) at \(a=0\)

To find the linear approximation of a function at a point, we will use the following formula: \[L(x) = f(a) + f'(a) (x-a)\] In this case, our function is \(f(x)=\tanh x\). First, let's find the derivative of the function \(f'(x)\). The derivative of \(\tanh x\) is: \[\frac{d}{dx}(\tanh x) = \operatorname{sech}^2 x\] We are given that the linear approximation will be done at \(a=0\). For this, we need to evaluate the original function and its derivative at \(a=0\): \[f(0) = \tanh 0 = 0\] \[f'(0) = \operatorname{sech}^2 0 = 1\] Now, we can plug these values into the linear approximation formula to find \(L(x)\): \[L(x) = 0 + 1(x - 0) = x\] So, the linear approximation to \(f(x) = \tanh x\) at \(a = 0\) is indeed \(L(x) = x\).
02

b. Shallow water velocity equation

We are given the velocity of a surface wave on the ocean as: \[v=\sqrt{\frac{g\lambda}{2 \pi} \tanh \frac{2 \pi d}{\lambda}}\] Here, \(g\) is the acceleration due to gravity, \(\lambda\) is the wavelength, and \(d\) is the water depth. In shallow water where \(d / \lambda < 0.05\), we will use the linear approximation of \(\tanh x\) from part (a) to simplify the equation. When \(d / \lambda < 0.05\), the argument inside \(\tanh\) is close to zero: \[\frac{2 \pi d}{\lambda} < 0.1 \pi\] Therefore, we can use the result from part (a) and approximate \(\tanh\) as follows: \[\tanh \frac{2 \pi d}{\lambda} \approx \frac{2 \pi d}{\lambda}\] Now, we replace the original \(\tanh\) term in the velocity equation with its approximation: \[v=\sqrt{\frac{g\lambda}{2 \pi} \cdot \frac{2 \pi d}{\lambda}}\] Notice that \(2 \pi\) and \(\lambda\) cancel out: \[v=\sqrt{gd}\] We have shown that the shallow water velocity equation is \(v=\sqrt{gd}\) using the linear approximation of \(\tanh x\) from part (a).
03

c. Waves slowing down near the shore

Now, we use the shallow-water velocity equation from part (b) to explain why waves slow down as they approach the shore: \[v=\sqrt{gd}\] As waves approach the shore, the depth of water "d" decreases because the seabed becomes more shallow. Since "d" is in the square root part of the equation and is multiplied with "g", the overall value of \(v\) will decrease when "d" decreases. In other words, the wave velocity is directly proportional to the square root of the water depth. When waves get closer to the shore, the water depth gets smaller, so the wave velocity also gets smaller, causing waves to slow down as they approach the shore.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tanh Function Derivative
Understanding the derivative of the hyperbolic tangent function, otherwise known as \( \tanh x \), is essential in many calculus problems, including those involving linear approximations. \( \tanh x \) itself is a hyperbolic function representing the ratio of the hyperbolic sine and cosine functions. It shares properties with both the regular tangent function and the sigmoid function, important in many physics and engineering contexts.

The derivative of the \( \tanh x \) can be expressed as \( \operatorname{sech}^2 x \), where \( \operatorname{sech} x \) is the hyperbolic secant function. This derivative is particularly relevant when making linear approximations, as seen in the provided exercise. The function's behavior near the origin (\( x=0 \)) allows us to create a simple linear model, which is a straight line representing the function over a small interval around the origin.
Shallow Water Wave Velocity
The calculation of shallow water wave velocity incorporates concepts from both calculus and fluid dynamics. Generally, the velocity of surface waves depends on gravitational acceleration, water depth, and wave properties.

The relevant formula for wave velocity on the ocean is \( v = \sqrt{\frac{g \lambda}{2 \pi} \tanh \frac{2 \pi d}{\lambda}} \), where \( g \) is the acceleration due to gravity, \( \lambda \) is the wave's wavelength, and \( d \) is the depth of the water. However, for shallow water, with \( d/\lambda < 0.05 \) meaning the water depth is much less than the wavelength, the \( \tanh \) term can be approximated linearly. This results in the much simpler equation \( v = \sqrt{gd} \) for shallow water wave velocity. This approximation significantly simplifies calculations and can be used to explain various phenomena, such as the decrease in wave velocity as it approaches the shoreline.
Fluid Dynamics in Calculus
Fluid dynamics plays a pivotal role when applying calculus to real-world problems, like predicting wave behavior or air flow. Calculus allows us to quantify changes in fluid systems, such as velocity and pressure gradients.

When examining fluids in motion, Bernoulli's equation and the continuity equation are two principles often used. They are derived from the conservation of energy and mass, respectively, and both involve differentials, the core of calculus. For example, the continuity equation, which asserts that mass flow rate must remain constant in a closed system, is an application of the concept of derivatives to fluid flow. These principles illustrate how calculus helps us understand and model fluid behavior in various scenarios, including the approximation of shallow water wave velocity.
Wave Behavior Near Shore
Wave behavior near the shore is visibly different from that in the open ocean, and calculus can help explain these differences. As waves travel from deeper into shallower waters, their properties change due to interactions with the seafloor. The water depth (\( d \) in our equations) is key in determining wave velocity.

According to the shallow water velocity equation \( v = \sqrt{gd} \) derived using calculus, as the depth \( d \) decreases, so does the velocity \( v \) since they are directly related through a square root function. This phenomenon is why we observe waves slowing down as they near the shore, eventually leading to the wave breaking. Knowledge of this behavior is crucial for various activities along coastlines, from surfing to the construction of coastal infrastructure.

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Most popular questions from this chapter

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A tsunami is an ocean wave often caused by earthquakes on the ocean floor; these waves typically have long wavelengths, ranging between \(150\) to \(1000 \mathrm{km} .\) Imagine a tsunami traveling across the Pacific Ocean, which is the deepest ocean in the world, with an average depth of about \(4000 \mathrm{m}.\) Explain why the shallow-water velocity equation (Exercise 71 ) applies to tsunamis even though the actual depth of the water is large. What does the shallow-water equation say about the speed of a tsunami in the Pacific Ocean (use \(d=4000 \mathrm{m}\) )?

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