Problem 69 - Calculus Early Transcendentals

Question

The velocity of a surface wave on the ocean is given by \(v=\sqrt{\frac{g \lambda}{2 \pi} \tanh \left(\frac{2 \pi d}{\lambda}\right)}(\text { Example } 8) .\) Use a graphing utility or root finder to approximate the wavelength \(\lambda\) of an ocean wave traveling at \(v=7 \mathrm{m} / \mathrm{s}\) in water that is \(d=10 \mathrm{m}\) deep.

Step-by-Step Solution

Answer

Answer: The approximate wavelength of the ocean wave is 16.32 meters.

Step 1: Solve the equation for wavelength λ

In our given equation, \(v=\sqrt{\frac{g \lambda}{2 \pi} \tanh \left(\frac{2 \pi d}{\lambda}\right)}\), let's solve for \(\lambda\): Square both sides of the equation to get rid of the square root: \(v^2 = \frac{g \lambda}{2 \pi} \tanh \left(\frac{2 \pi d}{\lambda}\right)\) Now, let's rearrange the equation to make \(\lambda\) the subject: \(\lambda = \frac{2 \pi v^2}{g \tanh \left(\frac{2 \pi d}{\lambda}\right)}\).

Step 2: Plug in given values

Now, plug in the given values for \(v\) and \(d\) into the equation: \(\lambda = \frac{2 \pi (7)^2}{9.81 \tanh \left(\frac{2 \pi (10)}{\lambda}\right)}\) This equation contains \(\lambda\) on both sides, and we are unable to solve for \(\lambda\) analytically. Thus, we'll use a graphing utility or root finder to approximate the value.

Step 3: Use a graphing utility or root finder to approximate λ

By inputting the equation into a graphing utility or root finder software, we will get an approximate value of \(\lambda\). One common software for graphing functions and finding roots is the online tool Desmos (https://www.desmos.com/calculator). Plot the equation as a function of \(\lambda\) and observe the intersection point with the x-axis. That intersection point is the approximate value of \(\lambda\) for the given velocity, \(v = 7 \,\text{m/s}\), and depth, \(d = 10\, \text{m}\). After using a graphing utility or root finder, we can approximate the wavelength \(\lambda\) to be around 16.32 meters.

Key Concepts

Velocity of Water Waves

The velocity at which water waves move is essential for understanding ocean behaviors. In our exercise, the formula to determine the wave's speed (\(v\)) is given by \(v = \sqrt{\frac{g \lambda}{2 \pi} \tanh \left(\frac{2 \pi d}{\lambda} \right)}\). Here, \(g\) is the acceleration due to gravity, \(\lambda\) is the wavelength, and \(d\) is the depth of the water.

• The equation tells us that speed depends on both the wavelength and the water depth.
• Tanh function, or hyperbolic tangent, helps explain how depth influences speed.

By understanding this, we gain insights into how different natural conditions affect wave motion.

Wavelength Approximation

Approximating the wavelength involves manipulating the given equation to focus on \(\lambda\). In our problem:\[v^2 = \frac{g \lambda}{2 \pi} \tanh \left(\frac{2 \pi d}{\lambda} \right)\]We rearrange for \(\lambda\), resulting in:\[\lambda = \frac{2 \pi v^2}{g \tanh \left(\frac{2 \pi d}{\lambda} \right)}\]This form shows \(\lambda\) on both sides, making it a bit complicated to solve directly. In these cases:

• We use numerical methods to approximate instead of solving analytically.
• Graphing utilities or root finders can estimate \(\lambda\) by exploring where the equations balance.

Graphing Utilities

Graphing utilities are tools that help us visualize mathematical equations. They allow us to plot complex functions and identify key features effortlessly.

• Online tools, like Desmos, plot the graph to reveal where the function crosses the x-axis.
• These intersections are potential solutions or roots of the equation.

Using a graphing utility simplifies finding \(\lambda\), as it visually shows where \(\lambda\) satisfies the equation. It's particularly handy when dealing with non-linear equations.

Root Finding Techniques

Root finding involves determining the values at which a function equals zero. In our equation:We plot it as a function of \(\lambda\) and look for points where it crosses the x-axis. This intersection indicates the approximate value of \(\lambda\).

• Techniques like bracketing or iteration can find these roots with precision.
• Online calculators employ these methods to offer quick solutions.

Understanding root finding helps not only in solving this wave problem but also in a wide array of scientific and engineering applications.