Problem 57 - Algebra and Trigonometry

Question

For a wave to be surfable it can't break all at once. Robert Guza and Tony Bowen have shown that a wave has a surfable shoulder if it hits the shoreline at an angle \(\theta\) given by $$\theta=\sin ^{-1}\left(\frac{1}{(2 n+1) \tan \beta}\right)$$ where \(\beta\) is the angle at which the beach slopes down and where \(n=0,1,2, \ldots\) (a) For \(\beta=10^{\circ},\) find \(\theta\) when \(n=3\) (b) For \(\beta=15^{\circ},\) find \(\theta\) when \(n=2,3,\) and \(4 .\) Explain why the formula does not give a value for \(\theta\) when \(n=0\) or 1

Step-by-Step Solution

Answer

For \(n\) values 0 and 1, the formula gives an undefined \(\theta\) because the sine function's output range is [-1,1].

Step 1: Understanding the Problem Statement

We need to find the angle \(\theta\) for a wave that is surfable, given specific values of \(\beta\) (the angle of the beach slope) and \(n\). The formula is \(\theta=\sin ^{-1}\left(\frac{1}{(2 n+1) \tan \beta}\right)\).

Step 2: Calculate \(\theta\) for \(\beta=10^{\circ}\) and \(n=3\)

First, calculate \(\tan(10^{\circ})\), then use the given formula: \[\theta = \sin^{-1}\left(\frac{1}{(2 \times 3 + 1) \tan(10^{\circ})}\right)\]Compute the expression inside the inverse sine to find \(\theta\).

Step 3: Calculate \(\theta\) for \(\beta=15^{\circ}\) and \(n=2\)

Calculate \(\tan(15^{\circ})\), then use:\[\theta = \sin^{-1}\left(\frac{1}{(2 \times 2 + 1) \tan(15^{\circ})}\right)\]Solve to find \(\theta\).

Step 4: Calculate \(\theta\) for \(\beta=15^{\circ}\) and \(n=3\)

Use the formula with \(\tan(15^{\circ})\):\[\theta = \sin^{-1}\left(\frac{1}{(2 \times 3 + 1) \tan(15^{\circ})}\right)\]Compute the expression and find \(\theta\).

Step 5: Calculate \(\theta\) for \(\beta=15^{\circ}\) and \(n=4\)

Again, calculate using:\[\theta = \sin^{-1}\left(\frac{1}{(2 \times 4 + 1) \tan(15^{\circ})}\right)\]Solve to get \(\theta\).

Step 6: Analysis for \(n=0\) and \(n=1\) with \(\beta=15^{\circ}\)

For \(n=0\) or \(n=1\), the expression inside the inverse sine could exceed 1, which is not possible for real numbers, as the sine function only returns outputs between -1 and 1.

Key Concepts

Inverse Sine Function

The inverse sine function, denoted as \( \sin^{-1} \) or \( \arcsin \), is the function that undoes the sine function. It helps us find an angle given the opposite value of the sine in some trigonometric expressions. The range of the inverse sine function is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians, or from \(-90^{\circ}\) to \(90^{\circ}\) in degrees.
It's important to note the domain of the inverse sine is limited between \(-1\) and \(1\). This means the value inside the \( \sin^{-1} \) function must fall within this range to find a real angle. When working with problems that involve \( \sin^{-1} \), start by checking the value inside to ensure it's valid.

Beach Slope Angle

The beach slope angle, represented by \( \beta \), is a critical factor in determining how waves approach the shore. The slope angle describes how steeply the beach descends into the water.
In our problem, \( \beta \) influences the calculation through the tangent function, \( \tan(\beta) \), as seen in the formula
\[ \theta = \sin^{-1}\left(\frac{1}{(2n+1)\tan(\beta)}\right) \].
This factor affects how the wave breaks, indirectly controlling if the wave will form a surfable shoulder, which is crucial for surfers seeking to ride the wave effectively.
If the beach slope is too steep, it might influence how quickly waves break, sometimes too quickly for surfing.

Wave Surfing Conditions

In wave surfing, conditions are immensely important. A wave becomes surfable when it hIts the shore at the right angle, producing what surfers call a "shoulder."

• The shoulder is the part of the wave that allows surfers to ride it with enough time and space.
• The primary focus is to avoid waves that "close out," meaning they break across their entire length at once.

By using the given formula \( \theta = \sin^{-1}\left(\frac{1}{(2n+1)\tan(\beta)}\right) \), we can determine the surfability under various conditions. The parameter \(n\) allows adjustment for different scenarios by affecting the wave angle \(\theta\).
If \(n\) is too small, the inside value for the \( \sin^{-1} \) function could exceed its maximum possible value of \(1\), leading to no valid solution.

Angle Calculation

Calculating angles for surfable waves involves substituting values into the formula \( \theta = \sin^{-1}\left(\frac{1}{(2n+1)\tan(\beta)}\right) \).
Here's a step-by-step of how to approach these calculations:

• First, determine \(\beta\), the beach slope angle. You will need \(\tan(\beta)\).
• Next, pick \(n\) based on environmental conditions or desired setup. Remember, \(n\) should be greater than 1 to ensure the inside function does not exceed 1.
• Substitute \(\beta\) and \(n\) into the formula and simplify.
• Finally, apply the inverse sine function to find \(\theta\), ensuring your calculator is set to the correct angle unit (degrees or radians).

Always cross-verify your results to make sure they fall within possible real-world values, and if the inside of \( \sin^{-1} \) calculation is out of bounds, consider adjusting \(n\).