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Chapter 4: Problem 28

A water break at the entrance to a harbor consists of a rock barrier with a 50.0 -m-wide opening. Ocean waves of 20.0 -m wavelength approach the opening straight on. At what angles to the incident direction are the boats inside the harbor most protected against wave action?

Short Answer

Expert verified
Boats inside the harbor are most protected against wave action at the angles corresponding to the minima of the diffraction pattern, which are approximately 23.6 degrees and 53.1 degrees on both sides of the incident wave direction.

Step by step solution

01

Write down the single slit diffraction equation

The equation for single slit diffraction is given by: \(n\lambda = d \sin{\theta}\) where, - n is the order of the diffraction (integer) - λ is the wavelength of the waves (20.0 m) - d is the width of the opening (50.0 m) - θ is the angle of diffraction. We are looking for the angles θ where the boats will be most protected, meaning the positions of minimum intensity or "dark spots" in the diffraction pattern. These occur at the integer values of n.
02

Calculate the angle of diffraction for n = 1

First, we will calculate the angle of diffraction for the first-order minimum (n = 1). Plugging in the given values into the equation: \(1 \times 20.0 = 50.0 \sin{\theta_1}\) Now, we need to solve for \(\theta_1\): \(\theta_1 = \arcsin{\frac{20.0}{50.0}}\) \(\theta_1 = \arcsin{0.4}\) \(\theta_1 \approx 23.6^\circ\) Thus, the angle at the first-order minimum is approximately 23.6 degrees.
03

Check if there are any higher-order minima

For the higher-order minima (n > 1), we must check if the sine function exceeds 1, making the angle no longer valid. If this occurs, there is no minimum at that order. Let's examine the case for n = 2. \(\theta_2 = \arcsin{\frac{2 \times 20.0}{50.0}}\) \(\theta_2 = \arcsin{0.8}\) Here, the sine function does not exceed 1, meaning there is a second-order minimum. \(\theta_2 \approx 53.1^\circ\) Thus, the angle at the second-order minimum is approximately 53.1 degrees. Now let's check n = 3: \(\theta_3 = \arcsin{\frac{3 \times 20.0}{50.0}}\) \(\theta_3 = \arcsin{1.2}\) In this case, the sine function exceeds 1, meaning there is no third-order minimum (nor any higher-order ones).
04

Write down the final answer

Boats inside the harbor are most protected against wave action at the angles corresponding to the minima of the diffraction pattern, which are approximately 23.6 degrees and 53.1 degrees on both sides of the incident wave direction.

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Most popular questions from this chapter

How many complete orders of the visible spectrum \((400 \mathrm{nm}<\lambda<700 \mathrm{nm})\) can be produced with a diffraction grating that contains 5000 lines per centimeter?

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