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Chapter 16: Problem 69

A wave on the surface of the ocean with wavelength \(44 \mathrm{~m}\) is traveling east at a speed of \(18 \mathrm{~m} / \mathrm{s}\) relative to the ocean floor. If, on this stretch of ocean surface, a powerboat is moving at \(15 \mathrm{~m} / \mathrm{s}\) (relative to the ocean floor), how often does the boat encounter a wave crest, if the boat is traveling (a) west, and (b) east?

Short Answer

Expert verified
(a) 0.75 Hz, (b) 0.068 Hz.

Step by step solution

01

Understanding the Scenario

In this problem, we have a wave moving east with a speed of \(18\, \text{m/s}\) and a powerboat moving at \(15\, \text{m/s}\). The boat experiences wave crests at a certain frequency depending on its direction (east or west). We need to find this frequency.
02

Determining Wave Frequency

The frequency \(f\) of the wave can be calculated using the speed of the wave \(v_w\) and its wavelength \(\lambda\). The formula is \(f = \frac{v_w}{\lambda}\). Substitute \(v_w = 18\, \text{m/s}\) and \(\lambda = 44\, \text{m}\):\[f = \frac{18}{44} \approx 0.409\, \text{Hz}.\]
03

Step 3a: Calculating Encounter Frequency (Boat moving West)

When the boat moves west, its speed relative to the wave becomes the sum of its speed and the wave speed, since they are moving in opposite directions. Thus, the relative speed \(v_{rel}\) is \(15 + 18 = 33\, \text{m/s}\). The frequency \(f_{encounter}\) at which the boat encounters wave crests is calculated as:\[f_{encounter} = \frac{v_{rel}}{\lambda} = \frac{33}{44} \approx 0.75\, \text{Hz}.\]
04

Step 3b: Calculating Encounter Frequency (Boat moving East)

When the boat moves east, its speed relative to the wave is the difference between the wave speed and boat speed, since both are moving in the same direction. Thus, \(v_{rel} = 18 - 15 = 3\, \text{m/s}\). The frequency \(f_{encounter}\) is:\[f_{encounter} = \frac{v_{rel}}{\lambda} = \frac{3}{44} \approx 0.068\, \text{Hz}.\]
05

Conclusion

The boat encounters wave crests at different frequencies depending on its direction: \(0.75\, \text{Hz}\) when moving west and \(0.068\, \text{Hz}\) when moving east.

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Key Concepts

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

Wave Frequency
Wave frequency refers to how often the wave crests (the highest points) pass a particular point in one second. It is an essential aspect of understanding wave motion, especially in the study of ocean waves like in the exercise. To calculate the frequency of a wave, you can use the formula:\[ f = \frac{v_w}{\lambda} \]- where \( f \) is the frequency,- \( v_w \) is the speed of the wave, and- \( \lambda \) is the wavelength. In the provided problem, the wave travels east at a speed of 18 meters per second with a wavelength of 44 meters. When you plug these values into the formula, you get a wave frequency of approximately 0.409 Hz.This means that about 0.409 wave crests pass a fixed point in one second under stationary conditions relative to the ocean floor.
Relative Motion
Relative motion considers how the velocity of an object is perceived in relation to another moving object or point. It plays a crucial role in determining how often the powerboat encounters wave crests. The speed of the boat relative to the waves differs based on the boat's direction: - **Moving West:** Here, the boat moves against the wave direction. The relative speed is the sum of both speeds (boat plus wave). Thus, the relative speed is 33 m/s (15 m/s + 18 m/s). - **Moving East:** The boat and waves travel in the same direction. Here, you subtract the boat's speed from the wave's speed, resulting in a relative speed of 3 m/s (18 m/s - 15 m/s). Understanding relative motion helps us to determine how often the boat meets wave crests during its journey.
Wavelength
Wavelength is the distance between two consecutive crests or troughs of a wave. It is a critical measure as it connects with both wave speed and frequency through the wave equation. Understanding wavelength allows us to measure: - **How compressed or stretched out a wave is**: Longer wavelengths mean the wave is stretched out; shorter wavelengths mean it's more compressed. - **Wave behavior**: Wavelength affects how waves interact with objects, like our boat. Longer wavelengths mean fewer crests encountered per distance traveled, impacting the boat's encounter frequency. In our scenario, the wavelength of 44 meters helps calculate the frequency of the wave and the encounter frequency based on relative motion. It is essential for understanding the pattern and timing of waves hitting the boat as it moves east or west.

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Most popular questions from this chapter

(I) What is the beat frequency if middle \(\mathrm{C}(262 \mathrm{Hz})\) and \(\mathrm{C}^{4}\) \((277 \mathrm{Hz})\) are played together? What if each is played two octaves lower (each frequency reduced by a factor of 4\() ?\)

(1I) Two piano strings are supposed to be vibrating at 220 \(\mathrm{Hz}\) , but a piano tuner hears three beats every 2.0 \(\mathrm{s}\) when they are played together. \((a)\) If one is vibrating at 220.0 \(\mathrm{Hz}\) what must be the frequency of the other (is there only one answer)? (b) By how much (in percent) must the tension be increased or decreased to bring them in tune?

When a player's finger presses a guitar string down onto a fret, the length of the vibrating portion of the string is shortened, thereby increasing the string's fundamental frequency (see Fig. 16-35). The string's tension and mass per unit length remain unchanged. If the unfingered length of the string is \(\ell=65.0 \mathrm{~cm},\) determine the positions \(x\) of the first six frets, if each fret raises the pitch of the fundamental by one musical note in comparison to the neighboring fret. On the equally tempered chromatic scale, the ratio of frequencies of neighboring notes is \(2^{1 / 12}\)

A Doppler flow meter is used to measure the speed of blood flow. Transmitting and receiving elements are placed on the skin, as shown in Fig. 16 - \(42 .\) Typical sound-wave frequencies of about \(5.0 \mathrm{MHz}\) are used, which have a reasonable chance of being reflected from red blood cells. By measuring the frequency of the reflected waves, which are Doppler-shifted because the red blood cells are moving, the speed of the blood flow can be deduced. "Normal" blood flow speed is about \(0.1 \mathrm{~m} / \mathrm{s}\). Suppose that an artery is partly constricted, so that the speed of the blood flow is increased, and the flow meter measures a Doppler shift of \(780 \mathrm{~Hz}\). What is the speed of blood flow in the constricted region? The effective angle between the sound waves (both transmitted and reflected) and the direction of blood flow is \(45^{\circ} .\) Assume the velocity of sound in tissue is \(1540 \mathrm{~m} / \mathrm{s}\)

An unfingered guitar string is \(0.73 \mathrm{~m}\) long and is tuned to play \(\mathrm{E}\) above middle \(\mathrm{C}(330 \mathrm{~Hz}) .(a)\) How far from the end of this string must a fret (and your finger) be placed to play A above middle C \((440 \mathrm{~Hz}) ?\) (b) What is the wavelength on the string of this \(440-\mathrm{Hz}\) wave? \((c)\) What are the frequency and wavelength of the sound wave produced in air at \(25^{\circ} \mathrm{C}\) by this fingered string?
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