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Chapter 16: Problem 6

A person fishing from a pier observes that four wave crests pass by in 7.0 s and estimates the distance between two successive crests to be 4.0 m. The timing starts with the first crest and ends with the fourth. What is the speed of the wave?

Short Answer

Expert verified
The wave speed is approximately 1.72 m/s.

Step by step solution

01

Determine the Time Period

First, calculate the time period of a single wave. The observation starts at the first crest and ends with the fourth. Therefore, the time period for three complete waves is 7.0 seconds. We divide this by 3 to find the time for one wave cycle: \( T = \frac{7.0\, \text{s}}{3} = 2.33\, \text{s} \).
02

Identify the Wavelength

The problem states that the distance between successive crests is 4.0 meters. This distance is known as the wavelength (\( \lambda \)). So we have \( \lambda = 4.0\, \text{m} \).
03

Calculate the Wave Speed

Wave speed can be determined using the formula \( v = \frac{\lambda}{T} \), where \( \lambda \) is the wavelength and \( T \) is the time period. Plugging in the values: \( v = \frac{4.0\, \text{m}}{2.33\, \text{s}} \approx 1.72\, \text{m/s} \).

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Key Concepts

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

Wave Crests
When observing waves, a 'crest' is the highest point of the wave. In the example problem, four wave crests pass by in 7.0 seconds during the observation period. This highlights the repeating nature of waves, forming distinct peaks. Understanding wave crests is crucial for gauging wave behavior over time.
  • Wave crests represent the points where the wave is at its maximum height.
  • Counting the number of crests helps determine the frequency and time period of waves.
  • In this context, passing four crests signifies the completion of three full waves.
By counting crests, we gain insight into how many complete wave cycles occur within a given time period.
Wavelength
The wavelength is the distance between two consecutive wave crests. It represents how long one complete wave cycle is, from crest to crest or trough to trough.In the exercise, the wavelength is given as 4.0 meters. This distance tells us how far the wave extends in one cycle.
  • Wavelength is denoted by the Greek letter lambda (\(\lambda\)).
  • Knowing the wavelength helps in calculating wave speed and understanding wave patterns.
  • A longer wavelength indicates a more extended wave cycle.
The understanding of wavelength is vital for exploring various wave phenomena, as it influences how waves propagate over distances.
Time Period
The time period of a wave measures the time it takes for one complete wave cycle to pass a fixed point. It's found by dividing the total time by the number of wave cycles.In the problem, the time period is calculated by observing that 7.0 seconds cover three wave cycles. So, \[ T = \frac{7.0 \, \text{s}}{3} = 2.33 \, \text{s} \]This means each wave cycle takes 2.33 seconds to pass.
  • Time period is an essential factor in determining wave speed.
  • A longer time period means cycles take longer to pass, moving slowly.
  • Time period is often inversely related to frequency: \( T = \frac{1}{f} \).
Understanding the time period provides insights into how frequently waves occur over time.
Wave Formula
Wave speed can be calculated using a standard formula relating wavelength and time period. The formula is given by\[ v = \frac{\lambda}{T} \]In this equation:
  • \( v \) is the wave speed, reflecting how fast the wave travels.
  • \( \lambda \) is the wavelength, the distance between two crests.
  • \( T \) is the time period, or the time for one full wave cycle.
Using the values from the exercise, we have a wavelength of 4.0 m and a time period of 2.33 s, resulting in\[ v = \frac{4.0 \, \text{m}}{2.33 \, \text{s}} \approx 1.72 \, \text{m/s} \]Thus, the wave travels at approximately 1.72 meters per second. This formula is fundamental in understanding wave mechanics, allowing us to link distance and time in the context of waves.

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Most popular questions from this chapter

When an earthquake occurs, two types of sound waves are gen- erated and travel through the earth. The primary, or P, wave has a speed of about 8.0 km/s and the secondary, or S, wave has a speed of about 4.5 km/s. A seismograph, located some distance away, records the arrival of the P wave and then, 78 s later, records the arrival of the S wave. Assuming that the waves travel in a straight line, how far is the seismograph from the earthquake?

When one person shouts at a football game, the sound intensity level at the center of the field is 60.0 dB. When all the people shout together, the intensity level increases to 109 dB. Assuming that each person generates the same sound intensity at the center of the field, how many people are at the game?

A rocket, starting from rest, travels straight up with an acceleration of 58.0 \(\mathrm{m} / \mathrm{s}^{2}\) . When the rocket is at a height of \(562 \mathrm{m},\) it produces sound that eventually reaches a ground-based monitoring station directly below. The sound is emitted uniformly in all directions. The monitoring station measures a sound intensity I. Later, the station measures an intensity \(\frac{1}{3} I\) There are no reflections. Assuming that the speed of sound is \(343 \mathrm{m} / \mathrm{s},\) find the time that has elapsed between the two measurements.

As a prank, someone drops a water-filled balloon out of a window. The balloon is released from rest at a height of 10.0 m above the ears of a man who is the target. Then, because of a guilty conscience, the prankster shouts a warning after the balloon is released. The warning will do no good, however, if shouted after the balloon reaches a certain point, even if the man could react infinitely quickly. Assuming that the air temperature is \(20^{\circ} \mathrm{C}\) and ignoring the effect of air resistance on the balloon, determine how far above the man's ears this point is.

An amplified guitar has a sound intensity level that is 14 dB greater than the same unamplified sound. What is the ratio of the amplified intensity to the unamplified intensity?
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