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Chapter 16: Problem 5

A sinusoidal wave travels along} a string. The time for a particular point to move from maximum displacement to zero is \(0.170 \mathrm{~s}\). What are the (a) period and (b) frequency? (c) The wavelength is \(1.40 \mathrm{~m}\); what is the wave speed?

Short Answer

Expert verified
(a) The period is 0.680 s; (b) frequency is 1.47 Hz; (c) wave speed is 2.06 m/s.

Step by step solution

01

Understand the Movement from Maximum Displacement to Zero

When a wave particle moves from its maximum displacement to zero, it has covered a quarter of the wave cycle. Thus, this time can help us calculate the period of the wave. Given this time is 0.170 s, this means \( \frac{T}{4} = 0.170 \) s.
02

Calculate the Period \( T \)

Using the relationship \( \frac{T}{4} = 0.170 \) s, we solve for the period \( T \). Multiply both sides by 4 to isolate \( T \):\[ T = 4 \times 0.170 \text{ s} = 0.680 \text{ s} \]
03

Determine the Frequency \( f \)

Frequency is the reciprocal of the period. Calculate the frequency \( f \) using \( f = \frac{1}{T} \):\[ f = \frac{1}{0.680 \text{ s}} \approx 1.47 \text{ Hz} \]
04

Calculate the Wave Speed \( v \)

Wave speed is the product of wavelength and frequency. Given the wavelength \( \lambda = 1.40 \text{ m} \), use the formula:\[ v = f \times \lambda = 1.47 \text{ Hz} \times 1.40 \text{ m} \approx 2.058 \text{ m/s} \]

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Key Concepts

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

Period of a Wave
The period of a wave is a fundamental concept in wave mechanics that refers to the time it takes for a wave to complete one full cycle. In simpler terms, if you imagine a wave as a repetitive motion, the period is the time it takes for this motion to start again from the same position.

In the given exercise, the time from maximum displacement to zero crossing is 0.170 seconds. This is a quarter of the wave cycle because when a point on a wave travels from the highest point to the middle, it鈥檚 merely transitioning through one-fourth of the wave path. Therefore, knowing this, the formula to find the period becomes:
  • Time for one complete cycle (Period, T) = 4 times the quarter cycle time
So, the period \( T \) can be calculated as:
  • \( T = 4 \times 0.170 \, \text{s} = 0.680 \, \text{s} \)
A longer period means the wave oscillates more slowly, and a shorter period means it has a quicker cycle.
Frequency Calculation
Frequency is another key parameter in understanding wave behavior. It tells us how often the events of a wave, such as crests or troughs, pass a specific point in one second.

Mathematically, frequency is the reciprocal of the period. It signifies how quickly a wave cycles in a given amount of time, typically measured in Hertz (Hz), where 1 Hz equals one cycle per second. The formula for frequency \( f \) based on the period \( T \) is:
  • \( f = \frac{1}{T} \)
From our earlier calculation of the period \( 0.680 \, \text{s} \), the frequency \( f \) is:
  • \( f = \frac{1}{0.680 \, \text{s}} \approx 1.47 \, \text{Hz} \)
Higher frequency means more cycles per second, leading to more energetic waves.
Wavelength and Wave Speed
Wave speed is a crucial concept that describes how fast a wave is traveling through a medium. It connects to both the wavelength, which is the distance between two equivalent points of consecutive cycles (such as crest to crest), and the frequency, as discussed above.

The formula linking wavelength \( \lambda \), frequency \( f \), and wave speed \( v \) is:
  • \( v = f \times \lambda \)
In our exercise, the wavelength is given as \( 1.40 \, \text{m} \). With the frequency calculated earlier as \( 1.47 \, \text{Hz} \), you can find the wave speed:
  • \( v = 1.47 \, \text{Hz} \times 1.40 \, \text{m} \approx 2.058 \, \text{m/s} \)
This resulting speed suggests how swiftly the wave propagates across the medium, whether it be a string, water, or air.

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Most popular questions from this chapter

(a) Write an cquation describing a sinusoidal transverse wave traveling on a cord in the positive direction of a \(y\) axis with an angular wave number of \(60 \mathrm{~cm}^{-1}\), a period of \(0.20 \mathrm{~s}\), and an amplitude of \(3.0 \mathrm{~mm}\). Take the transverse direction to be the \(z\) direction. (b) What is the maximum transverse speed of a point on the cord?

A human wave. During sporting events within large, densely packed stadiums, spectators will send a wave (or pulse) around the stadium (Fig. \(16-29)\). As the wave reaches a group of spectators, they stand with a cheer and then sit. At any instant, the width \(w\) of the wave is the distance from the leading edge (people are just about to stand) to the trailing edge (people have just sat down). Suppose a human wave travels a distance of 853 seats around a stadium in \(39 \mathrm{~s}\), with spectators requiring about \(1.8 \mathrm{~s}\) to respond to the wave's passage by standing and then sitting. What are (a) the wave speed \(v\) (in seats per second) and (b) width \(w\) (in number of seats)?

93 A traveling wave on a string is described by $$ y=2.0 \sin \left[2 \pi\left(\frac{t}{0.40}+\frac{x}{80}\right)\right] $$ where \(x\) and \(y\) are in centimeters and \(t\) is in seconds. (a) For \(t=0\), plot \(y\) as a function of \(x\) for \(0 \leq x \leq 160 \mathrm{~cm} .\) (b) Repeat (a) for \(t=0.05 \mathrm{~s}\) and \(t=0.10 \mathrm{~s}\). From your graphs, determine (c) the wave speed and (d) the direction in which the wave is traveling.

A sinusoidal wave of angular frequency \(1200 \mathrm{rad} / \mathrm{s}\) and amplitude \(3.00 \mathrm{~mm}\) is sent along a cord with linear density \(2.00 \mathrm{~g} / \mathrm{m}\) and tension \(1200 \mathrm{~N}\). (a) What is the average rate at which energy is transported by the wave to the opposite end of the cord? (b) If, simultaneously, an identical wave travels along an adjacent, identical cord, what is the total average rate at which energy is transported to the opposite ends of the two cords by the waves? If, instead, those two waves are sent along the same cord simultaneously, what is the total average rate at which they transport energy when their phase difference is (c) \(0,(\mathrm{~d}) 0.4 \pi \mathrm{rad}\), and \((\mathrm{e}) \pi \mathrm{rad} ?\)

A string oscillates according to the equation $$ y^{\prime}=(0.50 \mathrm{~cm}) \sin \left[\left(\frac{\pi}{3} \mathrm{~cm}^{-1}\right) x\right] \cos \left[\left(40 \pi \mathrm{s}^{-1}\right) t\right] $$ What are the (a) amplitude and (b) speed of the two waves (identical except for direction of travel) whose superposition gives this oscillation? (c) What is the distance between nodes? (d) What is the transverse speed of a particle of the string at the position \(x=1.5 \mathrm{~cm}\) when \(t=\frac{9}{8} \mathrm{~s} ?\)
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