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Chapter 15: Problem 46

A transverse wave on a rope is given by $$ y(x, t)=(0.750 \mathrm{~cm}) \cos \pi\left[\left(0.400 \mathrm{~cm}^{-1}\right) x+\left(250 \mathrm{~s}^{-1}\right) t\right] $$ (a) Find the amplitude, period, frequency, wavelength, and speed of propagation. (b) Sketch the shape of the rope at these values of \(t: 0,\) \(0.0005 \mathrm{~s}, 0.0010 \mathrm{~s} .\) (c) Is the wave traveling in the \(+x\) - or \(-x\) -direction? (d) The mass per unit length of the rope is \(0.0500 \mathrm{~kg} / \mathrm{m}\). Find the tension. (e) Find the average power of this wave.

Short Answer

Expert verified
a) Amplitude = 0.75 cm, Wavelength = 15.7 cm, Frequency = 39.8 Hz, Speed = 625 cm/s. b) The sketches would show a regular cos wave pattern that vertically shifts down the x-axis every 0.0005s. c) The wave is traveling in the -x direction. d) Tension in the rope is approximately 1.95 N. e) Average power carried by the wave is approximately 1.47 W.

Step by step solution

01

Identify Wave Features

Understanding how a transverse wave is represented mathematically is the key to identifying the wave's characteristics. The standard form is \(y(x, t) = A \cos(kx - wt)\), where \(A\) is the amplitude, \(k\) is the wave number (equal to \(2\pi/\lambda\) where \(\lambda\) is the wavelength), and \(w\) is the angular frequency (where \(w = 2\pi/f\), \(f\) being the frequency). From the given equation for our wave, we can see the amplitude \(A\) is 0.750 cm, \(k\) is 0.400 cm\(^{-1}\), and \(w\) is 250 s\(^{-1}\).
02

Calculate Key Characteristics

With these values from Step 1, we can calculate the other features. The frequency \(f\) is equal to \(w/2\pi\) which gives 39.8 Hz. The wavelength is equal to \(2\pi/k\) giving us approximately 15.7 cm. Finally, the speed of propagation \(v\) is given by the equation \(v = \lambda*f\) which results in approximately 625 cm/s.
03

Sketch wave at different time points

With the understanding of the wave's characteristics, we can sketch the wave at \(t = 0, 0.0005s, 0.0010s\). It's important to note that the amplitude and wavelength determined in the previous steps determine the shape and size of the wave. The cos function's specific value at these different time points would determine the vertical displacement of points along the x axis.
04

Identify wave direction

The sign in front of the \(t\) parameter in the cosine function would determine the direction of wave propagation. If the wave equation is \(y(x, t) = A \cos(kx - wt)\), then the wave travels in the positive x-direction, and if the equation is \(y(x, t) = A \cos(kx + wt)\), the wave is in the negative x-direction. In this case, our equation shows that the wave is traveling in the negative x-direction.
05

Calculate tension in the rope

Using the wave speed formula for a string under tension, \(v = sqrt(T/\mu)\), where \(T\) is the tension in the string, and \(\mu\) is the linear mass density of the string (mass per unit length). We know \(v = 625 cm/s\) or \(6.25 m/s\), and \(\mu = 0.0500 kg/m\). Rearranging the formula to find \(T\) gives \(T = v^2 * \mu\) which is approximately \(1.95 N\).
06

Calculate average power

Finally, we can calculate the average power of the wave using the formula \(P = 1/2 \mu v w^2 A^2\). Plugging in the known values of \(\mu = 0.0500 kg/m\), \(v = 6.25 m/s\), \(w = 250 s^(-1)\), and \(A = 0.0075 m\) gives approximately \(1.47 W\).

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Key Concepts

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

Transverse Waves
Transverse waves are a fascinating type of wave where the oscillation or motion of the particles is perpendicular to the direction of the wave's propagation. Imagine holding a rope and shaking it up and down. The wave travels horizontally, but each point on the rope moves up and down. This is the essence of a transverse wave. Transverse waves are common in various physical systems, such as waves on a string, electromagnetic waves, and even some seismic waves.

Some key characteristics to understand about transverse waves include:
  • Amplitude: The maximum extent of the vibration from the rest position. In our problem, this value is given as 0.750 cm.
  • Direction of Travel: Identified by the sign in front of the angular frequency in the wave equation. A positive sign indicates negative x-direction travel and vice-versa.
Understanding these concepts allows for a deeper comprehension of how energy and information are transmitted in many physical mediums.
Wave Properties
Wave properties are a crucial area of study that pertains to the behavior and characteristics of waves. In transverse waves, we'll discuss some important properties that help define their behavior.

  • Period and Frequency: The period is the time it takes for one complete wave cycle and is calculated as the inverse of the frequency. The frequency, calculated here as approximately 39.8 Hz, tells us how many cycles occur in a second.
  • Wavelength: This is the distance over which the wave's shape repeats. It's a spatial measurement and is computed as approximately 15.7 cm in our instance.
  • Wave Speed: The speed of propagation, which is the rate at which the wave travels through a medium. The wave speed here is computed as approximately 625 cm/s, defined by the relationship between wavelength and frequency.
These properties are deeply interrelated and define how the wave propagates through its medium.
Tension and Power in Waves
When examining the mechanics of waves, particularly on a string or rope, understanding tension and power is essential. Tension plays a vital role in determining the wave speed on a rope. The tension refers to the force exerted along the rope that helps transmit the wave.

  • Tension: Calculated using the formula \( T = v^2 \times \mu \), where \( v \) is the wave speed and \( \mu \) is the linear mass density. In our example, the tension is approximately 1.95 N, translating the wave's energy along the rope.
  • Power: This is the rate at which energy is transferred by the wave. Average power can be found using the equation \( P = \frac{1}{2} \mu v w^2 A^2 \). For our exercise, it results in about 1.47 W. This tells us how quickly energy is communicated through the wave, crucial in practical applications like signal transmission.
Understanding tension and power not only helps in the analysis of wave dynamics on strings but also in areas such as sound engineering, materials science, and even in architectural acoustics.

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Most popular questions from this chapter

Adjacent antinodes of a standing wave on a string are \(15.0 \mathrm{~cm}\) apart. A particle at an antinode oscillates in simple harmonic motion with amplitude \(0.850 \mathrm{~cm}\) and period \(0.0750 \mathrm{~s}\). The string lies along the \(+x\) -axis and is fixed at \(x=0 .\) (a) How far apart are the adjacent nodes? (b) What are the wavelength, amplitude, and speed of the two traveling waves that form this pattern? (c) Find the maximum and minimum transverse speeds of a point at an antinode. (d) What is the shortest distance along the string between a node and an antinode?

The wave function of a standing wave is \(y(x, t)=(4.44 \mathrm{~mm})\) \(\sin [(32.5 \mathrm{rad} / \mathrm{m}) x] \sin [(754 \mathrm{rad} / \mathrm{s}) t] .\) For the two traveling waves that make up this standing wave, find the (a) amplitude; (b) wavelength; (c) frequency; (d) wave speed; (e) wave functions. (f) From the information given, can you determine which harmonic this is? Explain.

\(\mathrm{A}\) jet plane at takeoff can produce sound of intensity \(10.0 \mathrm{~W} / \mathrm{m}^{2}\) at \(30.0 \mathrm{~m}\) away. But you prefer the tranquil sound of normal conversation, which is \(1.0 \mu \mathrm{W} / \mathrm{m}^{2}\). Assume that the plane behaves like a point source of sound. (a) What is the closest distance you should live from the airport runway to preserve your peace of mind? (b) What intensity from the jet does your friend experience if she lives twice as far from the runway as you do? (c) What power of sound does the jet produce at takeoff?

A piano wire with mass \(3.00 \mathrm{~g}\) and length \(80.0 \mathrm{~cm}\) is stretched with a tension of \(25.0 \mathrm{~N}\). A wave with frequency \(120.0 \mathrm{~Hz}\) and amplitude \(1.6 \mathrm{~mm}\) travels along the wire. (a) Calculate the average power carried by the wave. (b) What happens to the average power if the wave amplitude is halved?

Transverse waves on a string have wave speed \(8.00 \mathrm{~m} / \mathrm{s},\) amplitude \(0.0700 \mathrm{~m},\) and wavelength \(0.320 \mathrm{~m} .\) The waves travel in the \(-x\) -direction, and at \(t=0\) the \(x=0\) end of the string has its maximum upward displacement. (a) Find the frequency, period, and wave number of these waves. (b) Write a wave function describing the wave. (c) Find the transverse displacement of a particle at \(x=0.360 \mathrm{~m}\) at time \(t=0.150 \mathrm{~s}\). (d) How much time must elapse from the instant in part (c) until the particle at \(x=0.360 \mathrm{~m}\) next has maximum upward displacement?
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