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Transverse waves on a string have wave speed 8.00 m/s, amplitude 0.0700 m, and wavelength 0.320 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 \(x = 0.360\) m at time \(t = 0.150\) s. (d) How much time must elapse from the instant in part (c) until the particle at \(x = 0.360\) m next has maximum upward displacement?

Short Answer

Expert verified
Frequency: 25 Hz, period: 0.04 s, wave number: 19.63 m⁻¹. Next max at 0.190 s.

Step by step solution

01

Calculate Wave Frequency

The wave speed \(v\) is related to the wavelength \(\lambda\) and frequency \(f\) by the formula:\[ v = f \cdot \lambda \]Given the wave speed \(v = 8.00 \text{ m/s}\) and wavelength \(\lambda = 0.320 \text{ m}\), we can solve for the frequency:\[ f = \frac{v}{\lambda} = \frac{8.00}{0.320} = 25.0 \text{ Hz} \]
02

Calculate Wave Period

The period \(T\) of a wave is the inverse of the frequency:\[ T = \frac{1}{f} = \frac{1}{25.0} = 0.040 \text{ s} \]
03

Calculate Wave Number

The wave number \(k\) is given by the formula:\[ k = \frac{2\pi}{\lambda} \]Substituting the known wavelength:\[ k = \frac{2\pi}{0.320} \approx 19.63 \text{ m}^{-1} \]
04

Write the Wave Function

The general form of a wave traveling in the negative \(x\)-direction is:\[ y(x,t) = A \sin(kx + \omega t + \phi) \]where \(A = 0.0700 \text{ m}\), \(\omega = 2\pi f = 2\pi \times 25.0 = 50\pi \text{ rad/s}\), and the initial condition specifies maximum upward displacement at \(x = 0\) and \(t = 0\), so \(\phi = \frac{\pi}{2}\) (since sine of \(\pi/2\) gives maximum value). Thus the wave function becomes:\[ y(x,t) = 0.0700 \sin(19.63x + 50\pi t + \frac{\pi}{2}) \]
05

Find Transverse Displacement at Given Position and Time

Substitute \(x = 0.360 \text{ m}\) and \(t = 0.150 \text{ s}\) into the wave function:\[ y(0.360, 0.150) = 0.0700 \sin(19.63 \times 0.360 + 50\pi \times 0.150 + \frac{\pi}{2}) \]Calculating the argument:\[ 19.63 \times 0.360 + 50\pi \times 0.150 + \frac{\pi}{2} = 7.0676 + 23.5619 + 1.5708 \approx 32.2003 \text{ rad} \]\[ y = 0.0700 \sin(32.2003) \approx 0.026 \text{ m} \]
06

Calculate Time for Next Maximum Displacement

The time interval \(\Delta t\) between successive maximum displacements is the period \(T\), and the sine function reaches its maximum at one complete period later. Thus, from \(t = 0.150 \text{ s}\), the next maximum displacement occurs at:\[ t_{\text{next max}} = 0.150 + 0.040 = 0.190 \text{ s} \]Time elapsing from given time to next maximum:\[ \Delta t = 0.190 - 0.150 = 0.040 \text{ s} \]

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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 fascinating waves that move perpendicular to the direction of wave propagation. Imagine holding a rope and moving your hand up and down; the waves travel horizontally while the medium (rope) moves vertically.
This is exactly how transverse waves behave.
  • An example of transverse waves can be seen in water waves or electromagnetic waves.
  • In the context of waves on a string, the particles of the string move up and down as the wave travels along the string.
  • These waves are different from longitudinal waves where the movement of the medium is parallel to wave travel, like sound waves in air.
In our exercise, these transverse waves on the string have a specific wave speed, amplitude, and they travel in the opposite of the positive x-direction, or (-x)-direction, showcasing the versatile nature of how waves can move across mediums.
Wave Speed
Wave speed is a crucial aspect of understanding how fast a wave travels through a medium. It determines how quickly the wave displaces particles along its path.
To find wave speed in a real-world scenario, it's vital to understand the relationship between wave speed, frequency, and wavelength:
  • The formula to find wave speed is: \[ v = f \cdot \lambda \]Where:
    • \( v \) is the wave speed.
    • \( f \) is the frequency, or the number of wave cycles per second.
    • \( \lambda \) is the wavelength, or the space between repeating units of wave pattern.
In the original problem, we found that the wave speed of the transverse wave on the string is 8.00 m/s. Knowing this makes it easier to calculate related properties like frequency and period using given physical quantities.
Wavelength
Wavelength is a measure of the distance between two identical phases of a wave, such as from crest to crest or trough to trough. It defines the length of a full wave cycle in a particular medium.
  • Physically, the wavelength can affect how waves interact with each other and with obstacles.
  • In our exercise, the given wavelength is 0.320 m, an important starting parameter. It helps us calculate other properties like the wave number.
  • The wavelength also impacts the wave speed and frequency, based on the formula \[ v = f \cdot \lambda \], showing that all these characteristics are intertwined.
By understanding the wavelength, we gain insight into how far a wave must travel before repeating itself, which is essential in wave physics applications.
Wave Function
A wave function is a mathematical description of the wave's oscillation in a medium as it travels. It provides a detailed explanation of displacement over time and space.
  • The general form of the wave function is: \[ y(x, t) = A \sin(kx + \omega t + \phi) \]where:
    • \( A \) is the amplitude, or maximum displacement.
    • \( k \) is the wave number, related to wavelength.
    • \( \omega \) is the angular frequency, found from \( \omega = 2\pi f \).
    • \( \phi \) is the phase constant, accounting for initial displacement.
  • For our problem, the wave function quantifies how the wave travels in the negative x-direction: \[ y(x, t) = 0.0700 \sin(19.63x + 50\pi t + \frac{\pi}{2}) \]
  • By using the wave function, we can predict the position of any particle on the string at any time.
Understanding wave functions is fundamental in wave physics since they allow us to model and anticipate wave behaviors naturally observed or in laboratory settings.

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Most popular questions from this chapter

The wave function of a standing wave is \(y(x, t) = 4.44 \, \mathrm{mm} \, \mathrm{sin}[(32.5 \, \mathrm{rad/m})x] \mathrm{sin}[(754 \, \mathrm{rad/s})t]\). 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.

A wire with mass 40.0 g is stretched so that its ends are tied down at points 80.0 cm apart. The wire vibrates in its fundamental mode with frequency 60.0 Hz and with an amplitude at the antinodes of 0.300 cm. (a) What is the speed of propagation of transverse waves in the wire? (b) Compute the tension in the wire. (c) Find the maximum transverse velocity and acceleration of particles in the wire.

A 1.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 62.0 m/s. What are the wavelength and frequency of (a) the fundamental; (b) the second overtone; (c) the fourth harmonic?

A heavy rope 6.00 m long and weighing 29.4 N is attached at one end to a ceiling and hangs vertically. A 0.500-kg mass is suspended from the lower end of the rope. What is the speed of transverse waves on the rope at the (a) bottom of the rope, (b) middle of the rope, and (c) top of the rope? (d) Is the tension in the middle of the rope the average of the tensions at the top and bottom of the rope? Is the wave speed at the middle of the rope the average of the wave speeds at the top and bottom? Explain.

A transverse sine wave with an amplitude of 2.50 mm and a wavelength of 1.80 m travels from left to right along a long, horizontal, stretched string with a speed of 36.0 m/s. Take the origin at the left end of the undisturbed string. At time \(t = 0\) the left end of the string has its maximum upward displacement. (a) What are the frequency, angular frequency, and wave number of the wave? (b) What is the function \(y(x, t)\) that describes the wave? (c) What is \(y(t)\) for a particle at the left end of the string? (d) What is \(y(t)\) for a particle 1.35 m to the right of the origin? (e) What is the maximum magnitude of transverse velocity of any particle of the string? (f) Find the transverse displacement and the transverse velocity of a particle 1.35 m to the right of the origin at time \(t = 0.0625\) s.

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