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A certain transverse wave is described by $$y(x, t) = (6.50 \, \mathrm{mm}) \mathrm{cos} \, 2\pi \Big( \frac{x}{28.0 \mathrm{cm}} - \frac{t}{0.0360 \, \mathrm{s}} \Big)$$ Determine the wave's (a) amplitude; (b) wavelength; (c) frequency; (d) speed of propagation; (e) direction of propagation.

Short Answer

Expert verified
(a) 6.50 mm; (b) 28.0 cm; (c) 27.78 Hz; (d) 778.84 cm/s; (e) positive x-direction.

Step by step solution

01

Identify the Amplitude

The amplitude of a wave is the maximum displacement from its equilibrium position. In the given wave equation \( y(x, t) = (6.50 \, \mathrm{mm}) \cos \left( 2\pi \left( \frac{x}{28.0 \, \mathrm{cm}} - \frac{t}{0.0360 \, \mathrm{s}} \right) \right) \), the amplitude is the coefficient in front of the cosine function. Here, it is \(6.50 \, \mathrm{mm}\).
02

Determine the Wavelength

The wavelength \( \lambda \) is given by the reciprocal of the coefficient of \(x\) inside the cosine function multiplied by \(2\pi\). Here, the coefficient is \( \frac{1}{28.0 \, \mathrm{cm}} \). Therefore, \( \lambda = 28.0 \, \mathrm{cm} \).
03

Calculate the Frequency

Frequency \( f \) is the reciprocal of the coefficient of \( t \) inside the cosine function multiplied by \(2\pi\). Here, the coefficient is \( \frac{1}{0.0360 \, \mathrm{s}} \). Thus, frequency is \( f = \frac{1}{0.0360 \, \mathrm{s}} = 27.78 \, \mathrm{Hz}\).
04

Calculate the Speed of Propagation

The speed \( v \) of the wave is calculated using the formula \( v = f \lambda \). Substituting the values from previous steps: \( v = 27.78 \, \mathrm{Hz} \times 28.0 \, \mathrm{cm} = 778.84 \, \mathrm{cm/s}\) or equivalently \(7.7884 \, \mathrm{m/s}\).
05

Determine the Direction of Propagation

The direction of wave propagation depends on the sign of the term in the cosine function. The term is \( \frac{x}{28.0 \, \mathrm{cm}} - \frac{t}{0.0360 \, \mathrm{s}} \), where \(x\) is positive and \(t\) is negative, indicating that the wave is traveling in the positive \(x\)-direction.

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Key Concepts

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

Amplitude
Amplitude is a fundamental property of waves, defining their maximum displacement from rest. In a wave, amplitude depicts how far the particles of the medium move from their equilibrium position due to the wave's energy. It essentially measures how 'strong' or 'intense' the wave is.

For the given wave equation, \(y(x, t) = (6.50 \, \mathrm{mm}) \cos \left( 2\pi \left( \frac{x}{28.0 \, \mathrm{cm}} - \frac{t}{0.0360 \, \mathrm{s}} \right) \right)\), the amplitude is represented by the coefficient in front of the cosine function. Hence, in this example, the amplitude is \(6.50 \, \mathrm{mm}\).

Amplitude is crucial for understanding wave energy; the greater the amplitude, the more energy the wave carries. It is one of the key parameters when analyzing wave behavior in physics and engineering.
Wavelength
Wavelength is the distance between consecutive identical points, like crests, in a wave pattern. It directly relates to the wave's spatial periodicity, defining the space over which the wave's shape repeats.

In our wave equation, the term \(\frac{x}{28.0 \, \mathrm{cm}}\) reveals the spatial component. The wavelength \(\lambda\) is found by taking the reciprocal of this coefficient, then multiplying by \(2\pi\). This simplifies directly as \(\lambda = 28.0 \, \mathrm{cm}\).

Wavelength plays a vital role in wave interactions and behaviors such as interference and diffraction. Understanding the wavelength helps in visualizing how waves propagate through various mediums.
Frequency
Frequency, denoted as \(f\), measures how often the wave oscillates in a unit of time, usually per second (Hertz, \(\mathrm{Hz}\)). It tells us how fast the wave cycles back and forth.

From the wave equation, the frequency is linked to the reciprocal of the coefficient of \(t\). In this case, it is \(\frac{1}{0.0360 \, \mathrm{s}}\), resulting in \(f = 27.78 \, \mathrm{Hz}\).

Frequency is a key characteristic in describing waves, particularly in contexts like sound, where pitch is related to frequency, or in electromagnetic waves where it relates to energy levels. Understanding frequency helps to quantify wave properties in scientific and practical applications.
Wave Speed
Wave speed is how fast a wave moves through a medium. It is calculated by the formula \(v = f \lambda\), meaning the wave speed is the product of frequency and wavelength. This relationship makes sense because wave speed combines how quickly the wave cycles (frequency) and how far each cycle travels (wavelength).

For this wave, substituting the derived values for frequency and wavelength gives us \(v = 27.78 \, \mathrm{Hz} \times 28.0 \, \mathrm{cm} = 778.84 \, \mathrm{cm/s}\), or \(7.7884 \, \mathrm{m/s}\).

Understanding wave speed allows us to predict how waves will move through different environments, a critical concept in engineering and physics fields like telecommunications and acoustics.
Wave Direction
The direction of wave propagation illustrates where the wave is traveling. By examining the terms in the wave equation, \( \frac{x}{28.0 \, \mathrm{cm}} - \frac{t}{0.0360 \, \mathrm{s}} \), the signs give insights into direction.

In this case, the term \( x \) is positive while \( t \) is negative, reflecting that the wave travels in the positive \(x\)-direction. This positive direction indicates that as time progresses, the points \( x \) where the wave reaches certain phases move positively along the \(x\)-axis.

Identifying the direction is vital for understanding wave behaviors and planning deployments in technologies such as radar, sonar, and wireless communication.

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Most popular questions from this chapter

A light wire is tightly stretched with tension F. Transverse traveling waves of amplitude \(A\) and wavelength \(\lambda_1\) carry average power \(P_{av,1} = 0.400\) W. If the wavelength of the waves is doubled, so \(\lambda_2 = 2\lambda_1\), while the tension \(F\) and amplitude \(A\) are not altered, what then is the average power \(P_{av,2}\) carried by the waves?

A piano wire with mass 3.00 g and length 80.0 cm is stretched with a tension of 25.0 N. A wave with frequency 120.0 Hz and amplitude 1.6 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?

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 1750-N irregular beam is hanging horizontally by its ends from the ceiling by two vertical wires (\(A\) and \(B\)), each 1.25 m long and weighing 0.290 N. The center of gravity of this beam is one-third of the way along the beam from the end where wire A is attached. If you pluck both strings at the same time at the beam, what is the time delay between the arrival of the two pulses at the ceiling? Which pulse arrives first? (Ignore the effect of the weight of the wires on the tension in the wires.)

A 0.800-m-long string with linear mass density \(\mu = 7.50\) g/m is stretched between two supports. The string has tension \(F\) and a standing-wave pattern (not the fundamental) of frequency 624 Hz. With the same tension, the next higher standing-wave frequency is 780 Hz. (a) What are the frequency and wavelength of the fundamental standing wave for this string? (b) What is the value of \(F\)?

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