Problem 8 - University Physics with Modern Physics

Question

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.

Step-by-Step Solution

Answer

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

Step 1: 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 \(6.50 \, \mathrm{mm}\).

Step 2: 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, \( \lambda = 28.0 \, \mathrm{cm} \).

Step 3: 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 \( f = \frac{1}{0.0360 \, \mathrm{s}} = 27.78 \, \mathrm{Hz}\).

Step 4: 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 \(7.7884 \, \mathrm{m/s}\).

Step 5: 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.

Key Concepts

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.