Problem 42 - University Physics with Modern Physics

Question

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.

Step-by-Step Solution

Answer

The amplitude is 2.22 mm; wavelength is 0.193 m; frequency is 120 Hz; wave speed is 23.2 m/s; wave functions are given. Harmonic can't be determined without medium length.

Step 1: Understanding the Standing Wave Equation

The given wave function is \(y(x, t) = 4.44 \, \text{mm} \, \sin[(32.5 \, \text{rad/m})x] \sin[(754 \, \text{rad/s})t]\). This is a standing wave, which can be expressed as a sum of two traveling waves moving in opposite directions. By using the trigonometric identity: \( \sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)] \), the given wave can be decomposed into two components.

Step 2: Finding the Amplitude of Traveling Waves

The amplitude of the full standing wave is 4.44 mm. For the two traveling waves, the amplitude is half of the standing wave amplitude since it combines two waves with the same amplitude: \( A = \frac{4.44}{2} = 2.22 \, \text{mm}\).

Step 3: Determining the Wavelength

The wavenumber \( k \) in the equation \( k = 32.5 \, \text{rad/m} \) can be used to find the wavelength with \( \lambda = \frac{2\pi}{k} \). Substituting \( k \), we get \( \lambda = \frac{2\pi}{32.5} \approx 0.193 \, \text{m} \).

Step 4: Calculating the Frequency

The angular frequency \( \omega = 754 \, \text{rad/s} \) relates to the frequency by \( f = \frac{\omega}{2\pi} \). Calculating this gives \( f = \frac{754}{2\pi} \approx 120 \, \text{Hz} \).

Step 5: Finding the Wave Speed

The wave speed \( v \) is found using the formula \( v = f \cdot \lambda \). Substitute the known values: \( v = 120 \, \text{Hz} \times 0.193 \, \text{m} \approx 23.2 \, \text{m/s} \).

Step 6: Writing the Wave Functions

The wave functions for the two traveling waves are \( y_1 = 2.22 \, \text{mm} \, \sin[(32.5 \, \text{rad/m})x - (754 \, \text{rad/s})t] \) and \( y_2 = 2.22 \, \text{mm} \, \sin[(32.5 \, \text{rad/m})x + (754 \, \text{rad/s})t] \). These represent the forward and backward traveling components.

Step 7: Determining the Harmonic

The wavelength \( \lambda = 0.193 \, \text{m} \) can suggest the harmonic if we know the length \( L \) of the medium (string, tube, etc.) containing the wave. If \( L \) were provided, \( n \lambda/2 \) such that \( n \) is an integer would help determine the harmonic. Without \( L \), the harmonic cannot be specified.

Key Concepts

Amplitude

Amplitude is a crucial attribute of a wave that defines its maximum displacement from the rest position. For standing waves, the amplitude is typically noted at the antinodes, the points where the wave is at its highest energy. In the given standing wave equation, the amplitude is seen to be 4.44 mm. However, this value represents the combined amplitude of both traveling waves that form the standing wave.

• To find the amplitude of each individual traveling wave, we divide the standing wave's amplitude by two.
• This division occurs because the standing wave results from two identical waves moving in opposite directions, each contributing equally to the resultant standing wave amplitude.
• Thus, each traveling wave has an amplitude of 2.22 mm.

Wavelength

Wavelength helps determine the size of the wave along the medium. It is the distance between consecutive points like crests or troughs. The wavelength defines the spatial period of the wave, dictating how far a wave travels before repeating its cycle.

• For a wave described by the equation \(2\pi k\), the wavenumber "k" is instrumental in deriving the wavelength \(\lambda\).
• Through the relation \(\lambda = \frac{2\pi}{k}\), we find that the given standing wave has a wavenumber of 32.5 rad/m.
• Plugging in this value, we calculate the wavelength to be approximately 0.193 meters.

Frequency

Frequency refers to the number of oscillations that occur in one second. It determines how often particles of the medium vibrate as the wave passes through. In wave functions, the angular frequency \(\omega\) is pivotal in computing the frequency of oscillations.

• Given \(\omega = 754 \, \text{rad/s}\), the conversion from angular frequency to frequency \(\(f\)\) uses \(f = \frac{\omega}{2\pi}\).
• Upon calculating, the frequency of each traveling wave forming the standing wave emerges as approximately 120 Hz, meaning 120 cycles per second.
• This frequency defines how swiftly or slowly the wave pattern repeats over time.

Wave Speed

Wave speed indicates the rate at which energy or information is transmitted through the medium. It is the velocity at which a wave propagates and is influenced by both its frequency and wavelength.

• The fundamental formula \(v = f \lambda\) is used to calculate wave speed.
• Using known values, \(v = 120 \, \text{Hz} \times 0.193 \, \text{m}\), we deduce the wave speed as approximately 23.2 m/s.
• This speed reveals how quickly waves vibrate and spread across the medium.

Wave Functions

Wave functions are equations that describe the motion and properties of waves. In the case of standing waves, two traveling waves superimpose to create a wave pattern that appears stationary. This superposition results from waves moving in opposite directions and, when perfectly aligned, leads to a standing wave.

• Trigonometric identities like \(\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]\) help express a standing wave in terms of these traveling waves.
• The provided wave functions \(y_1 = 2.22 \, \text{mm} \, \sin[(32.5 \, \text{rad/m})x - (754 \, \text{rad/s})t]\) and \(y_2 = 2.22 \, \text{mm} \, \sin[(32.5 \, \text{rad/m})x + (754 \, \text{rad/s})t]\) denote these components.
• This description encapsulates the forward and backward movement that forms the stationary appearance of a standing wave.