Problem 67 - Fundamentals of Physics

Question

Two sinusoidal waves, identical except for phase, travel in the same direction along a string, producing the net wave \(y^{\prime}(x, t)=(3.0 \mathrm{~mm}) \sin (20 x-4.0 t+0.820 \mathrm{rad})\), with \(x\) in meters and \(t\) in seconds. What are (a) the wavelength \(\lambda\) of the two waves. (b) the phase difference between them, and (c) their amplitude \(y_{m}\) ?

Step-by-Step Solution

Answer

(a) \(0.314 \text{ m}\); (b) \(0.820 \text{ rad}\); (c) \(3.0 \text{ mm}\).

Step 1: Understanding the Net Wave Equation

The given net wave equation is \( y'(x, t) = (3.0 \text{ mm}) \sin (20x - 4.0t + 0.820 \text{ rad}) \). This can be compared to the general form of a wave equation \( y(x, t) = y_m \sin(kx - \omega t + \phi) \), where \( k \) is the wave number, \( \omega \) is the angular frequency, \( \phi \) is the phase constant, and \( y_m \) is the amplitude.

Step 2: Extracting the Wave Number

From the equation, \( 20x \) is the term for \( kx \), where \( k \) is the wave number. Therefore, \( k = 20 \).

Step 3: Calculating the Wavelength

The relationship between the wave number \( k \) and the wavelength \( \lambda \) is given by \( k = \frac{2\pi}{\lambda} \). Solve for the wavelength: \[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{20} = \frac{\pi}{10} \approx 0.314 \text{ m} \]

Step 4: Identifying the Phase Constant

The net wave equation includes a phase constant \( 0.820 \text{ rad} \), which is used here for calculation of the phase difference between the two waves.

Step 5: Determining the Phase Difference

Because the waves are identical except for phase, the phase difference \( \Delta \phi \) between them results in the observed phase constant. Thus, \( \Delta \phi = 0.820 \text{ rad} \).

Step 6: Extracting the Wave Amplitude

The wave amplitude \( y_m \) is the coefficient of the sine function in the wave equation, given as \( 3.0 \text{ mm} \).

Key Concepts

Wave Equation

One of the first things we need to do when dealing with wave interference is to understand the wave equation itself. In this exercise, the net wave equation is given as \( y'(x, t) = (3.0 \text{ mm}) \sin (20x - 4.0t + 0.820 \text{ rad}) \).
This equation is crucial as it helps us understand how the wave behaves as it travels along a medium, such as the string mentioned. The wave equation expresses the wave in terms of its amplitude \( y_m \), its wave number \( k \), its angular frequency \( \omega \), and the phase constant \( \phi \).

• Amplitude \( y_m \): Dictates the height of the wave's peak from its equilibrium position.
• Wave Number \( k \): Tells us how many cycles of the wave fit into a unit of space. Here, it is \( 20 \), indicating its spatial frequency.
• Angular Frequency \( \omega \): Describes how quickly the wave oscillates in time. This is reflected as \( 4.0 \text{ rad/s} \) in the equation.
• Phase Constant \( \phi \): Shifts the wave along the length of the string, either forward or backward.

Wavelength Calculation

The wavelength \( \lambda \) is another fundamental property of waves and provides information about the spatial period of the wave.
Wavelength represents the distance over which the wave pattern repeats itself. Calculating the wavelength involves the wave number \( k \), which in this case is found within the wave equation as \( 20 \).
To calculate the wavelength \( \lambda \), we use the relationship:
\[ \lambda = \frac{2\pi}{k} \]
Substitute the given wave number into this equation:
\[ \lambda = \frac{2\pi}{20} = \frac{\pi}{10} \approx 0.314 \text{ meters} \]
This means the wave repeats its pattern approximately every 31.4 cm along the string.

Phase Difference

In wave interference, the phase difference \( \Delta \phi \) between two waves is crucial as it determines their interactive behavior. Phase difference refers to the offset between similar points of two sinusoidal waves.
In the given net wave equation, the phase constant \( 0.820 \text{ rad} \) represents the phase difference. This means despite being identical in every aspect, there is a lead of \( 0.820 \text{ rad} \) in one wave's crests, troughs, and zero points relative to the other.
Such a phase difference can lead to interference effects like constructive and destructive interference. Depending on whether the phase difference results in the waves reaching the same point in phase or out of phase, they can amplify each other or cancel each other out partially or completely.

Wave Amplitude

Amplitude is an essential characteristic of waves as it represents the maximum extent of a wave measured from its equilibrium position. In other words, it describes how "tall" or "strong" a wave is.
In the wave equation \( y'(x, t) = (3.0 \text{ mm}) \sin (20x - 4.0t + 0.820 \text{ rad}) \), the term \( 3.0 \text{ mm} \) is the wave amplitude.
Amplitude is related to the intensity of the wave. Higher amplitudes typically mean more energy is being carried by the wave.
In many physical waves, such as sound or light waves, amplitude affects how loud or bright a wave is perceived.