Problem 65 - Fundamentals of Physics

Question

The equation of a transverse wave traveling along a string is $$ y=(2.0 \mathrm{~mm}) \sin \left[\left(20 \mathrm{~m}^{-1}\right) x-\left(600 \mathrm{~s}^{-1}\right) t\right] . $$ Find the (a) amplitude, (b) frequency, (c) velocity (including sign), and (d) wavelength of the wave. (e) Find the maximum transverse speed of a particle in the string.

Step-by-Step Solution

Answer

Amplitude: 2.0 mm Frequency: 95.5 Hz Velocity: +30 m/s Wavelength: 0.314 m Maximum transverse speed: 1.2 m/s

Step 1: Identify the Wave Equation Components

The given wave equation is \( y = (2.0 \text{ mm}) \sin\left[ (20 \text{ m}^{-1}) x - (600 \text{ s}^{-1}) t \right] \).The general form of the wave equation is \( y = A \sin(kx - \omega t) \).Where \( A \) is the amplitude, \( k \) is the wave number, and \( \omega \) is the angular frequency.

Step 1: Determine Amplitude

The amplitude \( A \) is the coefficient in front of the sine function.Here, \( A = 2.0 \text{ mm} \).

Step 2: Determine Frequency

Frequency \( f \) is calculated from the angular frequency \( \omega \).Given \( \omega = 600 \text{ s}^{-1} \), use the formula \( f = \frac{\omega}{2\pi} \).\[ f = \frac{600}{2\pi} \approx 95.5 \text{ Hz} \]

Step 3: Determine Wave Velocity

Wave velocity \( v \) can be found using \( v = \frac{\omega}{k} \).Given \( k = 20 \text{ m}^{-1} \) and \( \omega = 600 \text{ s}^{-1} \),\[ v = \frac{600}{20} = 30 \text{ m/s} \].The negative sign in the equation indicates the wave is traveling in the positive x-direction, so velocity is \(+30 \text{ m/s}\).

Step 4: Determine Wavelength

Wavelength \( \lambda \) is related to the wave number \( k \) by the formula \( \lambda = \frac{2\pi}{k} \).\[ \lambda = \frac{2\pi}{20} \approx 0.314 \text{ m} \]

Step 5: Find Maximum Transverse Speed

The maximum transverse speed \( v_{\text{max}} \) of a particle in the string is given by the maximum rate of change of the displacement, which is \( v_{\text{max}} = A \omega \).Convert amplitude to meters: \( A = 2.0 \times 10^{-3} \text{ m} \).\[ v_{\text{max}} = 2.0 \times 10^{-3} \times 600 = 1.2 \text{ m/s} \]

Key Concepts

Amplitude

In the context of wave motion, the amplitude refers to the height of the wave's crest or depth of its trough from the central axis. It indicates the maximum extent of the wave's oscillation.
This is crucial as it often relates to the energy carried by the wave; a larger amplitude means more energy.

• The amplitude is always a positive quantity.
• In our example, the given wave equation is in the form of a sine wave: \( y = A \sin(kx - \omega t) \).
• Here, the amplitude \( A \) is directly given ahead of the sine function as \( 2.0 \text{ mm} \).
• This value means the displacement of the wave from its mean position reaches a maximum height of 2 mm.

Understanding amplitude helps us grasp how waves interact with their surroundings, such as sound waves influencing air particles or ocean waves crashing with great force.

Frequency

Frequency defines how many cycles of the wave pass a given point per unit time, usually counted in Hertz (Hz), where 1 Hz equals one cycle per second. It is an indicator of the wave's oscillation rate.
The given equation involves angular frequency \( \omega \), noted as \( 600 \text{ s}^{-1} \).

• The relation between angular frequency and frequency is \( f = \frac{\omega}{2\pi} \).
• Using this relationship, we find the frequency to be approximately \( 95.5 \text{ Hz} \).

Higher frequencies correspond to more cycles in a given timeframe and are associated with higher pitches in sound or more rapid oscillations in light.
Comprehending frequency is pivotal to understanding wave behavior in practical applications, from tuning instruments to assessing signal transmissions.

Velocity and Wavelength

Velocity and wavelength are intertwined properties that dictate the wave's movement and length in space.
Wave velocity signifies how quickly a wave propagates through a medium.

• To find velocity, we use \( v = \frac{\omega}{k} \), resulting in \( 30 \text{ m/s} \) in our exercise.
• The positive velocity indicates motion in the positive x-direction.
• A constant velocity means consistent wave travel across the medium.

Wavelength, denoted by \( \lambda \), is the distance between two consecutive crests or troughs in a wave.
For determination, the relation is \( \lambda = \frac{2\pi}{k} \).

• In this wave, the wavelength calculates to \( 0.314 \text{ m} \).

Both velocity and wavelength are key to understanding the spatial characteristics and speed of wave travel, impacting areas such as acoustics, optics, and oceanography.

Transverse Wave

Transverse waves feature wave motion where particles move perpendicularly to the direction of energy transfer.
These are commonly visualized with up-and-down undulations, like ripples on a water surface or light waves traveling through a vacuum.

• In our equation, the displacement involves a sine function culmination in a transverse wave representation.
• Since the wave equation is given in the form \( y = A \sin(kx - \omega t) \), this is a classic indicator of a transverse wave.

Understanding transverse waves helps in applying this knowledge to real-world contexts like seismic wave studies, electromagnetic waves, and design models in engineering.
Grasping their qualities can assist in interpreting how energy is transported via varying materials, from solid earth to the vast reaches of space.