Question

The equation of a transverse wave traveling along a very long string is given by $$ y=(6.0 \mathrm{~cm}) \sin [(2.0 \pi \mathrm{rad} / \mathrm{m}) x+(4.0 \pi \mathrm{rad} / \mathrm{s}) t] $$ Calculate \((a)\) the amplitude, \((b)\) the wavelength, \((c)\) the frequency, \((d)\) the speed, \((e)\) the direction of propagation of the wave, and \((f)\) the maximum transverse speed of a particle in the string.

Short answer

The amplitude is \(6.0 cm\), the wavelength is \(1m\), the frequency is \(2 Hz\), the speed is \(2 m/s\), the wave is moving in the negative x direction, and the maximum transverse speed of a particle in the string is \(24\pi cm/s\).

Step-by-step solution

Finding the amplitude

The amplitude of a wave is the maximum displacement from the equilibrium position. From the given equation, the amplitude, A is clearly indicated in front of the sine function. Here, it is \(6.0 \, cm\).

Finding the wavelength

The wave number \(k\) is given by \(2\pi/\lambda\), where \(\lambda\) is the wavelength. In the equation, it is given as \(2.0\pi\ rad/m\). Therefore, by isolating \(\lambda\) we get \(\lambda = 2\pi / (2\pi rad/m)\) = \(1m\).

Finding the frequency

The angular frequency \(w\) is given by \(2\pi f\), where \(f\) is the frequency. In the equation, it is \(4.0\pi rad/s\). Isolating \(f\), we find \(f = 4.0\pi rad/s / 2\pi\) = \(2 Hz\).

Finding the speed

The speed \(v\) of a wave can be calculated by the formula \(v=\lambda f\). Substituting the values from previous steps, we find \(v = 1m * 2Hz = 2 m/s\).

Finding the direction of propagation of the wave

The direction of the wave propagation is given by the sign in front of the time \(t\). In our case it is positive, so the wave is moving in the negative x direction.

Finding the maximum transverse speed of a particle in the string

The maximum transverse speed is given by the product of the amplitude and the angular frequency, \(Aw\). Thus the maximum transverse speed is \(6cm * 4.0 \pi rad/s = 24\pi cm/s\).

Key concepts

Amplitude

Amplitude is a key characteristic of a wave that defines the maximum extent of a vibration or oscillation measured from the position of equilibrium. In simple terms, it represents how tall or strong the wave appears. For the given transverse wave equation:

• The amplitude is indicated directly as the coefficient in front of the sine function.
• Here, the amplitude is given as 6.0 cm.
• This tells us that the wave reaches up to 6.0 cm above and below its equilibrium position as it propagates.

You could think of amplitude as the 'height' of the wave peaks or valleys, determining the wave's overall energy.

Wavelength

Wavelength is the distance between two consecutive points in phase on the wave, like crest to crest or trough to trough. It is usually denoted by the Greek letter lambda (\( \lambda \)). In the wave equation:

• The wave number \( k \) is related to wavelength as \( k = \frac{2\pi}{\lambda} \).
• Given \( k = 2.0\pi \) rad/m, we can solve for \( \lambda \):
• \( \lambda = \frac{2\pi}{2.0\pi} = 1 \text{ m} \).

Thus, the wavelength here is 1 meter, meaning the wave repeats every 1 meter along the string.

Frequency

Frequency describes how many waves pass a given point in one second. It is measured in hertz (Hz), where 1 Hz equals one cycle per second. For our wave:

• The angular frequency \( \omega \) is related to frequency by \( \omega = 2\pi f \).
• The equation gives an angular frequency of \( 4.0\pi \) rad/s.
• Solving for \( f \), we find \( f = \frac{4.0\pi}{2\pi} = 2 \text{ Hz} \).

This indicates that two complete waves pass by a point every second. Frequency helps us understand the wave's oscillation rate.

Wave Speed

Wave speed is the rate at which the wave propagates through the medium. It can be calculated using the formula \( v = \lambda f \), where \( v \) is the wave speed, \( \lambda \) is the wavelength, and \( f \) is the frequency. In this problem:

• The wavelength \( \lambda \) is 1 m.
• The frequency \( f \) is 2 Hz.
• Thus, the wave speed \( v = 1 \text{ m} \times 2 \text{ Hz} = 2 \text{ m/s} \).

This means the wave travels along the string at a speed of 2 meters per second.

Transverse Wave

A transverse wave is a type of mechanical wave where the motion of the medium is perpendicular to the direction of the wave. This is different from longitudinal waves where the oscillations are parallel to wave propagation. In our transverse wave:

• The displacement of the string particles is vertical while the wave travels horizontally along the x-axis.
• This particular wave equation describes how the peaks and troughs on the string move over time.
• The positive sign in the time component indicates it is traveling in the negative x-direction.

Transverse waves are often visualized as up-and-down movements, like waves on a rope.