Question

Consider a monochromatic wave on a string, with amplitude \(A\) and wavelength \(\lambda\), traveling in one direction. Find the relationship between the maximum speed of any portion of string, \(v_{\max },\) and the wave speed, \(v\)

Short answer

Answer: The maximum speed of any portion of the string, \(v_{\max}\), is related to the wave speed, \(v\), through the equation: $$ v_{\max} = 2\pi A f $$ where \(A\) is the amplitude of the wave, and \(f\) is the frequency. The frequency can be expressed in terms of the wave speed and wavelength through the equation \(v = f\lambda\).

Step-by-step solution

Understanding the properties of a monochromatic wave on a string

A monochromatic wave on a string is a wave with a single frequency. The wave is characterized by its amplitude, \(A\), and wavelength, \(\lambda\). The amplitude is the maximum displacement of the string from its equilibrium position, and the wavelength is the distance between two consecutive points with the same phase (e.g. two consecutive peaks or troughs) in the wave.

Writing down the wave equation

The wave equation that describes a monochromatic wave on a string traveling in one direction can be written as: $$ y(x,t)=A\sin\left(\frac{2\pi}{\lambda}(x-vt)\right) $$ where \(y(x,t)\) represents the displacement of the string as a function of position, \(x\), and time, \(t\), and \(v\) is the wave speed.

Calculating the velocity of a point on the wave

To find the velocity of a point on the wave, we need to determine the rate of change of the displacement with respect to time. We can do this by taking the partial derivative of the wave equation with respect to time: $$ v_y(x,t) = \frac{\partial y(x,t)}{\partial t} $$ Applying the chain rule and differentiating the wave equation with respect to time: $$ v_y(x,t) = A \frac{2\pi}{\lambda} (-v) \cos\left(\frac{2\pi}{\lambda}(x-vt)\right) $$

Finding the maximum value of the velocity

To find the maximum speed of any portion of the string, we need to find the maximum value of \(v_y(x,t)\). Since the maximum value of the cosine function is \(1\), the maximum value of \(v_y(x,t)\) is obtained when \(\cos\left(\frac{2\pi}{\lambda}(x-vt)\right)=1\). Hence, $$ v_{\max} = A \frac{2\pi}{\lambda}(-v) $$

Expressing the maximum speed in terms of the wave speed

To find the relationship between the maximum speed, \(v_{\max}\), and the wave speed, \(v\), we can rearrange the equation we derived in the previous step: $$ v_{\max} = 2\pi A \frac{-v}{\lambda} $$ Since the frequency, \(f\), is related to the wave speed and wavelength by \(v=f\lambda\), we can rewrite the above equation in terms of frequency: $$ v_{\max} = 2\pi A f $$ This equation shows the relationship between the maximum speed of any portion of the string, \(v_{\max}\), and the wave speed, \(v\).

Key concepts

Monochromatic Wave

A monochromatic wave is a wave with a single, well-defined frequency. It is one of the simplest forms of waves and can be visualized as a sinusoidal pattern.
It consists of repeating cycles of crests (high points) and troughs (low points).

• **Frequency**: The number of cycles that pass a point per unit time.
• **Wavelength**: The length of one complete cycle, typically measured from crest to crest.
• **Amplitude**: The maximum distance a point on the wave moves from its equilibrium rest position.

The dynamics of a monochromatic wave are crucial in understanding phenomena such as light and sound, which behave similarly in various media.

Wave Equation

The wave equation is a mathematical representation that describes the form and movement of a wave. For a monochromatic wave traveling along a string, this can be expressed as:\[ y(x,t) = A \sin\left(\frac{2\pi}{\lambda}(x-vt)\right) \]Here,

• \(y(x,t)\) is the displacement of the wave at any position \(x\) and time \(t\).
• \(A\) is the amplitude of the wave, indicating the maximum displacement.
• \(\lambda\) and \(v\) are the wavelength and wave speed, respectively.

This equation captures how a wave propagates through a medium, with each point on the medium oscillating harmoniously.

Maximum Speed

The maximum speed of any portion of a string due to a wave is crucial for understanding wave dynamics.
It tells us how fast a point can move as the wave passes through. The velocity of a point on a wave can be found by differentiating the wave equation with respect to time.
Using the derivative:\[ v_y(x,t) = A \frac{2\pi}{\lambda} (-v) \cos\left(\frac{2\pi}{\lambda}(x-vt)\right) \]The maximum speed, \(v_{\max}\), occurs when the cosine term is maximized (i.e., reaches 1). This results in the expression:\[ v_{\max} = 2\pi A f \] where \(f\) is the frequency related to wave speed via \(v = f\lambda\). This underscores the link between the maximum speed of the string and fundamental wave characteristics.

String Wave

A string wave is a type of mechanical wave that travels along a string or similar medium. For a wave to form and propagate on a string, it must have tension and elasticity.

• **Tension**: Provides the restoring force that allows the wave to travel.
• **Elasticity**: The string's ability to return to its original shape after being disturbed.

In string waves, each particle of the string moves perpendicular to the direction of the wave.
This transverse movement contrasts with longitudinal waves like sound in air, where particles move parallel to wave propagation. These properties create rich patterns of waves essential in musical instruments and various technological applications.

Wave Speed

Wave speed represents how quickly a wave travels through a medium. For waves on strings, this speed depends on the tension \(T\) in the string and the linear density \(\mu\) (mass per unit length) of the string. The formula for wave speed \(v\) is:\[ v = \sqrt{\frac{T}{\mu}} \] This shows that:

• Higher tension results in faster wave speeds.
• Greater mass per length slows the wave.

Wave speed is crucial for determining how fast information, energy, or signals are transmitted along the string, thereby influencing everything from stringed musical instruments to sophisticated scientific experiments.