Question

Consider a 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, \(v_{\text {max }}\), and the wave speed, \(v\)

Short answer

Answer: The relationship between the maximum speed of any portion of the string (v_max) and the wave speed (v) is given by the equation \(v_{\text{max}} = Av\frac{2\pi}{\lambda}\), where A is the amplitude and \(\lambda\) is the wavelength.

Step-by-step solution

Write down the equation for the wave on the string

We will use the equation for a sinusoidal wave on a string: \(y(x, t) = A \sin(\omega t - kx)\), where \(\omega\) is the angular frequency, \(k\) is the wave number, and \(y(x, t)\) is the displacement of the string at position \(x\) and time \(t\).

Calculate the wave number \(k\) and angular frequency \(\omega\)

The wave number \(k\) is related to the wavelength \(\lambda\) by the equation \(k = \frac{2\pi}{\lambda}\). The wave speed \(v\) is related to the angular frequency \(\omega\) and the wave number \(k\) by the equation \(v = \frac{\omega}{k}\). So, we can write \(\omega\) in terms of these variables: \(\omega = vk\).

Find the velocity of any portion of the string

The velocity of any portion of the string at position \(x\) and time \(t\) can be found by taking the time derivative of the displacement \(y(x, t)\). Using the chain rule, we have: \(v(x, t) = \frac{dy(x, t)}{dt} = A \frac{d}{dt}\sin(\omega t - kx) = A\omega \cos(\omega t - kx)\)

Find the maximum velocity of any portion of the string

The maximum velocity of any portion of the string, \(v_{\text{max}}\), occurs when the cosine term in the equation for the velocity is at its maximum value, which is 1. Thus: \(v_{\text{max}} = A\omega\)

Substitute the expression for angular frequency \(\omega\)

As we already found the expression for the angular frequency \(\omega\) in terms of the wave speed \(v\), we can substitute this expression into the equation for the maximum velocity: \(v_{\text{max}} = A(vk)\)

Simplify the equation for the relationship between \(v_{\text{max}}\) and \(v\)

Now, we only need to simplify this equation to find the relationship between the maximum velocity of any portion of the string and the wave speed: \(v_{\text{max}} = Av\frac{2\pi}{\lambda}\) Thus, the relationship between the maximum speed of any portion of the string and the wave speed is given by the equation \(v_{\text{max}} = Av\frac{2\pi}{\lambda}\).

Key concepts

Wave Amplitude

The amplitude of a wave is the maximum displacement of the wave from its rest position. In other words, it's the height of the wave's crest or the depth of its trough, in relation to the equilibrium position. When we discuss waves on a string, the amplitude, typically denoted by the letter \(A\), represents how far the string moves from its undisturbed position.

Amplitude is a critical factor because it's directly related to the energy carried by the wave—greater amplitudes mean more energy. When calculating wave phenomena, such as velocity or energy, you'll often find amplitude playing a key role in the equations.

Wavelength

Wavelength, denoted by the Greek letter lambda \(\lambda\), is the distance between two consecutive points that are in phase on the wave—typically measured from crest to crest or trough to trough. It's one of the fundamental characteristics of any wave, including waves on a string, sound waves, and electromagnetic waves.

Understanding the wavelength is crucial in solving many wave-related problems because it's directly linked to the wave's frequency and speed. The wavelength is inversely proportional to the frequency—the higher the frequency, the shorter the wavelength, assuming the wave speed remains constant.

Maximum Velocity

The maximum velocity on a string wave, often denoted by \(v_{\text{max}}\), is the peak speed achieved by any point on the string as the wave passes through. It is not to be confused with the wave speed itself, which is the speed at which the wave propagates through the medium.

The significance of finding the maximum velocity lies in understanding the dynamics of the wave's movement. This value can provide insights into the wave's kinetic energy and is essential for calculations in various physical contexts, such as material stresses on the string and energy transfer.

Angular Frequency

Angular frequency, denoted by \(\omega\), is a measure of how quickly the wave oscillates in terms of radians per second. It's related to the wave's frequency (\(f\)) by the equation \(\omega = 2\pi f\). Angular frequency is a pivotal part of wave equations because it connects the temporal aspect of wave motion to its spatial characteristics through the wave number (\(k\)).

In the context of a string wave, the angular frequency helps us determine how fast the string's particles vibrate as the wave travels along the string. Higher angular frequency indicates quicker oscillations, which, in turn, affects the maximum velocity of the wave's particles.