Animated Solution for Physics - Waves: A travelling wave in a stretched string is described by the equation ; $y=A\sin (kx-\omega t)$ The maximum particle velocity is

When a wave travels through a medium, such as a stretched string, it carries energy from one point to another.

However, a common point of confusion for students is the distinction between the speed of the wave itself and the speed of the individual particles of the medium.

In this article, we will dissect the mathematics of a progressive wave to understand how these two velocities are defined, how they relate to each other, and how to find the maximum speed of a vibrating particle.

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A progressive transverse wave traveling along the positive $x$-direction is mathematically represented by the wave function:

Here, $y$ represents the vertical displacement of a particle located at a horizontal position $x$ at any time $t$.

The parameter $A$ is the amplitude of the wave, representing the maximum displacement of any particle from its equilibrium position.

The term $k$ is the wave number ($k=\frac{2\pi }{\lambda }$), and $\omega$ is the angular frequency ($\omega =2\pi f$).

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Particle Velocity

It is vital to distinguish between these two physical quantities:

1. Wave Velocity ($v_w$): This is the speed with which the wave disturbance (or phase) travels through the medium. It depends solely on the properties of the medium (like tension and linear mass density of the string) and is given by:

2. Particle Velocity ($v_p$): This is the velocity of the physical particles of the string as they oscillate vertically about their mean positions. Unlike the wave velocity, which is constant, the particle velocity is continuously changing as the particles execute Simple Harmonic Motion (SHM).

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To find the velocity of a particle at a specific position $x$, we take the partial derivative of the displacement $y$ with respect to time $t$:

We use a partial derivative because the displacement $y$ depends on both space $x$ and time $t$. By taking the partial derivative with respect to $t$, we are effectively "freezing" the position $x$ to look at one specific particle.

Let's differentiate:

Using the chain rule of differentiation:

Since $x$ is treated as a constant, the derivative of $kx$ with respect to $t$ is $0$. The derivative of $-\omega t$ is $-\omega$. Therefore:

This is the general equation for the velocity of any particle in the medium at any position $x$ and time $t$.

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The particle velocity varies sinusoidally because of the $\cos (kx-\omega t)$ term.

To find the maximum particle velocity ($v_{p,\max }$), we look for the maximum magnitude of this expression.

Since the maximum absolute value of a cosine function is $1$:

Substituting this back into our velocity equation gives:

Thus, the maximum speed of any particle on the string is simply the product of the amplitude and the angular frequency.

This matches Option (a).

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Let's look at the ratio of the maximum particle velocity to the wave velocity:

What does $kA$ represent physically?

If we differentiate the wave equation with respect to $x$, we get the slope of the wave profile at any point:

The maximum slope of the wave is therefore $kA$.

Thus, we arrive at a fundamental relation in wave mechanics:

This elegant connection helps us visualize how the physical motion of the medium's particles directly dictates the geometric shape of the wave traveling through it.