Animated Solution for Physics - Waves: A travelling wave has the frequency $\nu$ and the particle displacement amplitude $A$. For the wave the particle velocity amplitude is ...... and the particle acceleration amplitude is ......

Imagine standing by a calm pond and dropping a single pebble into the water. You immediately see concentric ripples expanding outward, carrying energy across the surface.

But if you look closely at a leaf floating on the water, you will notice something fascinating: the leaf does not travel outward with the ripples. Instead, it simply bobs up and down in place.

This is the fundamental beauty of wave motion. The wave itself—the disturbance—propagates forward, but the individual particles of the medium merely oscillate about their fixed equilibrium positions.

In this problem, we are invited to explore the kinematics of these oscillating particles. We will derive the exact expressions for their maximum velocity and maximum acceleration, unlocking the deep connection between wave propagation and Simple Harmonic Motion (SHM).

To describe a travelling wave mathematically, we use a function of two variables: position $x$ and time $t$. For a transverse wave propagating in the positive $x$-direction, the vertical displacement $y$ of a particle is given by the classic wave equation:

Here, $A$ represents the displacement amplitude, which is the maximum distance a particle can move from its equilibrium position. The parameter $k$ is the wave number, and $\omega$ is the angular frequency.

Before we proceed, we must connect the given linear frequency $ u$ to the angular frequency $\omega$. The linear frequency tells us how many full cycles occur per second, while the angular frequency represents the rate of phase change in radians per second. The bridge between them is:

This simple relation will be our key tool when we write our final answers.

Now, let us find how fast a particle is moving at any given instant. Because the particle only moves vertically along the $y$-axis while its horizontal position $x$ remains strictly constant, we must find the rate of change of displacement with respect to time at a fixed $x$.

In calculus, this is represented by the partial derivative of $y$ with respect to $t$:

Let us differentiate our wave equation:

Using the chain rule, the derivative of the sine function is cosine, and the derivative of the inner term $(kx-\omega t)$ with respect to $t$ is $-\omega$. Multiplying these together, we get:

This equation describes the instantaneous velocity of any particle in the medium. To find the particle velocity amplitude (the maximum speed), we look at the cosine term. Since the maximum value of a cosine function is $\pm 1$, the maximum magnitude of velocity is:

Substituting our relation $\omega = 2\pi u$, we obtain the first part of our answer:

Next, let us determine the acceleration of the particle. Just as velocity is the rate of change of displacement, acceleration is the rate of change of velocity. Mathematically, this is the derivative of velocity with respect to time:

Let us differentiate our velocity expression:

The derivative of cosine is negative sine, and applying the chain rule again brings out another factor of $-\omega$. Combining these terms, we get:

Notice something extraordinary here: since $y=A\sin (kx-\omega t)$, we can rewrite this as:

This is the defining equation of Simple Harmonic Motion! It proves that every single particle in the medium is executing SHM, with an acceleration that is always directly proportional to its displacement and pointing back toward the equilibrium position.

To find the particle acceleration amplitude (the maximum acceleration), we look at the sine term. Since the maximum value of sine is $\pm 1$, the maximum magnitude of acceleration is:

Substituting $\omega = 2\pi u$ into this expression, we get:

This is the second part of our answer.

By carefully differentiating the wave equation, we have successfully filled in the blanks:

1. The particle velocity amplitude is $2\pi u A$. 2. The particle acceleration amplitude is $4\pi^2 u^2 A$.

These results reveal a profound physical truth: even though a wave can travel thousands of kilometers across an ocean or through the air, the individual particles of the medium never stray far from home. They simply dance in place, their maximum speeds and accelerations governed entirely by the frequency and amplitude of the wave.