Animated Solution for Physics - Waves: The amplitude of wave disturbance propagating in the positive x-direction is given by $y=\frac{1}{1+x^2}$ at time $t=0$ and $y=\frac{1}{1+(x-2{)}^2}$ at $t=1\text{ s}$, where $x$ and $y$ are in metre. The shape of wave does not change during the propagation. The velocity of the wave will be ...... m/s.

The journey of a wave is one of the most mesmerizing phenomena in physics. Imagine a ripple traveling across a pond or a pulse moving down a stretched string. In this problem, we are given the mathematical snapshot of a wave pulse at two different moments in time, and our mission is to uncover its velocity.

Let's dive into the elegant mathematics that describes this motion!

When a wave propagates through a medium without losing its energy or changing its shape, it is called a non-dispersive traveling wave.

The fundamental principle here is that the displacement $y$ of any particle in the medium depends on both its position $x$ and the time $t$. For a wave moving in the positive x-direction, this relationship is beautifully captured by the general wave equation:

Here, $v$ represents the wave velocity. The negative sign in the argument $(x-vt)$ is crucial—it dictates that as time $t$ increases, the position $x$ must also increase to maintain the same phase or shape. In other words, the wave moves to the right!

The problem provides us with the shape of the wave at the very beginning, when the stopwatch reads $t=0$.

At this instant, the wave equation simplifies to:

This is a classic bell-shaped curve, often called a Lorentzian function. Its peak is perfectly centered at the origin, $x=0$. Because this is the shape of our wave at $t=0$, we can confidently say that our base function $f(x)$ is exactly $\frac{1}{1+x^2}$.

Now, let's let the clock tick. According to our general wave equation, at any arbitrary time $t$, the shape of the wave is found by replacing every $x$ in our base function with $(x-vt)$.

So, the dynamic equation for our traveling wave becomes:

This equation is a mathematical time machine. It tells us exactly what the wave looks like at any given second.

The problem gives us a second snapshot. We are told that exactly one second later, at $t=1\text{ s}$, the wave looks like this:

Notice how the peak of the wave has shifted from $x=0$ to $x=2$.

Let's use our dynamic wave equation to see what it predicts for $t=1\text{ s}$. By substituting $t=1$ into $y(x,t)$, we get:

Now, we have two expressions for the wave at $t=1\text{ s}$—one given by the problem, and one derived from the physics of traveling waves. For the universe to make sense, these two expressions must be identical!

Let's set them equal to each other:

By comparing the denominators, it is immediately obvious that the arguments must match:

The $x$ terms gracefully cancel out, leaving us with the final, elegant result:

The wave is traveling at a crisp speed of 2 meters per second. By simply observing how far the mathematical function shifted over a specific time interval, we unlocked the dynamic secret of the wave. Keep visualizing the math, and the physics will always reveal itself!