Animated Solution for Physics - Waves: The displacement $y$ of a wave travelling in the $x$-direction is given by $y=10^{-4}\sin \left(600t-2x+\frac{\pi }{3}\right)$ metre where, $x$ is expressed in metres and $t$ in seconds. The speed of the wave-motion, in ${\text{ms}}^{-1}$ is

Imagine you are standing by a calm pond, and you toss a pebble into the water. The ripples that travel outward are a perfect example of a progressive wave. In physics, we love to capture this beautiful motion using mathematics.

The problem hands us a rather intimidating-looking equation:

Don't let the numbers scare you! This equation is simply the DNA of our wave. It tells us exactly how much any particle in the medium is displaced ($y$) at any given position ($x$) and at any specific time ($t$).

Standard Wave Equation

To unlock the secrets hidden inside this equation, we need our master key. This key is the standard equation of a progressive wave traveling in the positive x-direction.

It is written as:

Let's break down what these symbols mean. $A$ is the amplitude, the maximum height of the wave. $\omega$ is the angular frequency, telling us how fast the particles are oscillating in time. $k$ is the angular wave number, which relates to how the wave is spread out in space. Finally, $\varphi$ is the initial phase, which just tells us where the wave cycle started.

Now, we play a simple game of pattern matching. We place our given equation right next to the standard equation and compare the corresponding parts.

First, look at the term multiplied by time $t$. In our standard equation, this is $\omega$. In the given equation, it is $600$. Therefore, we instantly know:

Next, we look at the term multiplied by position $x$. In the standard equation, this is $k$. In our given equation, it is $2$. So, we have:

We have successfully extracted the angular frequency and the wave number. But the question asks for the wave speed, $v$.

How do we connect these dots? There is a beautiful and fundamental relationship in wave mechanics that links these three quantities. The speed of a wave is simply the ratio of how fast it oscillates in time to how fast it oscillates in space.

Mathematically, this is expressed as:

We are at the final stretch! All that is left is to substitute the values we extracted into our speed formula.

Performing this simple division, we get:

And there we have it! The wave is traveling through the medium at a brisk speed of $300\text{ m/s}$. By simply understanding the anatomy of the wave equation, we were able to extract the exact information needed to find its speed.