Animated Solution for Physics - Waves: A travelling harmonic wave is represented by the equation $y(x,t)=10^{-3}\sin (50t+2x)$, where $x$ and $y$ are in metre and $t$ is in second. Which of the following is a correct statement about the wave?

Imagine you are standing by the ocean, watching a perfect, continuous wave roll in. In physics, we describe such a wave using a mathematical function that depends on both space ($x$) and time ($t$). The problem gives us the equation of a travelling harmonic wave:

Our goal is to determine two things: how fast the wave is moving (its speed) and in which direction it is travelling.

To find the speed of the wave, we need to track a specific point on it—say, the very top of a crest. As the wave moves, this crest moves with it. For this specific point, the value inside the sine function, which we call the phase, must remain constant.

Let's write this down mathematically:

This equation is the secret key to unlocking the wave's velocity. It tells us how the position $x$ of our crest must change as time $t$ ticks forward to keep the phase constant.

Velocity is simply the rate of change of position with respect to time, or $\frac{dx}{dt}$. To find this, we differentiate our constant phase equation with respect to time $t$:

Remember, the derivative of a constant is zero. Applying the chain rule, we get:

Now, we just solve for $\frac{dx}{dt}$:

We found that the velocity $v=-25\text{ m/s}$. What does this mean physically?

The magnitude, $25\text{ m/s}$, is the speed of the wave. The negative sign is crucial—it tells us the direction. It means that as time increases, the position $x$ of our crest must decrease to keep the phase constant. Therefore, the wave is propagating along the negative X-axis.

Once you understand the physics, you can use a powerful shortcut. The general equation for a harmonic wave is:

By comparing our given equation $y=10^{-3}\sin (50t+2x)$ with the standard form, we can immediately identify the angular frequency $\omega =50\text{ rad/s}$ and the wave number $k=2{\text{ m}}^{-1}$.

The wave speed $v$ is given by the ratio of $\omega$ to $k$:

Furthermore, the sign between the $\omega t$ and $kx$ terms dictates the direction. A positive sign ($+$) means the wave travels in the negative $x$-direction, while a negative sign ($-$) means it travels in the positive $x$-direction. Since we have a plus sign, the wave is moving along the negative X-axis. Both methods lead us beautifully to the same conclusion!