Animated Solution for Physics - Waves: A sound wave of frequency $245\text{ Hz}$ travels with the speed of $300{\text{ ms}}^{-1}$ along the positive X-axis. Each point of the wave moves to and fro through a total distance of $6\text{ cm}$. What will be the mathematical expression of this travelling wave?

Imagine a sound wave travelling smoothly along the positive X-axis. As the wave moves forward, any single particle in the medium oscillates back and forth. We are given the frequency $f=245\text{ Hz}$, the wave speed $v=300\text{ m/s}$, and the total distance of this to-and-fro motion which is $6\text{ cm}$. Let's note down these values as they form the foundation of our mathematical model.

To find the mathematical expression, we need the standard equation of a travelling wave. Since it is moving in the positive X-direction, the equation is:

Our goal now is to find the three critical constants: the amplitude $A$, the wave number $k$, and the angular frequency $\omega$.

Let's start with the amplitude. The problem states that each point moves to and fro through a total distance of $6\text{ cm}$. This total distance represents the full span of oscillation, from the positive extreme to the negative extreme, which is exactly twice the amplitude ($2A$).

So, the amplitude $A$ is half of $6\text{ cm}$, which is $3\text{ cm}$. Converting this to standard SI units, we get:

Next, let's calculate the angular frequency, $\omega$. We know the formula:

Substituting the given frequency of $245\text{ Hz}$, we get $\omega =2\pi (245)$. Calculating this gives us approximately:

Now for the wave number, $k$. The wave speed $v$ is the ratio of angular frequency $\omega$ to the wave number $k$. Rearranging this, $k=\frac{\omega }{v}$. Let's plug in our values:

Finally, let's assemble our wave equation. Substituting the values of $A$, $k$, and $\omega$ back into our general equation, we get:

Looking at our options, this perfectly matches option (d), where $1.54\times 10^3$ is approximated to $1.5\times 10^3$.

Always pay close attention to the direction of propagation! If the wave was travelling in the negative X-direction instead, the minus sign inside the sine function would become a plus, giving us $(kx+\omega t)$.