A sound wave of frequency 245 Hz travels with the speed of 300 m s$$^{-1}$$ along the positive x-axis. Each point of the wave moves to and fro through a total distance of 6 cm. What will be the mathematical expression of this travelling wave?

The total distance of to-and-fro motion is 6 cm, so the amplitude is $$A = \frac{6}{2} = 3$$ cm $$= 0.03$$ m.

The angular frequency is $$\omega = 2\pi f = 2\pi \times 245 \approx 1539.4 \approx 1.5 \times 10^3$$ rad s$$^{-1}$$.

The wave number is $$k = \frac{\omega}{v} = \frac{2\pi \times 245}{300} = \frac{490\pi}{300} \approx 5.13 \approx 5.1$$ rad m$$^{-1}$$.

The general equation for a wave travelling in the positive x-direction is $$Y(x,t) = A \sin(kx - \omega t)$$. Substituting the values: $$Y(x,t) = 0.03[\sin 5.1x - (1.5 \times 10^3)t]$$.