# Simple harmonic waves

When a simple harmonic progressive wave is travelling through medium,then each succeeding particle lags in phase before the preceding particle.Can anyone expain how does it lag? Thanks…

Let us imagine that the disturbance that causes the wave is located at position x=0. If the disturbance is oscillatory then it could be of the form $\displaystyle y(x=0,t)=A\sin\omega t$, in which y denotes the displacement of the particle of the medium, $\omega$ is the angular frequency of oscillation. The quantity $\displaystyle\omega t$ is called the phase of the vibration at x=0. The disturbance at x=0 propagates through the medium with a speed v, say. So the displacement of the particle of the medium located at x=x at time t is the displacement at x=0 at time $\displaystyle t-\frac{x}{v}$ which is $\displaystyle A\sin\omega(t-\frac{x}{v})$. Thus the phase of the vibration at x=x at time t is $\displaystyle \omega (t-\frac{x}{v})$. But the phase of the vibration at x=0 at time t is $\displaystyle\omega t$. So the phase at x=x lags the phase at x=0 by an amount $\displaystyle\frac{\omega x}{v}$.