The displacement of particle O at any instant is given by y(0,t) = A sin ωt The disturbance is handed over from one particle to next and will reach point P at a distance x from O, a bit later. Therefore, phase of particle at P lags behind the phase of particle at O by
                           where   
        Therefore the displacement of particle at P is,
               
                           

Therefore simple harmonic progressive wave propagating towards positive direction of x-axis is,
y(x,t) = A sin (ωt – kx) If wave propagates towards left direction then replacing x by -x, we get
y(x,t) = A sin (ωt + kx).

To consider the propagation of longitudinal waves in a media, consider nine particles of media on a reference line AB. Let the particles vibrate perpendicular to line AB with amplitude ‘a’ and the wave also propagates perpendicular to AB from left to right. When the wave propagates, the different particles of media vibrate in different phase because it takes some time to transfer the disturbance (momentum and energy) from one particle to next particle. For sake of simplicity, let disturbance take 778 seconds to travel from one particle to next. At t = 0, all the particles are at mean position.
At t = T/8 sec, particle 1 gets displaced by 0.707a distance in right direction, while the disturbance reaches the particle 2.
At t = 2T/8 sec, particle 1 reaches positive extreme position, the particle 2 gets displaced by 0 707a distance in right direction and the disturbance reaches the particle 3.
At f = 3T/8 sec, particle 1 after completing three eighth of vibration, comes back to 0.707a, the particle 2 reaches positive extreme position, particle 3 undergoes the displacement of 0.707a and the disturbance reaches the particle 4.


In this way the disturbance continues and the position of different particles at 4778, 5778, 6778 and 7778 seconds is as shown in figure.
After T seconds, the particle 1 completes one vibration and particle 9 is just at the point to start its first vibration. Thus the particle 1 leads the particle 9 in phase by angle 
If we draw the instantaneous position of different particles and their relative displacement from their mean positions after the particle 1 has completed on vibration, the situation will be as shown in figure.


After one complete vibration, the particles 1,5,9 are at mean position, particles 2,3 and 4 move close towards particle 1, particles 6,7 and 8 move towards particle 9. i.e. there is crowdedness of particles near 1 and 9. Thus the positions of particles 1 and 9 are the positions of condensation. On the other hand, the particles 2,3 and 4 move away towards left from particle 5 and particles 6,7 and 8 move away towards right from particle 5. Thus the position of particle 5 is the position of rarefaction.
Thus there is alternate formation of compression and rarefaction in media and hence the wave propagates in the form of compression and rarefaction. In longitudinal wave, the distance between the two consecutive position of maximum compression or rarefaction is equal to wavelength of wave.