What is the difference between wave velocity and particle velocity?

Wave velocity:

Wave velocity is the velocity with which progressive wave front travels forward. It is constant throughout the media if the media is homogenous and given by,

                      v = v λ.

Particle velocity:

Particle velocity is the velocity with which particles vibrate during the wave propagation. The particle velocity is changing and given by, 

               v = rω cos(ωt – kx).

Velocity of a wave is given by,

                     

∴                                     ...(1) 

Also,  

                              

∴             ...(2) 

From (1) and (2), 

                    

                

                            

Increase in wavelength is, 

           

Given,

Frequency of the wave, v = 400HZ

Velocity of the wave, v = 340m/s 

∴ Distance between two nearest points which are out by phase 120^o is given by the wavelength of the wave, 

                   

Given,

 

Therefore velocity of sound in sea water, 

 

Let d be the depth of the sea.

As the echo is heard after 1.8s, therefore the wave travels 2d distance in water in time 1.8s. 

i.e.