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Mechanical Properties Of Fluids

Question
CBSEENPH11018726

Discuss the propagation of longitudinal wave in a media. Show that the longitudinal wave propagates in the form of compression and rarefaction.

Solution

Propagation of longitudinal waves in a media:
i) Consider nine particles of media on a reference line AB.
ii) Let the particles vibrate perpendicular to line AB with amplitude ‘a’ and the wave propagates perpendicular to AB from left to right.
iii) 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 the next particle.
iv) Let disturbance take 778 seconds to travel from one particle to next.
v) At t = 0, all the particles are at mean position. 
vi) At t = T/8 sec, particle 1 gets displaced by 0.707a distance in the right direction, while the disturbance reaches the particle 2. 
vii) At t = 2T/8 sec, particle 1 reaches positive extreme position, the particle 2 gets displaced by 0 707a distance in the right direction and the disturbance reaches the particle 3. 
viii) 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. 
The disturbance of the particles is as shown below: 

viii) In this way, the disturbance continues and the position of different particles at 4778, 5778, 6778 and 7778 seconds is as shown in fig. above. 
ix) 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 space space 2 straight pi. 
The longitudinal wave propagates in the form of compression and rarefaction. Let us see how:
i) Draw the instantaneous position of different particles and their relative displacement from their mean positions after the particle 1 has completed on vibration.
The illustration of the disturbance is as shown below: 

ii) After one complete vibration, the particles 1,5,9 are at the mean position; particles 2,3 and 4 move close towards particle 1 and particles 6,7 and 8 move towards particle 9. That is the particles crowd nearby 1 and 9. Thus the positions of particles 1 and 9 are the positions of compression.
iii) 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. 
iv) Thus, there is an alternate formation of compression and rarefaction in media and hence the wave propagates in the form of compression and rarefaction.
v) In the longitudinal wave, the distance between the two consecutive position of maximum compression or rarefaction is equal to the wavelength of wave.