Explain the propagation of sound waves in:
$(i)$ air
$(ii)$ solids

(N/A) $(i)$ As the sound wave passes through air, compression and expansion of small regions of the air medium take place periodically, turn by turn, along the direction of propagation of the wave. A small region of the air medium in which air particles tend to move towards each other causes an increase in density $(\Delta \rho)$ and hence an increase in pressure $(\Delta P)$, because according to the ideal gas equation, we have $PV = nRT \Rightarrow P(\frac{M}{\rho}) = nRT \Rightarrow P = (\frac{nRT}{M}) \rho$. At constant temperature, $\Delta P \propto \Delta \rho$.
In such a high-pressure region, where density is increased temporarily, a "Condensation" or "Compression" is said to be formed. It is shown by the symbol $C$ in the figure.
Since pressure is the force exerted normally on a unit area, a restoring force is developed in this region, which is directly proportional to the displacement of a particle from its mean position. Consequently, air particles in this region tend to move outward (parallel to the direction of propagation of the wave) towards the adjoining regions on the left and right, causing compressions in them. When this happens, the density of air in the middle region (from where air particles have moved out) decreases temporarily.
In such a low-pressure region, a "rarefaction" or "expansion" is said to have been formed. It is shown by the symbol $R$ in the figure.
As explained above, each small region of the air medium coming successively in a direction away from the origin of the sound undergoes compression and rarefaction periodically. This is how the disturbance moves away from the source of sound, which indicates the propagation of sound waves, which are mechanical and longitudinal.
$(ii)$ Crystalline solids possess a lattice structure in which atoms or molecules are arranged with a definite geometric periodic pattern. In the normal condition, without any disturbance, all these atoms or molecules are under equilibrium because the forces exerted from the surroundings balance each other.
Now, under this equilibrium condition, if a disturbance is produced in any atom or molecule, it gets displaced from its equilibrium position. Here, this atom or molecule behaves as if it is elastically connected to neighboring atoms or molecules. Hence, a restoring force is developed in such imaginary elastic springs. Such a force becomes responsible for producing very small oscillations in the atoms or molecules, turn by turn, along the direction of motion of the disturbance, which ultimately results in the propagation of the wave in the solid medium from one end to the other.