When an object which execute vibration motion is slightly displaced from its stable position and is left then due to the effect of restitution force it vibrates with its natural (fundamental) frequency. This type of vibration is called the ‘free vibration”. In ideal free vibration the amplitude of the object is constant state of. But in reality when the object excutes free vibrations then it experiences some frictional force in the medium in which it is vibrating. As a result some part of its energy is lost as heat. Hence, the amplitude of the object decreases with time. Those types of vibrations whose amplitude decreases with time are called the “damped vibrations.”
If somehow we can provide the same energy to the vibrating object as to which it has lost; then the object keeps on vibrating with the same (constant) amplitude. This type of vibration is called the “maintained vibrations”.
A vibrating object forces another object (which has the tendency to vibrate) to vibrate. It is also possible if their fundamental frequencies are different. This type of vibration is called the “forced vibrations”. This type of vibrations end very soon because in this situation the second vibrating object does not vibrate with its fundamental frequency but vibrates with the frequency of the first vibrating object.
When the frequency of first vibrating object is equal to the fundamental frequency of the second vibrating object then the amplitude of forced vibrations increases relatively. This is called the “resonance”. Every impulse of frequency force helps the system state to vibrate. Due to this the amplitude of the system keeps on increasing.
As a result of Laplace’s correction we know that the velocity of sound in air is given by the following formula;
\(v = \sqrt{\frac{\gamma P}{\rho}}\ ...(1)\)
where, γ = \(\frac{C_P}{C_v}\) = constant
P = pressure
ρ = denstiy of gas
We also know that for any gas if density and pressure are kept constant then its density is inversely proportional to the temperature.
ρ ∝ \(\frac{1}{T}\) ...(2)
We can say from equations (1) and (2) that in air or in any gas the velocity of sound is directly proportional to the square root its absolute temperature.
v ∝ \(\sqrt{T}\)
or \(\frac{v_1}{v_2} = \sqrt{\frac{T_1}{T_2}}\ ..(3)\)
If at 0°C the velocity of sound is v0 and at t°C temperature it is vt then by equation (3):
It is clear from this equation that if there is 1°C change in temperature then the velocity of sound is changed by 0.61 m/s.
Effect of Density: The velocity of sound in a gas is inversely proportional to the square root of density of the gas.
\(\because \ v = \sqrt{\frac{\gamma \rho}{\rho}}\)
\(\therefore\ \frac{v_1}{v_2} = \sqrt{\frac{\rho _2}{\rho _2}}\)
Effect of pressure: The formula for velocity of sound in a gas is:
\(v =\sqrt{\frac{\gamma \rho}{\rho}}\)
Put \(\rho = \frac{M}{V}\)
\(\therefore\ v = \sqrt{\frac{\gamma P V}{M}}\)
When T = constant
then PV = constant
∴ v = constant
Hence velocity of sound is independent of the change pressure of gas, provided temperature remains constant.
Effect of humidity: The presence of water vapour in air changes in its density. That is why the velocity of sound changes with humidity of air.
∴ ρm < ρd
∴ vm > vd
