Question

According to Laplace correction, the propagation of sound in gas takes place under
A) Isothermal Condition.
B) Isobaric Condition.
C) Isochoric Condition
D) Adiabatic Condition

Answer 1

Hint: The speed of sound like the speed of light varies differently for different mediums. The speed of the sound depends on the medium through which it is travelling. This is similar to light when light travels from a rarer medium to a denser medium.

Complete step-by-step answer:
Newton was the scientist that worked on the propagation of sound waves through the medium of air. The assumption that Newton made was that the propagation of sound is isothermal in nature. The exchange of heat during the compression and the rarefaction would cancel each other out and therefore the temperature remains constant throughout the process.
So, according to the law of Boyle’s:
PV = Constant; …(P = Pressure; V = volume)
If we differentiate each variable then by product rule we get:
\[PdV{\text{ }} + {\text{ }}VdP{\text{ }} = {\text{ }}0\]; …(differentiation of a constant is zero)
\[ \Rightarrow PdV{\text{ }} = {\text{ - }}VdP{\text{ }}\];
Write the above equation in terms of P:
\[ \Rightarrow P{\text{ }} = {\text{ - }}V\dfrac{{dP}}{{dV}}{\text{ }}\];
Divide the RHS in the above equation with V;
\[ \Rightarrow P{\text{ }} = {\text{ - }}\dfrac{{dP}}{{\left( {\dfrac{{dV}}{V}} \right)}}{\text{ }}\];
\[ \Rightarrow P{\text{ }} = B\];
In the above relation $B = {\text{ - }}\dfrac{{dP}}{{\left( {\dfrac{{dV}}{V}} \right)}}{\text{ }}$which is the bulk modulus of air. (Here bulk modulus is described as the capacity of an object to resist compression)
The velocity of the sound is different in different materials. The velocity of sound depends upon the property of the medium.
$v = \sqrt {\dfrac{{{\text{Elastic Property }}}}{{{\text{Inertial Property}}}}} $ ;
The speed of waves through a string is equal to $v = \sqrt {\dfrac{{{F_r}}}{\mu }} $, where ${F_r}$ = The restoring force as tension in the string ${F_T}$ and $\mu $= linear density which is the inertial property. Similarly, speed of sound in the fluid (gas is also a fluid). It is dependent upon the bulk modulus and the density.
$v = \sqrt {\dfrac{B}{\rho }} $ ;
We have P = B.
 $ \Rightarrow v = \sqrt {\dfrac{P}{\rho }} $;
The speed of sound in a gas is not right Laplace corrected this, he assumed that there is no exchange of heat as the time is not enough for the molecules of gasses to exchange heat. So, the temperature is not constant. He correctly postulated that the sound wave in air is an adiabatic process.

Final Answer: Option “D” is correct.

Note: Here we need to explain the origin of sound propagation in different mediums and how the speed of sound is dependent upon the properties of the medium through which the sound is travelling. Then show Laplace correction in the concept.