Question

If the volume elasticity (i.e. bulk modulus) of fresh water and the sea water are assumed to be the same. It is necessary that for the velocity of the sound to be the same,
A. fresh water must be at a higher temperature
B. sea water must be at a higher temperature
C. both must be at the same temperature
D. fresh water must have higher refractive index

Answer 1

Hint: The velocity of sound in sea water and fresh water is equal to the square root of their respective bulk modulus and density ratio. We will use this concept to write the respective equation of fresh water and sea water to find which option is correct.

Complete step by step answer:
It is given that bulk modulus of fresh water and sea water is the same. We have to determine which one of the given options is correct for the velocity of sound to be the same in sea water and fresh water.

Assume:
The density of freshwater is \[{\rho _w}\].
The density of seawater is \[{\rho _s}\].
The temperature of freshwater is \[{T_w}\].
The temperature of sea water is \[{T_s}\].

The velocity of sound in fresh water is expressed as:
\[{v_w} = \sqrt {\dfrac{{{B_w}}}{{{\rho _w}}}} \]……(1)

And, the velocity of sound in seawater is expressed as:
\[{v_s} = \sqrt {\dfrac{{{B_s}}}{{{\rho _s}}}} \]……(2)

Divide equation (1) and equation (2)
\[\begin{array}{c}
\dfrac{{{v_w}}}{{{v_s}}} = \dfrac{{\sqrt {\dfrac{{{B_w}}}{{{\rho _w}}}} }}{{\sqrt {\dfrac{{{B_s}}}{{{\rho _s}}}} }}\\
 = \sqrt {\dfrac{{{B_w}}}{{{\rho _w}}} \times \dfrac{{{\rho _s}}}{{{B_s}}}}
\end{array}\]……(3)

It is given that the bulk modulus of fresh water and sea water are equal.
 \[{B_w} = {B_s}\]

Substitute \[{B_s}\] for \[{B_w}\] in equation (3).
\[\begin{array}{c}
\dfrac{{{v_w}}}{{{v_s}}} = \sqrt {\dfrac{{{B_s}}}{{{\rho _w}}} \times \dfrac{{{\rho _s}}}{{{B_s}}}} \\
 = \sqrt {\dfrac{{{\rho _s}}}{{{\rho _w}}}}
\end{array}\]……(4)

But we know that the density of seawater is greater than the density of freshwater.
\[{\rho _s} > {\rho _w}\]

We also know that if we increase the temperature of fresh water, its density will decrease. So we will decrease this density till it becomes equal to the density of freshwater.

Substitute \[{\rho _w}\] for \[{\rho _s}\] in equation (4).
\[\begin{array}{l}
\dfrac{{{v_w}}}{{{v_s}}} = \sqrt {\dfrac{{{\rho _w}}}{{{\rho _w}}}} \\
{v_w} = {v_s}
\end{array}\]

Therefore, sea water must be at a higher temperature so that the velocity of sound in sea water and fresh water becomes equal

So, the correct answer is “Option B”.

Note:
 We have to be extra careful while dividing the velocities equations in sea water and fresh water.The density of a liquid is inversely proportional to its volume, increasing with temperature, resulting in decreased density.