91Ó°ÊÓ

The ratio is, $\frac{f_s}{f_i}=0.333$.

The expression for the speed of the sound in terms of frequency and wavelength is given by,

$v=\mathrm{λ}f$…… (i)

Here v is the speed of the sound, $\mathrm{λ}$is the wavelength and f is the frequency

The speed of the sound in terms of bulk modulus and density is given by,

$v=\sqrt{\frac{B}{\mathrm{ÒÏ}}}$…… (ii)

Here B is the bulk modulus and $\mathrm{ÒÏ}$is the density.

The expression for bulk modulus is given by,

localid="1662381277405" $B=-\frac{dp/dV}{V}$…… (iii)

Since, v,$\mathrm{λ}and\mathrm{ÒÏ}$ are not changed appreciably, the frequency ratio becomes,

$$\begin{gathered} \frac{f_s}{f_i}=\frac{v_s/\mathrm{λ}}{v_i/\mathrm{λ}} \\ ⇒\frac{f_s}{f_i}=\frac{v_s}{v_i} \end{gathered}$$

Using the concept of equation (ii) we can write,

$$\begin{gathered} \frac{f_s}{f_i}=\sqrt{\frac{B_s/\mathrm{ÒÏ}}{B_i/\mathrm{ÒÏ}}} \\ ⇒\frac{f_s}{f_i}=\sqrt{\frac{B_s}{B_i}} \end{gathered}$$

Using the concept of equation (iii) we can write,

$$\begin{gathered} \frac{f_s}{f_i}=\sqrt{\frac{{\left(-\frac{dp/dV}{V}\right)}_s}{{\left(-\frac{dp/dV}{V}\right)}_i}} \\ ⇒\frac{f_s}{f_i}=\sqrt{\frac{{\left(dp/dV\right)}_s}{{\left(dp/dV\right)}_i}} \end{gathered}$$

But,
$\frac{f_s}{f_i}=\sqrt{\frac{B_s}{B_i}}$

Thus,

$$\begin{gathered} \frac{f_s}{f_i}=\sqrt{\frac{B_s}{B_i}} \\ =\sqrt{\frac{{\left(dp/dV\right)}_s}{{\left(dp/dV\right)}_i}} \end{gathered}$$

Squaring,

$$\begin{gathered} {\left(\frac{f_s}{f_i}\right)}^2=\frac{B_s}{B_i} \\ =\frac{{\left(dp/dV\right)}_s}{{\left(dp/dV\right)}_i} \end{gathered}$$

Thus,

$\frac{{\left(dp/dV\right)}_s}{{\left(dp/dV\right)}_i}={\left(\frac{f_s}{f_i}\right)}^2$

It is given that the,$\frac{f_s}{f_i}=0.333$,

$∴\frac{{\left(dp/dV\right)}_s}{{\left(dp/dV\right)}_i}=9.00$

Hence, the ratio is,$\frac{{\left(dp/dV\right)}_s}{{\left(dp/dV\right)}_i}=9.00$ .