91Ó°ÊÓ

1. Distance between molecule A and B is $d=15.0\text{ c³¾or}0.15\text{″¾}$.
2. Distance between molecule C and D is$d=15.0\text{ c³¾or}0.15\text{″¾}$.
3. Velocity of sound, $v=343\text{″¾}/\text{s}$.

Using the fact that the positions of the molecule A and B are given, we can find the wavelength. From the velocity for sound and wavelength, we can find the frequency of the wave using the corresponding relation. The same procedure can be used for molecules C and D.

Formula:

The frequency of the oscillation wave,

$\mathit{f}\mathbf{=}\mathit{v}\mathbf{/}\mathit{λ}$ …(¾±)

The molecule A is at maximum displacement in the negative direction of the axis and molecule B is at the equilibrium position. Hence, the wavelength is given as:

$\begin{array}{c}d=\frac{\mathrm{λ}}{4}\\ \mathrm{λ}=4d\end{array}$

Substitute all the value in the above equation.

role="math" localid="1661574315309" $\begin{array}{l}\mathrm{λ}=4×0.15\text{″¾}\\ \mathrm{λ}=0.6\text{″¾}\end{array}$

Using equation (i), the frequency of the wave for molecules A and B is given as:

$\begin{array}{l}f=\frac{343\text{″¾}/\text{s}}{0.6\text{″¾}}\\ =571.67\text{ H³ú}\\ ~572\text{ H³ú}\end{array}$

Hence, the value of the frequency of the sound wave is $572\text{ H³ú}$.

The molecule C is at maximum displacement in the positive direction of the axis, and Molecule D is at maximum displacement in the negative direction of the axis. From this we can write, the value of wavelength as:

$\begin{array}{c}d=\frac{\mathrm{λ}}{2}\\ \mathrm{λ}=2d\end{array}$

Substitute all the value in the above equation.

role="math" localid="1661574883069" $\begin{array}{l}\mathrm{λ}=2×0.15\text{″¾}\\ \mathrm{λ}=0.3\text{″¾}\end{array}$

Using equation (i), the frequency of the wave for molecules C and D is given as:

$\begin{array}{c}f=\frac{343\text{″¾}/\text{s}}{0.3\text{″¾}}\\ =1143.33\text{ H³ú}\\ ~1144\text{ H³ú}\end{array}$

Hence, the value of the frequency of the sound wave is $1144\text{ H³ú}$.