Step 1: In single-slit diffraction, the condition for the \( m \)-th diffraction minimum is given by:
\[ a \sin \theta_m = m\lambda, \]
where \( a \) is the width of the slit, \( \lambda \) is the wavelength of light, and \( m \) is the diffraction order.
Step 2: For the 2nd order minimum of \( \lambda_1 \) and the 3rd order minimum of \( \lambda_2 \), we can set the angle for both minima equal since they coincide:
\[ 2\lambda_1 = 3\lambda_2 \]
Thus:
\[ \frac{\lambda_1}{\lambda_2} = \frac{3}{2}. \]
Consider the sound wave travelling in ideal gases of $\mathrm{He}, \mathrm{CH}_{4}$, and $\mathrm{CO}_{2}$. All the gases have the same ratio $\frac{\mathrm{P}}{\rho}$, where P is the pressure and $\rho$ is the density. The ratio of the speed of sound through the gases $\mathrm{v}_{\mathrm{He}}: \mathrm{v}_{\mathrm{CH}_{4}}: \mathrm{v}_{\mathrm{CO}_{2}}$ is given by