Question

Sound waves with frequency $\mathbf{3000}\mathbf{}\mathbf{Hz}$and speed $\mathbf{343}\mathbf{}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$diffract through the rectangular opening of a speaker cabinet and into a large auditorium of length $\mathbf{d}\mathbf{}\mathbf{=}\mathbf{}\mathbf{100}\mathbf{}\mathbf{m}$. The opening, which has a horizontal width of $\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{cm}$, faces a wall $\mathbf{100}\mathbf{}\mathbf{m}$away ($\mathit{F}\mathit{i}\mathit{g}\mathbf{.}\mathbf{}\mathbf{36}\mathbf{-}\mathbf{36}$). Along that wall, how far from the central axis will a listener be at the first diffraction minimum and thus have difficulty hearing the sound? (Neglect reflections.)

Short answer

The distance between first order minima and central minima is$41.2\text{m}$.

Step-by-step solution

Identification of given data

The frequency of the sound wave is $f=3000\text{Hz}$

The speed of the sound wave is $v=343\text{m}/\text{s}$

The length of auditorium is $D=100\text{m}$.

The width of opening is $d=30\text{cm}$.

The order of first minima is $m=1$.

Concept used

The diffraction is the phenomenon in which the light bends on the edges of the slit and pattern of light is formed on a screen

Determination of wavelength of sound wave

The wavelength of sound wave is given as:

$v=f·\lambda$

Substitute all the values in equation.

$$\begin{gathered} 343\text{m}/\text{s}=\left(3000\text{Hz}\right)\lambda \\ \lambda =0.1143\text{m} \end{gathered}$$

Determination of distance of first diffraction minima from central diffraction minima

The distance between first order minima and central minima is given as:

$w=\frac{D}{\sqrt{{\left(\frac{d}{\lambda }\right)}^2-1}}$

Substitute all the values in equation.

$$\begin{gathered} w=\frac{100\text{m}}{\sqrt{{\left(\frac{\left(30\text{cm}\right)\left(\frac{1\text{m}}{100\text{cm}}\right)}{\left(0.1143\text{m}\right)}\right)}^2-1}} \\ w=41.2\text{m} \end{gathered}$$

Therefore, the distance between first order minima and central minima is $41.2\text{m}$.