Question

Sound waves with frequency $3000Hz$ and speed $343m/s$ diffract through the rectangular opening of a speaker cabinet and into a large auditorium of length $d=100m.$ The opening, which has a horizontal width of $30.0cm,$ faces a wall $100m$ away (Fig. 36-36). 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.)

Answer 1

Let the first minimum be a distance $y$ from the central axis that is perpendicular to the speaker.
Then $\sin \theta =y/{(D^2+y^2)}^{1/2}=m\lambda /a=\lambda /a(\text{for}m=1)$
Therefore, $y=\frac{D}{\sqrt{(a/\lambda {)}^2-1}}=\frac{D}{\sqrt{{\left(af/v_s\right)}^2-1}}=\frac{100m}{\sqrt{\left[\right(0.300m\left)\right(3000Hz\left)/\right(343m/s){]}^2-1}}=41.2m$