Question

Both drawings show the same square, each of which has a side of length L = 0.86 m. An observer O is stationed at one corner of each square. Two loudspeakers are located at corners of the square, as in either drawing 1 or drawing 2. The speakers produce the same single-frequency tone in either drawing and are in phase. The speed of sound is 343 m/s. Find the single smallest frequency that will produce both constructive interference in drawing 1 and destructive interference in drawing 2.

Answer 1

Fig.1: path difference Δ₁=0, since distances between each loudspeaker and point O are L
For constructive interference Δ₁=2kλ/2=kλ (k(min)=0) => Δ₁=0.
Fig.2:
path difference Δ₂=L√2 – L= L(√2 –1)= 0.41L=0.353 m
For destructive interference
Δ₂=(2k+1)λ/2= λ/2,
k(min)=0 => Δ₁=λ/2 =0.353 m
=> λ =2•0.353 =0.706 m
λ=v/f => f= v/λ=
=343/0.706 = 485.8 Hz