In the figure, two loudspeakers, separated by a distance of d 1 = 2.93 m, are in

Question

In the figure, two loudspeakers, separated by a distance of d1 = 2.93 m, are in phase. Assume the amplitudes of the sound from the speakers are approximately the same at the position of a listener, who is d2 = 4.12 m directly in front of one of the speakers. Consider the audible range for normal hearing, 20 Hz to 20 kHz. (a) What is the lowest frequency that gives the minimum signal (destructive interference) at the listener's ear? (b) What is the lowest frequency that gives the maximum signal (constructive interference) at the listener's ear? (Take the speed of sound to be 343 m/s.)

Explanation / Answer

a) for minimum signal (destructive interference) at the listener's ear

the phase difference has to be 180 degrees

distance travelled by wave from loudspeaker 1 = d1 + d2

distance travelled by wave from loudspeaker 2 = d2

difference in the distances = (d1+d2) - d2 = d1

If d1 = (n+1)* wavelength /2 , where n= 0,1,2, ...... then we have destructive interference

frequency = velocity of sound / wavelength

for minimum frequency, wavelength has to be maximum

so, n = 0 for wavelength has to be maximum in d1 = (n+1)* wavelength /2

so d1 = wavelength /2

2.93 m = wavelength /2

wavelength = 2*2.93 = 5.86 m for minimum signal (destructive interference) at the listener's ear

frequency = velocity of sound / wavelength

frequency = 343 / 5.86 = 58.532423 Hz

58.532423 Hz is the lowest frequency that gives the minimum signal (destructive interference) at the listener's ear.

b) for maximum signal (constructive interference) at the listener's ear

the phase difference has to be 0 degrees

distance travelled by wave from loudspeaker 1 = d1 + d2

distance travelled by wave from loudspeaker 2 = d2

difference in the distances = (d1+d2) - d2 = d1

If d1 = (n+1)* wavelength , where n= 0,1,2, ...... then we have constructive interference

frequency = velocity of sound / wavelength

for minimum frequency, wavelength has to be maximum

so, n = 0 for wavelength has to be maximum in d1 = (n+1)* wavelength

so d1 = wavelength

2.93 m = wavelength

wavelength = 2.93 m for maximum signal (constructive interference) at the listener's ear

frequency = velocity of sound / wavelength

frequency = 343 / 2.93 = 117.06484 Hz

117.06484 Hz is the lowest frequency that gives the maximum signal (constructive interference) at the listener's ear.

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.