Two loudspeakers are located 3.0 m apart on the stage of an auditorium. A listener at point P is seated 19.0 m from one speaker and 15.0 m from the other. A signal generator drives the speakers in phase with the same amplitude and frequency. The wave amplitude at P due to each speaker alone is A. The frequency is then varied between 30 Hz and 400 Hz. The speed of sound is 343 m/s. At what frequency or frequencies will the listener at P hear a maximum intensity?

The frequencies that the listener at P will hear a maximum intensity are: 85.75 Hz, 171.5 Hz, 257.25 Hz and 343 Hz.

Explanation:

Path Difference Δx is given as: nλ

where λ =

Δx can be re-written as: n×

where;

n = integer

v = speed of sound = 343 m/s

f = frequency

Δx  = 19.0 m - 15.0 m

Δx  = 4.0 m

f =

f = n × 85.75 Hz

Now;  n = 1, 2, 3, 4 ; the frequencies that the listener at P will hear the maximum intensity will be

when n= 1

f = 1 × 85.75 Hz

f = 85.75 Hz

when n= 2

f = 2 × 85.75 Hz

f = 171.5 Hz

when n= 3

f = 3  × 85.75 Hz

f = 257.25 Hz

when n= 4

f = 4 × 85.75 Hz

f = 343 Hz

∴ the frequencies that the listener at P will hear the maximum intensity will be : 85.75 Hz, 171.5 Hz, 257.25 Hz and 343 Hz for n =1,2,3, and 4 respectively.

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