Two loudspeakers placed 6.00 m apart are driven in phase by an audio oscillator having a frequency range from 1595 Hz to 2158 Hz.A point P is located 4.70 m from one loudspeaker and 3.60 m from the other speaker.The speed of sound in the room is 344 m/s.At what frequency (or frequencies)of the oscillator does the sound reaching point P interfere destructively? (There could be more than one correct choice. )
A)2030 Hz
B)2130 Hz
C)2100 Hz
D)1720 Hz
E)1630 Hz

Question

Quizplus · Accepted Answer

The path length difference between the two loudspeakers to point P is:$$6.00	ext{ m}-4.70	ext{ m}-3.60	ext{ m}= -2.30	ext{ m}$$For destructive interference, the path length difference must be an odd multiple of half a wavelength:$$\frac{1}{2}\lambda=\frac{c}{2f}$$$$2.30	ext{ m}=(2n+1)\frac{344	ext{ m/s}}{2f}$$$$f=\frac{344	ext{ m/s}}{4.60	ext{ m}}(2n+1)$$$$f=(74.8	ext{ Hz})(2n+1)$$The lowest frequency that produces destructive interference is:$$f=(74.8	ext{ Hz})(1)=74.8	ext{ Hz}$$This frequency is not within the given range, so we can ignore it.The next frequency that produces destructive interference is:$$f=(74.8	ext{ Hz})(3)=224.4	ext{ Hz}$$This frequency is also not within the given range, so we can ignore it.The next frequency that produces destructive interference is:$$f=(74.8	ext{ Hz})(5)=374.0	ext{ Hz}$$This frequency is within the given range, so it is a correct answer.The next frequency that produces destructive interference is:$$f=(74.8	ext{ Hz})(7)=523.6	ext{ Hz}$$This frequency is also within the given range, so it is a correct answer.The next frequency that produces destructive interference is:$$f=(74.8	ext{ Hz})(9)=673.2	ext{ Hz}$$This frequency is also within the given range, so it is a correct answer.The next frequency that produces destructive interference is:$$f=(74.8	ext{ Hz})(11)=822.8	ext{ Hz}$$This frequency is also within the given range, so it is a correct answer.The next frequency that produces destructive interference is:$$f=(74.8	ext{ Hz})(13)=972.4	ext{ Hz}$$This frequency is also within the given range, so it is a correct answer.The next frequency that produces destructive interference is:$$f=(74.8	ext{ Hz})(15)=1122.0	ext{ Hz}$$This frequency is also within the given range, so it is a correct answer.The next frequency that produces destructive interference is:$$f=(74.8	ext{ Hz})(17)=1271.6	ext{ Hz}$$This frequency is also within the given range, so it is a correct answer.The next frequency that produces destructive interference is:$$f=(74.8	ext{ Hz})(19)=1421.2	ext{ Hz}$$This frequency is also within the given range, so it is a correct answer.The next frequency that produces destructive interference is:$$f=(74.8	ext{ Hz})(21)=1570.8	ext{ Hz}$$This frequency is also within the given range, so it is a correct answer.The next frequency that produces destructive interference is:$$f=(74.8	ext{ Hz})(23)=1720.4	ext{ Hz}$$This frequency is also within the given range, so it is a correct answer.The next frequency that produces destructive interference is:$$f=(74.8	ext{ Hz})(25)=1870.0	ext{ Hz}$$This frequency is also within the given range, so it is a correct answer.The next frequency that produces destructive interference is:$$f=(74.8	ext{ Hz})(27)=2019.6	ext{ Hz}$$This frequency is also within the given range, so it is a correct answer.The next frequency that produces destructive interference is:$$f=(74.8	ext{ Hz})(29)=2169.2	ext{ Hz}$$This frequency is also within the given range, so it is a correct answer.Therefore, the correct answers are:$$f=374.0	ext{ Hz, }523.6	ext{ Hz, }673.2	ext{ Hz, }822.8	ext{ Hz, }972.4	ext{ Hz, }1122.0	ext{ Hz, }1271.6	ext{ Hz, }1421.2	ext{ Hz, }1570.8	ext{ Hz, }1720.4	ext{ Hz, }1870.0	ext{ Hz, }2019.6	ext{ Hz, }2169.2	ext{ Hz}$$However, the given range is from 1595 Hz to 2158 Hz, so the only correct answers within the given range are:$$f=1720.4	ext{ Hz, }2019.6	ext{ Hz}$$

Two Loudspeakers Placed 6