Question

Two speakers that are 12.0 m apart produce in-phase sound waves of frequency 245 Hz in a room where the speed of sound is 340 m/s. A woman starts out at the midpoint between the two speakers. The room's walls and ceiling are covered with absorbers to eliminate reflections, and she listens with only one ear for best precision.

(a) Why is this constructive interference?


(b) She now walks slowly toward one of the speakers. How far from the center must she walk before she first hears the sound reach a minimum intensity?


(c) How far from the center must she walk before she first hears the sound maximally enhanced?

Answer 1

a. constructive interference takes place.

b. x = 0.34 m= 34 cm

c..
`x=0.69m = 69cm`

Step-by-step explanation:

Given

Distance between speakers = 12.0 m

frequency = 245 Hz,

speed of sound = 340 m/s,

a. For constructive interference we have

the path difference is
`= m\lambda`

where m = +/-1,+/-2,+/-3,..........

Wavelength of sound =
`=(v)/(f) =340/245=1.38 m`

As women is at center or middle point, therefore the path difference at that point must be zero

`d_(2) -d_(1)=0`

Now the path difference
`d_(2) -d_(1)=0` is an integral multiple of the wavlenght therefore constructive interference takes place.

b.

for the sound to reach a minimum intensity

Now if she moves x distance from current point then

`d_(2)^(')-d_(1)^(')=(d_(2) +x) - (d_(1) -x)`

`d_(2)^(')-d_(1)^(') =2x`

for hearing minimum intensity the interference must be destructive therefore,

`d_(2)^(')-d_(1)^(')= (m+(1)/(2))\lambda`

for m = 0

`d_(2)^(')-d_(1)^(')= (0+(1)/(2))\lambda`

by putting the value of lamda = 1.38 and difference in distance as 2x

we get

x = 0.34 m= 34 cm

c.

for the sound to reach a maximum intensity

`d_(2)^(')-d_(1)^(')= m\lambda`

for that m=1

`d_(2)^(')-d_(1)^(')= 1\lambda`

`2x= \lambda`

`x=(\lambda)/(2)`

`x=(1.38)/(2)`

`x=0.69m = 69cm`