Question

You are walking around a large open field.The only objects nearby are two identical speakers set some distance apart, which are both producing sound, in unison with a wavelength of 4 m. As you wander around the field, you notice that at certain locations, the sound you hear is surprisingly loud, while at other is seems unusually quiet. You conclude that you are observing the effects of interference between the two sources of sound waves.indicate what sort of interference woud occur at that point. Constructive interference, destructive interference or in between.8 m from each speaker.11 m from one speaker and 7 m from the other.10 m from one speaker and 8 m from the other11 m from one speaker and 14 m from the other20 m from one speaker and 12 m from the other13 m from one speaker and 19 m from the other19 m from one speaker and 14 m from the other

Answer 1

• 8 m from both : Constructive
• 11 m and 7 m : Constructive
• 10 m and 8 m : Destructive
• 11 m and 14 m : In between.
• 20 m and 12 m : Constructive
• 13 m and 19 m : Destructive
• 19 m and 14 m : In between.

Step-by-step explanation:

Equation of the wave

We know that the amplitude of a wave starting at
`\vec{x}_0` measured at position
`\vec{x}` at time t is

`y(\vec{x},t) = A sin ( \vec{k} (\vec{x}-\vec{x}_0) - \omega t + \phi)`

where
`\vec{k}` is the wavevector, ω the angular frequency, and φ the phase angle.

If we measure for a time
`t_0` we get

`y(x,t_0) = A sin ( k (x-x_0) - \omega t_0 + \phi)`

Now, we can use:

`\Theta =  - \omega t_0 + \phi`

`y(x,t_0) = A sin ( \vec{k} (\vec{x}-\vec{x}_0) + \Theta )`

Finally, we can write this in term of the distance d, as
`\vec{k}` is parallel to the displacement vector for a sound wave:

`\vec{k} (\vec{x}-\vec{x}_0) = k |\vec{x}-\vec{x}_0| = k d`

where k is the wavenumber

`y(d,t_0) = A sin ( k d + \Theta )`

this is the amplitude of a sound wave measured at a distance d at time
`t_0`

Interference

Measuring two identical waves at the same time, one starting at distance d and the other at distance d', the amplitude measured is:

`Amp = y(d,t_0) + y(d',t_0)`

`Amp =  A \ sin ( k d + \Theta ) +  A  \ sin ( k d' + \Theta )`

Constructive interference

We get constructive interference when both sines equals one, or minus one, so, we need a phase difference of
`2 n \pi`, where n is an integer :

`k d + \Theta  = k d' + \Theta  + 2 \ n \ \pi`

`k d - k d'=  2 \ n \ \pi`

`k (d -  d')=  2 \ n \ \pi`

`(d - d') =  (2 \ n \ \pi)/(k)`

as the wavenumber is

`k = (2\pi)/(\lambda)`

where
`\lambda` is the wavelength,

`(d - d') =  ( \lambda \ 2 \ n \pi)/(2 \pi)`

`(d - d') =  n \lambda`

so, the difference between the distances must be a multiple of the wavelength to obtain constructive interference.

Destructive interference

We get destructive interference when one sin equals one, and the other minus one, so, we need a phase difference of
`(2 \ n + 1) \ \pi`, where n is an integer

`k d + \Theta  = k d' + \Theta  + (2 \ n + 1) \ \pi`

`k d - k d'=  (2 \ n + 1) \ \pi`

`k (d -  d')=  (2 \ n + 1) \ \pi`

`(d - d') =  ((2 \ n + 1) \ \pi)/(k)`

`(d - d') =  ( \lambda (2 \ n + 1) \ \pi)/(2 \pi)`

`(d - d') =  (n + (1)/(2)) \lambda`

Problem

Knowing that
`\lambda = 4 \ m`

so, for the first

8 m from both :

`d - d ' = 8  \ m - 8 \ m =  0  \ \lambda`

Constructive

11 m and 7 m:

`d - d ' = 11  \ m - 7 \ m = n \ \lambda`

`4 \ m = 1 * 4 \ m`

Constructive

10 m and 8 m

`d - d ' = 10  \ m - 8 \ m = n \ \lambda`

`2 \ m = (1)/(2) * 4 \ m`

Destructive

11 m and 14 m

`d - d ' = 14  \ m - 11 \ m = n \ \lambda`

`( d - d ' )/(\lambda) = ( 3 \ m )/(4 \ m) = (3)/(4)`

In between.

20 m and 12 m

`d - d ' = 20  \ m - 12 \ m = n \ \lambda`

`( d - d ' )/(\lambda) = ( 8 \ m )/(4 \ m) = 2`

Constructive

13 m and 19 m

`d - d ' = 19  \ m - 13 \ m = n \ \lambda`

`( d - d ' )/(\lambda) = ( 6 \ m )/(4 \ m) = 1 + (1)/(2)`

Destructive.

14 m and 19 m

`d - d ' = 19  \ m - 14 \ m = n \ \lambda`

`( d - d ' )/(\lambda) = ( 5 \ m )/(4 \ m) = 1 + (1)/(4)`

In between.