Finding Points of Destructive Interference for Two In-Phase Speakers

[zack7]

Two speakers that are in phase output sound at a frequency of 825 Hz. The speakers are separated by 5.00 m,
with one speaker at the origin and the other at (0, 5) m. Find points of totally destructive interference along the
line y = 5, for x > 0.
How do you get the points?

I would have to use change in Points= (n+1/2)lamda but what do I put for L after inserting all the info?
Thank you for all the help

wavelength=23/55
(L-x)-x=n(23/55)
L-2x=n(23/55)
L-n(23/55)=2x
x=L/2 -n(23/110)

Now what do I put for the L ?

 

[technician]

You need the wavelength but they have given you frequency. It looks like you are expected to know the speed of sound (330 m/s) so that you can calculate the wavelength

 

[zack7]

I have updated with my calcluations and the part that I am not sure

Thank you

 

[PeterO]

I have updated with my calcluations and the part that I am not sure

Thank you

Why did you say the wavelength was 23/55?

Edit: perhaps you were told to use a speed of sound of 345 m/s?

 

[PeterO]

I have updated with my calcluations and the part that I am not sure

Thank you

The minimum path difference (on the y=5 line) occurs at (2.5,5) where it is zero.

The maximum path difference is is the source separation, which is 5m

Once you have a wavelength - you say it is 23/55; which is 0.42 m , so when the path difference is 0.21, 0.63, 1.05, ... , up to ≤5 you will have destructive interference.

Note: 5m is almost 12 wavelengths, so you get interference at λ/2, 3λ/2, 5λ/2, 7λ/2, ... , 23λ/2, that means 12 on each side of the central maximum point (2.5,5)
There is a chance that the first few of them, on the left side of the central maximum, are still in the x>0 region and need to be counted.

Checking that.

from (0,0), to (0,5) is 5m
from (5,0) to (0.5) is 7.07m - a path difference of 2.07 m, but more importantly 4.95λ .
SO the minima associated with a path difference of λ/2, 3λ/2, 5λ/2, 7λ/2 & 9λ/2 will occur between (0,5) & (2.5,5) plus another 12 of them from there.

SOLVING:
Lets choose a point on the y=5 line .. the best being (x,5) so we keep it general.

Distance to the origin = √[x² + 25]

Distance from (5,0) = √[(x-5)² + 25]

The path difference is therefore √[x² + 25] - √[(x-5)² + 25]

That will then be set to equal λ/2, 3λ/2, 5λ/2, 7λ/2, ... in turn - up to 23λ/2

 

[PeterO]

Two speakers that are in phase output sound at a frequency of 825 Hz. The speakers are separated by 5.00 m,
with one speaker at the origin and the other at (0, 5) m. Find points of totally destructive interference along the
line y = 5, for x > 0.
How do you get the points?

I would have to use change in Points= (n+1/2)lamda but what do I put for L after inserting all the info?
Thank you for all the help

wavelength=23/55
(L-x)-x=n(23/55)
L-2x=n(23/55)
L-n(23/55)=2x
x=L/2 -n(23/110)

Now what do I put for the L ?

SIDE NOTE:

You will only get "totally destructive interference" if the two waves are out of phase, and equal in intensity.
If the two sound sources are exactly the same in phase, frequency and intensity (at the source) then the fact that we need a path difference, means that we are further from one speaker than the other. That means the intensity will be slightly less, so you would never get "totally destructive interference" - but I don't expect that technicality is part of this question.