Sound Waves-Destructive Interference hw help please

[reality99]

Homework Statement

Two loudspeakers are placed a distance of 5m above and below one another and driven by the same source. A detector is positioned a perpendicular distance of 12m from the two speakers and a vertical height 3.2m from the lower speaker. This results in constructive interference at the first order maximum. What minimum distance (in any direction0 should the top speaker be moved in order to create destructive interference between the two speakers?
*Velocity of sound=345m/s

Homework Equations

This is what the final equation comes out to be, I just don't know how to derive it.

distance apart=dA
detector placement distance=dD
detector vertical height=dH

((SQRT(dD^2+((dH-dA/2)+dA/2)^2)-(SQRT(dD^2+((dH-dA/2)-dA/2)^2)/2)

The Attempt at a Solution

I honestly have no clue how to do this or even where to start. I know the final equation and the answer, just not how to get to it. Any help is appreciated. (equation is above)

 

[Redbelly98]

Welcome to PF.

I'd start with the conditions for constructive and destructive interference to happen.

 

[thomate1]

I would approach this problem by first understanding the concept of destructive interference. In simple terms, destructive interference occurs when two sound waves with opposite phases meet and cancel each other out, resulting in a decrease in amplitude or volume. In this case, the two loudspeakers are producing sound waves that are interfering with each other.

To determine the minimum distance the top speaker should be moved to create destructive interference, we need to consider the distance between the two speakers, the distance from the speakers to the detector, and the vertical height of the detector.

Using the given information, we can set up the following equation:

dA + dD = 5m (distance between the two speakers)
dD = 12m (detector placement distance)
dH = 3.2m (detector vertical height)

We can then use the formula for the path difference, which is the difference in the distance traveled by the sound wave from each speaker to the detector:

Path difference = dD + sinθ * dA (where θ is the angle between the two speakers)

For destructive interference, the path difference must be an integer multiple of the wavelength of the sound wave. Therefore, we can set up the following equation:

Path difference = n * λ (where n is an integer)

Plugging in the values for path difference and λ (which is equal to the velocity of sound divided by the frequency), we get:

dD + sinθ * dA = n * (v/f)

We can rearrange this equation to solve for dA, which gives us:

dA = (n * (v/f) - dD)/sinθ

Substituting in the values for dD, θ, and v (given in the homework statement), we get:

dA = (n * (345/f) - 12)/sin(90)

Finally, to determine the minimum distance the top speaker should be moved, we need to find the smallest value of n that satisfies this equation. This will give us the minimum distance for destructive interference.

I hope this helps in understanding the problem and how to solve it. It is important to understand the concept of destructive interference and the equations involved in order to derive the final equation. If you are still struggling, it might be helpful to consult with your teacher or a tutor for further assistance.