Question

For what frequencies does their sound at the speakers produce constructive interference?

Answer 1

Constructive interference of sound at speakers occurs when sound waves are in phase, resulting in louder sounds at certain frequencies. These frequencies depend on the distance between the source and the point of interference relative to the sound's wavelength.

Sound waves from speakers can create points of constructive interference when the waves meet in phase, combining to produce a louder sound. This occurs at frequencies where the path length difference between the two speakers to a point is an integer multiple of the wavelength of the sound. The regions of constructive interference are where two crests or two troughs overlap, leading to higher intensity sounds.

Conversely, destructive interference happens when the sound waves from two speakers are out of phase, such as when a crest meets a trough, and they cancel each other out. This results in regions of lower intensity or 'dull areas.' In a typical room with speakers, wave interference can create a pattern of varying sound intensities as the sound waves reflect off walls and objects.

constructive interference in a room equipped with speakers occurs at specific frequencies that depend on the wavelength of the sound and the relative positions of the speakers and the listener. These positions determine whether the sound waves arrive in phase to constructively combine and amplify the sound.

Answer 2

You need to find the path difference. That is, how much further must sound waves from the more distant speaker travel than the close speaker, to reach the mike.
Use Pythagoras to find the distance of the further speaker: it is √(2.00²+4.50²)=4.924m so the path difference is 4.924-4.50=0.424m.
You will get constructive interference when this path difference is an integer number of wavelengths, because the waves will arrive at the mike in phase.
The speed of sound is 340m/s so the lowest frequency that will produce an antinode at the mike is the one that makes 0.424=λ
v=fλ so f=v/λ
f=340/0.424=801Hz.
The next one will be when 0.424m = 2λ => λ=0.212m
f=340/0.212=1602Hz
and so-on according to f=340n/0.424 where n is an integer.

For destructive interference the path difference must be (n-½)λ because that will make the waves arrive at the mike 180° out of phase.
f=340(n-½)/0.424