How Does Speaker Arrangement Affect Sound Intensity at a Point?

[Firben]

Homework Statement

Three speakers with the same effect are connected to the same tone generator are placed along the y-axis and sends out sound in equal directions. The distance a and b are in the same magnitude as the wave length of the sound.

http://s716.photobucket.com/user/Pitoraq/media/Fys21_zps1596efc1.png.html

a)
The point P is located on a distance L >> a from the speakers along the x-axis. If only one speaker is connected the intensity in the point P is 15.0 mW/m^2. Which is the total intensity when all speakers are connected ?

b)
If the distance a = λ/4, which is the smallest distance b that gives maximum intensity along the y-axis from a distance L ?

Homework Equations

I = P/A
P(0) = (2vρI)^(1/2)
I = P(0)/(2vρr^2)

The Attempt at a Solution

I = P/A
ρ = 1.2929 kg/m^2 density of air
P(0) = (2vρI)^(1/2) = 115.5 pa
I = P(0)/(2vρr^2) <==> r = P(0)/(2vρI)^(1/2) = 31.62 m

If r is the distance from the point P to one of the three speakers, i don't know which one. And what do they mean with "The distance a and b are in the same magnitude as the wave length of the sound." ?

 

[user3]

The r can be the distance from any of the three speakers since L >> a. Imagine standing 1 Km away from 3 water bottles separated by 1 cm. The distance between you an any of the bottles is practically the same.

An order of magnitude is the power of 10. It just means the distances are almost similar. For example, 1*10^2 and 5*10^2 are in the same order of magnitude; 1*10^3 and 1*10^9 are NOT in the same order of magnitude.

 

[Firben]

I wonder if it should be constructive or destructive interference ?

 

[Firben]

I solved a) but on b) How can i find the maximum intensity on the y-axis, is it destructive interference ? or destructive ?

 

[user3]

Since the Maximum Intensity of the wave is just the Amplitude squared: I=|A|^2 , we can deduce that the amplitude of the wave coming out of one speaker is $\sqrt{15m}$. We also know that the Total Maximum Intensity (I say maximum because we know the in the middle -along the x-axis- there's always constructive interference) is the Sum of the Amplitudes from all three speakers squared: $I_{total} = |A_1 + A_2 + A_3|^2$, but the three speakers have the same amplitude since they are connected to the same tone generator. Thus, $$ I_{total} = |3\sqrt{15}|^2 = 9*15=135mw/m^2 $$

 

[Firben]

I think i got the same result. I did this to get the total intensity

P₀ = √(2vρI) = 3.65 PaI = P(₀)/(2vρr^2) <==> r = P₀/√(2vρI) = 1m

I = P/A <==>

3 speakers => P = 4π*(3*1)^2*15*10^-3 = 1.69 W

and the total intensity then becomes

I = 1.69 /(4π*1^2) = 0.135 w/m^2

On b)

How can i find the maximal intensity on the y-axis if a = λ/4 ?