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I am trying to determine the validity of a formula I came across... I have not seen it anywhere else and I can't be sure wether it's valid or not...

Consider a number of sound pointsources positioned on a line. The soundwaves coming from each sound source are all in phase (at time t = 0) and they all have the same amplitude. (So they are perfectly coherent).

Consider a point P at some distance away from the pointsources, where I want to use superposition to add up the sounds.

Because the soundsources are not all in the same location, the soundwaves from one sound source will have travelled a short distance longer than the soundwave from another sound source; resulting in a phase difference.

I have found a formula to calculate exactly the sound pressure on point P, but I don't know if it's valid...:

[tex]p_{tot} = \left| \sum_{n=1}^N \frac{p_0}{r_n} e^{ikr_n} \right|[/tex]

Here, [itex]p_0[/itex] is the (effective*) sound pressure of a single sound source at 1 meter from the source. The wavenumber [itex]k[/itex] is [itex]\frac{2 \pi}{\lambda}[/itex] with [itex]\lambda[/itex] the wavelength of the soundwave.

*N*is the number of sound sources, and finally [itex]r_n[/itex] is the distance from point P to a single sound source

*n*.

Is this formula valid? If yes, how can it be derived?

I know the general equation for a spherical wave is given by:

[tex]\psi = \frac{A}{r} e^{i(kr - \omega t)}[/tex]

First of all, I suppose we are only looking at a instanteneous time, for simplicity t = 0, which eliminates the omega*t term?

Secondly, the sum is obvious, it is simply summing all waves, which each have a different phase [itex]kr_n[/itex] due to the difference in distance travelled.

Then, the modulus (abs. value) is taken after summation. I guess that makes sence too, this will give us the maximum (effective) sound pressure (invariant of time) at the point P, right?

But, the thing that troubles me most, is the units. They are not right, are they? Since the exponential is obviously dimensionless, we get:

[tex]\text{[Pressure]} = \frac{ \text{[Pressure]} }{ \text{[Distance]}} \text{ } \rightarrow \text{ } \text{Pa} = \frac{ \text{Pa}}{\text{m}}[/tex]. Huh?

Can someone help me out here? Thanks!

*By effective sound pressure I mean [itex]p_{\text{eff}} = \frac{1}{2} \sqrt{2} p_{\text{max}}[/itex]