Confusion over textbook formulae for directional characteristics of sound sources

Question

I'm currently reading Olson's Elements of Acoustical Engineering, in which formulae are given for the directional characteristics of various arrays of sound sources, all of which are stated in terms of ratios $R_\alpha$ of pressure at a given angle $\alpha$ to pressure normal to the array.

For example, the ratio given for an array of point sources is:

$$R_\alpha=\frac{sin(\frac{n\pi d}{\lambda}sin\alpha)}{nsin(\frac{\pi d}{\lambda}sin\alpha)}$$

where: $n$ is the number of sources; $d$ is the distance between the sources; and $\lambda$ is the wavelength

I've plotted this function in desmos (substituting $343/x$ for $\lambda$ where $x$ is frequency and 343m/s is the speed of sound in air at room temperature).

The resultant graph shows a periodic function with both positive and negative values.

However, if such an array were approximated by real sound sources and sound pressures measured with a microphone at a given angle $\alpha$, you'd expect to measure something resembling the absolute value of the same function, i.e. no negative values (see the second function in the Desmos link, above).

What I'm interested in understanding is: What does a negative pressure ratio mean, physically, in this context?

And, if it has physical meaning, how would you measure it?

Would anyone be able to point me in the right direction?

Answer 1

The sound pressure does go negative, that is out of phase with the reference signal.

Incidentally your plot is confusing to me, as x axis seems to be either frequency or wavelength(?), so you are looking at the response in a particular direction at all frequencies. Here's a more typical way to plot it for $n = 5$, $\lambda = d = 1$, at just one frequency but for all directions.

Here's a comparison for $n = 2$ $\lambda = 1$ or $2$ and $d = 1$

"if it has physical meaning, how would you measure it?"

A reference microphone at $0$ degrees, and a second microphone at the same distance from the centre of the array at the angle of interest. You could then use a scope or an FFT analyser to compare the two waveforms.