# Confusion over textbook formulae for directional characteristics of sound sources

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?

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.

• Hi Greg, many thanks for your comment. That's right, I'm plotting the output at a specified angle as a function of frequency (I explain why in the OP). There seems to be something obvious that I'm missing here! Clearly, any time the pressure ratio is less than one, this must be the result of mixed and destructive interference, i.e. the sources being out of phase. But when sound pressure is measured, destructive interference results in a measurement of zero pressure - not negative pressure. This is what I'm struggling with and hoping someone is able to explain...
– apk
Nov 27, 2022 at 11:12
• I suggest you think about intensity (spatial vector of sound power/area). If you plot the intensity around a radiator you'll find places where the sound power is looping back to the radiator. Nov 27, 2022 at 20:04