# Ultrasonic phased array speaker spacing for high directivity and efficiency

(Short version of question(s) below.)

I want to make an ultrasonic parametric speaker at 40kHz, I want to drive them all at the same constant frequency to keep the electronics cheap.

When simulating the wave behavior in GIMP on one axis (so 2D), I found that by far the best way to do it was to haven them all at exactly half the wavelength. The speakers/transducers however are around 1.6 cm while the wavelength is around 0.8575 cm.

In Gimp I couldn't as easily simulate it in many different parameters, I also could only test it on one axis while in reality I will be building it on a plane (2 axis) (a plate) so a 2D array rather than a 1D array of sound sources which I couldn't simulate in Gimp. The speakers will all be outputting the same signal.

How do I know what distances to use and how sensitive will the distances be? I noticed the moving element inside of the transducers is slightly smaller than 2 wavelengths in diameter, or slightly smaller than 1 wavelength in radius. Would this mean I would need to see the ultrasonic transducers as areas of many small emitting points instead of small emitting points (which would make distancing give almost no difference), or would I for example need to keep them at a specific distance like 1 times the wavelength (or 1.5 or 2 times) and why so?

Is there also a specific optimal one, the directivity of the transducers is rated at 75 degrees, so you shouldn't get the huge peaks to the sides when using full wavelengths, but still due to my inexperience in this area I ask:

1. What would be the optimal distancing for these 40 kHz transducers with a directivity of 75 degrees and an diameter of around 1.6c (outer) and slightly smaller inside?

2. Why/how and is there a simple FOSS software to simulate it or similar?

3. Where my assumptions and estimations right: 1) Can I just use multiples of the wavelength due to the output directivity of 75 degrees? 2) Would the spacing not be as sensitive due to the transmitters using small ultrasonic horn like membranes which aren't a exact multiples of the wavelength which would make it like more or a more diffused sound source rather than a single point? Is that last part even true, would it act less like a point and more like many small emitting points?

Hopefully this will give you a better understanding of beam pattern basics.
The -3 dB beamwidth (β), in degrees, for a circular piston transducer is (numerically derived from a cylindrical array equation that contains a first order Bessel function):

$$\beta = 2sin^{-1}\left(1.61634 \over {\pi d} \right)$$ where d = diameter in wavelengths.

For a 75° beamwidth the transducer diameter is 0.85 λ. To use an array to create such a wide beamwidth, you need small diameter transducers spaced much closer than 0.5 λ. With this array, you can weight the element amplitudes to reduce side lobes. For a single circular transducer, you can expect your first side lobe to be down by about 18 dB (13 dB for a rectangular array).

The reason for limiting the element spacing to < 0.5 λ in a transducer array is to eliminate grating lobes. 1 λ element spacing will give you a grating lobe. The grating lobe(s) can be strong enough where you won't be able to tell if sound is coming from the grating lobe or the intended direction.

Grating lobe demonstration:
The following beam patterns are for a 10 & 5 element line array (close to the same length) with about 0.5 λ (left plot) and about 1.0 λ (right plot) element spacing. The 0.5 λ spacing is well behaved, the 1.0 λ spacing has a nasty grating lobe which tripped up the beamwidth calculation. The individual element sizes are 10% of the element spacing which gives a wide beamwidth for the individual elements. Sorry, the program that generated these plots is unavailable.

For a simple circular transducer (not an array), the following image shows the theoretical beampattern for a 75° (0.8452 λ) diameter transducer (blue trace). You only get grating lobes for transducer arrays, not single solid elements.

• So using sizes of 1 or more wavelength will indeed make it a lot less usable, sadly I don't have smaller transducers, but might perhaps try to make smaller ones myself once. the transducers where labeled as having a directivity of 75degrees, this seems to be a smaller angle as that what the biggest lobes are located in, this would make it less problematic right, since the huge lobes are exactly to the sides at 180 degrees. Commented May 19, 2022 at 8:52
• You will not get grating lobes for a single element, only on arrays if the element spacing is too large. All transducers have side lobes which are different from grating lobes. I added a theoretical beam plot of a 75° circular transducer.
– qrk
Commented May 19, 2022 at 23:01
• Note that there are array configurations with elements spaced 0.5 lambda or greater along an axis that do not exhibit significant grating lobes, though the average sidelobe level may be higher. The easiest one to analyze is a triangular grid. Or an array with aperiodic, or even random spacings. You can also limit your scan volume to keep grating lobes out of real space. Commented May 19, 2022 at 23:45