# Phase grids of Helmholtz resonators

Phase grids consisting of differently tuned Helmholtz resonators.

Top: same perforation, varying cavity depths.

Bottom: varying perforation, same cavity depth.*g _{s}*: structure period

Periodic structures with varying wall impedance are called phase grids. Besides the wall impedance, the reflection factor can be used to describe the acoustic characteristics of a surface. Both quantities are represented by complex numbers, and for locally reacting surfaces they can be easily converted into each other. As for the following considerations, the reflection factor allows a more graphic description. The absolute value of the reflection factor corresponds to the amplitude of the reflected wave related to the amplitude of the incident wave. So if the absolute value of the reflection factor is 1 this means that the incident wave is completely reflected. An absolute value of 0, on the other hand, means that the incident wave is completely absorbed. The phase of the reflection factor mirrors the phase jump the reflected wave experiences at the surface.

To achieve diffuse reflection of the greatest possible part of the incident sound energy, the different areas of the wall surfaces should be only little absorbent and cause the greatest possible phase differences of the reflected waves. These conditions can be realized using Helmholtz resonators which are tuned accordingly. Around the resonance frequency the phase difference between the reflected wave and the incident wave amounts to up to 180 degrees. The lesser a Helmholtz resonator is damped, the larger is the frequency range over which this phase shift occurs. At the same time, the resonator’s absorption capacity constantly decreases. The necessary great phase difference of adjacent wall areas can be realized by means of two adjoining resonators that are tuned to different frequencies.

Meyer, Kuttruff and Rischbieter investigated how large the individual wall areas should be and to which frequencies the resonators should be tuned to. They limited themselves to structures consisting of two different resonators arranged in stripes. The resonators are made of a perforated plate connected to an air volume. Different resonance frequencies are either produced by stripes with the same perforation pattern and different air volumes, or by stripes with different perforation patterns and the same air volume.

The two resonance frequencies should be about 3 to 4 third octaves apart. Third octave means that the ratio of two frequencies is 1:2⅓ = 1:1.26. Good scattering coefficients are achieved in the frequency range between about one third octave below the lower resonance frequency and about one third octave above the upper resonance frequency, that is over a total of about 2 octaves. Octave means that the ratio of two frequencies is 1:2.

The highest scattering coefficients are obtained if the two different stripes have the same width.