Absorbency Models
Equivalent fluid models like Delany-Bazley (and others) begin to fail when the micro-structural frame of the absorbent can no longer be considered rigid, but mechanically resonates. This affects the low frequency accuracy. Poroelastic models such as those discussed by Biot[3] do account for mechanical behaviour, but also require careful attention to experimental details such as the fixing conditions of the specimens. In practice, this can generate a lot of experimental overhead trying to get models and experiments to align which can often lead to only a small return on investment.
As an example, in the headphone model I have chosen in the first instance to use a description of volume absorbency that required only the flow resistivity to be prescribed, which would initially be an estimate, but could be developed to a more specific equivalent fluid model, populating frequency dependent tables of effective density and bulk modulus.
In determining whether mechanical vibration of the absorbent frame was a factor, I measured the material velocity using a laser Doppler vibrometer anticipating that if a local peak could be detected, such as a drum-like membrane response, we could conclude that a poro-elastic approach was justified.
The graph shows velocity spectra of the diaphragm and frame vent damping layer in the centre of a frame vent hole, when measured in a vacuum chamber. An additional measurement of the frame damping was taken when air was re-introduced to the chamber to observe the change. The velocity recorded on the frame damping is significantly less than that of the diaphragm, which is expected and there is a localised resonance at ~530Hz. In the air-loaded case, we can see that the frequency and magnitude of this resonance drops. It does not appear to follow the idea that air oscillating in the frame vents is causing mechanical resonance in the damping layer however, because air was removed which would surely reduce the resonance peak, not increase it.
Taking a measurement at the rear of the motor (a mechanical entity coupled to the frame) in the vacuum chamber produces the purple trace in the graph below.
The 530Hz peak in velocity remains in the data when the rear of the motor is measured. This is understood to be a mechanical resonance of the driver frame itself, driven by the harmonic Lorentz force developed in the motor, when the frame is restrained at the outer edge.
Because the resonance at 530Hz is not strictly the result of a non-rigid frame condition of the absorbent layer, but rather a more general parasytic vibration of the driver, it would appear that an equivalent fluid model would suffice as a reasonable approximation, particularly because at this stage, the driver frame and motor are assumed to be rigid. In later developments, where the coupled vibro-acoustic details of the headphone components in addition to the diaphragm and voice coil are considered, more complex absorbency models might add value, and I believe this to be particularly interesting for the foam cushions.
References:
[1] Dossi, Martino et al, "An Inverse Method to Determine Acoustic Parameters of Polyurethane Foams", Inter-Noise Madrid, 2019.
[2] Taraldsen, G, "The Delany-Bazley Impedance Model and Darcy’s Law", ACTA ACUSTICA UNITED WITH ACUSTICA, Vol. 91 (2005) 4150.
[3] M. A. Biot, “Theory of Propagation of Elastic Waves in a Fluid‐Saturated Porous Solid. I. Low‐Frequency Range”, J. Acoust. Soc. Am., vol. 28, 1956.