Passive acoustic dampers can effectively be utilized to counter thermoacoustic instabilites. A classic example of a passive acoustic damper is the Helmholtz resonator. Near its resonance, the resonator experiences large amplitudes of acoustic oscillation in its neck. The high amplitude oscillations in the neck dissapate acoustic energy by the means of vortex shedding at the neck duct interface and by the means of viscous effects of fluid and wall interaction. The figure below depicts a ducted resonator system.

A Helmholtz resonator can be modeled with the concept of acoustic impedence. Acoustic impedence is analogous to the electrical impedence. The acoustic impedence is quantified as the ratio of the pressure- to velocity perturbations similar to the electrical impedence which is quantified as the ratio of voltage to the current. That is, the ratio is of the flow driver to the actual flow. By the means of acoustic impedence the resonator can essentially by broken down into a RLC curcuit. Where the viscous neck and fluid interaction and the vortex shedding at the aperture are analogous to the resistance. Likewise, the neck length and the resonator volume is similar to the inductance and the capacitance respectively.

Substituting the harmonic pressure perturbation solution into conservation equations across the control volume displayed in the figure above gives the following solution to the ducted resonator system.

The real and imaginary frequency solutions to the acoustic's of the ducted resonator system are marked by "x" on the figure. Please note that the substituted conservation equations provied a homogenous set of equations, therefore only the system frequencies can be solved for and not the amplitudes of the acoustic oscillations.

The figure above displays frequency solutions of an open-end duct/ resonator system with a mean flow. One obvious point depicted by the figure is the importance of the placement of the resonator. The damping provided by the resonator is driven by the pressure oscillations the openning of the resonator experiences. That is, the higher the pressure oscillations at the openning the greater the damping. The figure clearly agrees with the this assesment such that at the first mode (an anti-node in an open-end duct) the resonator provides significant amounts of damping while at the second mode (a node in an open-end duct) the resonator provides no damping at all.

Moreover, these results give an insight on how the resonance of the resonator impacts the system. Due to the location of the resonator the anti-nodes and node alternate, that is, all the odd modes are anti-nodes while all the even modes are nodes. The first and the third modes are both anit-nodes yet the damping provided by the frist is greater than the third. Same is true for the third and fifth modes and so forth. Physically, a resonator is most effective at it's resonance due to the large amplitude of oscillations experienced by the system. Whether the first mode is closer to the resonace over the third mode and so forth can be seen by examining the impedance provided by the resonator at these modes, please see the figure below.

The impedance is closest to resonance at mode 1 and diverges with mode increment, satisfying yet another physical aspect of a Helmholtz resonator.

Varying the location of the resonator along the duct for the first and the second modes.