  Assigning homogeneous boundary conditions, such as acoustic impedance, to the
thermoviscous wave equations (TWE) derived by transforming the linearized
Navier-Stokes equations (LNSE) to the frequency domain yields a so-called
Helmholtz solver, whose output is a discrete set of complex eigenfunction and
eigenvalue pairs. The proposed method -- the inverse Helmholtz solver (iHS) --
reverses such procedure by returning the value of acoustic impedance at one or
more unknown impedance boundaries (IBs) of a given domain via spatial
integration of the TWE for a given real-valued frequency with assigned
conditions on other boundaries. The iHS procedure is applied to a second-order
spatial discretization of the TWEs derived on an unstructured grid with
staggered grid arrangement. The momentum equation only is extended to the
center of each IB face where pressure and velocity components are co-located
and treated as unknowns. One closure condition considered for the iHS is the
assignment of the surface gradient of pressure phase over the IBs,
corresponding to assigning the shape of the acoustic waveform at the IB. The
iHS procedure is carried out independently for each frequency in order to
return the complete broadband complex impedance distribution at the IBs in any
desired frequency range. The iHS approach is first validated against Rott's
theory for both inviscid and viscous, rectangular and circular ducts. The
impedance of a geometrically complex toy cavity is then reconstructed and
verified against companion full compressible unstructured Navier-Stokes
simulations resolving the cavity geometry and one-dimensional impedance test
tube calculations based on time-domain impedance boundary conditions (TDIBC).
The iHS methodology is also shown to capture thermoacoustic effects, with
reconstructed impedance values quantitatively in agreement with thermoacoustic
growth rates.
