  In this work, we study wave propagation in generic Hermitian local periodic
baths, and investigate the effects of anisotropy and quasi-breaking of
periodicity on resonant emission into the band of the bath. We asymptotically
decompose the Green's function into long-range travelling waves composed of all
wavevectors (near-)resonant at the emitter frequency, and rapidly decaying
evanescent waves. Our approximation then converges exponentially with
increasing source-receiver separation ${\rho}$ when resonant wavepackets with
group velocity parallel to ${\rho}$ exist. In hyperbolic media this condition
may not be satisfied, and we find that the exponential decay length of
oscillating evanescent waves locally around caustics generally depends as a
power law with exponent 3/2 on the angle made between ${\rho}$ and the caustic.
For ${\rho}$ beyond the caustic we observe that the Green's function can become
almost imaginary, which results in exclusively incoherent emitter-emitter
interactions and allows the simulation of purely dissipative systems with
short-range interactions. Here the interaction length is tunable via the
separation vector of the emitters. We finally probe the hyperbolic dispersion
beyond the previous regimes by applying an artificial gauge field on the
lattice. We find that emission resonant with the corresponding open orbits in
the Brillouin zone is quasi-one dimensional, in contrast to an isotropic
environment. The quasi-1D emission is further topologically protected against
local and global lattice perturbations and periodically refocussing, offering a
robust bi-directional transport of excitations in higher-dimensional media.
