Transfer functions are constantly used in both Seismology and Geotech- nical Earthquake Engineering to relate seismic displacement at di erent depths within strata. In the context of Di usive Theory, they also appear in the expression of the imaginary part of 1D Green's functions. In spite of its remarkable importance, their mathematical structure is not fully un- derstood yet, except in the simplest cases of two or three layers at most. This incomplete understanding, in particular as to the e ect of increasing number of layers, hinders progress in some areas, as researchers have to resort to expensive and less conclusive numerical parametric studies. This text presents the general form of transfer functions for any number of lay- ers, overcoming the above issues. Owing to the formal connection between seismic wave propagation and other phenomena that, in essence, repre- sent di erent instances of wave propagation in a linear-elastic medium, one can extend the results derived elsewhere [ Garcia-Suarez, Joaquin. 2021. \Trace Spectrum of 1D Transfer Matrices for Wave Propagation in Layered Media." engrXiv. June 24. doi:10.31224/osf.io/ygt8z. ] in the context of longitudinal wave propagation in modular rods to seismic response of strati ed sites. The knowledge of the general closed-form ex- pression of the transfer functions allows to analytically characterize the long-wavelength asymptotics of the horizontal-to-vertical spectral ratio for any number of layers. 1 Introduction The study of elastic-wave propagation phenomena is of utmost interest to both the Seismology and the Geotechnical Earthquake Engineering communities: the sudden release of energy created by a fault rupturing generates a deformation eld that propagates towards the free surface and it is ultimately felt as an earthquake. This wave eld is usually studied using the tools of Linear Elasticity 1 𝑧ଵ ଵଵ ℎଵ 𝑧௠ 𝑧ேାଵ௠௠ ௛௔௟௙௛௔௟௙ℎ௠ ℎே→∞𝑢ଵ 𝑢௠ … 𝑢ே …Figure 1: Scheme of the layered system and local coordinates. [1], except when the propagation takes place within soft soils, what requires considering material non-linear response [2]. It is also the case that when the front arrives to the shallow layers, assuming the earthquake source is located deep enough, it does so almost-vertically (the decreasing sti ness bends the rays towards the normal of the free surface). Moreover, for the length-scales of interest in applications, the local curvature of the wave eld is relatively small compared to the window of observation, so the impinging excitation can be considered to be made up of plane waves. Under the assumption of one-dimensional propagation perpendicular to the free surface, the di erent components decouple and SH, SV and P waves can be studied separately and then summed up, as there is no mode conversion. A customary method to study the evolution of the wave elds across the layers is the so-called \Propagator Matrix Method", also referred as \Method of Matrizants" in older texts [3]. An equivalent matrix approach for the 1D case is presented in [2], and used often in geotechnical engineering. The classic Seismology texts [1, 4] present a general treatment for inclined SH waves and inclined P-SV waves; however, in this work, we will consider no inclination, thus the expressions that appear in the prior