  Holmboe (1962) postulated that resonant interaction between two or more
progressive, linear interfacial waves produces exponentially growing
instabilities in idealized (broken-line profiles), homogeneous or density
stratified, inviscid shear layers. In this paper, we generalize Holmboe's
mechanistic picture of linear shear instabilities by (i) not initially
specifying the type of the waves, and (ii) by providing the option for
non-normal growth. We demonstrate the mechanism behind linear shear
instabilities by proposing a purely kinematic model consisting of two linear,
Doppler-shifted, progressive interfacial waves moving in opposite directions.
Moreover, we have found a necessary and sufficient (N&S) condition for the
existence of exponentially growing instabilities in idealized shear flows. The
two interfacial waves, starting from arbitrary initial conditions, eventually
phase-lock and resonate (grow exponentially), provided the N&S condition is
satisfied. The instability mechanism occurring prior to reaching steady state
is non-modal, favouring rapid transient growth. The theoretical underpinnings
of our wave interaction model is analogous to that of synchronization between
two coupled harmonic oscillators. Our proposed model is used to study three
well known types of shear instabilities - Rayleigh/Kelvin-Helmholtz, Holmboe
and Taylor-Caulfield. We show that the N&S condition provides a range of
unstable wavenumbers for each instability type, and this range matches the
predictions of the canonical normal-mode based linear stability theory.
