  The linear evolution of disturbances due to a ribbon vibrating at frequency
$\omega_0$ in plane Poiseuille flow is computed by solving the associated
initial boundary value problem in the Fourier-Laplace plane, followed by
inversion. A novel algorithm for identifying the temporal modes of the
Orr-Sommerfeld equation (OSE) in the complex wavenumber plane, which are
required in the inversion, is presented. Unlike in many prior studies, the
performance of the Laplace integral first, not only avoids complicated
causality arguments and confusion, in locating upstream and downstream modes,
that is prevalent in literature but also yields a spatio-temporally uniform
solution. It also reveals that the solution consists of a time-periodic part at
$\omega_0$ , associated with the relevant spatial mode (the
Tollmein-Schlichting wave) and a transient wavepacket, associated mainly with
the saddle points of the OSE and is computed by the method of steepest
descents, which can also include contributions from the spatial pole. Which of
these parts dominates depends on the Reynolds number and {\omega}0. A secondary
stability analysis of this dominant part is seen to explain the disturbance
growth observed in the seminal experiments of Nishioka, Iida & Ichikawa (J.
Fluid Mech., vol.72 , 1975, p.731) and Nishioka, Iida & Kanbayashi (NASA
TM-75885, 1981). Threshold amplitudes for instability at a subcritical Reynolds
number Re = 5000 are obtained from the time-averaged three dimensional
disturbances, by combining the secondary base states and the growing Floquet
modes. The observed minima of the threshold amplitude curves in the experiments
are explained in terms of the instabilities of these two base states.
Computations, for another subcritical (4000) and a supercritical (6000)
Reynolds number, are also validated with the experimental data.
