The modulation equations for the small-amplitude resonant
motion of the system are the well-known three-wave equations.
We use Hamiltonian reduction to determine a complete analytical
solution. The amplitudes and phases are expressed in terms of both
Weierstrass and Jacobi elliptic functions. The strength of
the pulsation may be computed from the invariants of the equations.
Several analytical formulas are found for the precession angle.
We deduce simplified approximate expressions, in terms of elementary
functions, for the pulsation amplitude and precession angle
and demonstrate their high accuracy by numerical experiments.
Thus, for given initial conditions, we can describe the envelope
dynamics without solving the equations.
Conversely, given the parameters which determine the envelope,
we can specify initial conditions which, to a high level of
accuracy, yield this envelope.
When the frequencies of the elastic and pendular oscillations
of an elastic pendulum or swinging spring are in the ratio two-to-one,
there is a regular exchange of energy between the two modes of
oscillation. We refer to this phenomenon as pulsation.
Between the horizontal excursions, or pulses,
the spring undergoes a change of azimuth which we call the
precession angle. The pulsation and stepwise precession are the
characteristic features of the dynamics of the swinging spring.