  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.
  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.
