  We examine the spectrum of gravitational radiation emitted by a network of
cosmic strings, with emphasis on the observational constraints and the
opportunities for detection. The analysis improves over past work, as we use a
phenomenological model for the radiation spectrum emitted by a cosmic string
loop. This model attempts to include the effect of the gravitational
back-reaction on the radiation emission by an individual loop with a high
frequency cut-off in the spectrum. Comparison of the total spectrum due to a
network of strings with the recently improved bound on the amplitude of a
stochastic gravitational wave background, due to measurements of noise in
pulsar signal arrival times, allows us to exclude a range of values of $\mu$,
the cosmic string linear mass density, for certain values of cosmic string and
cosmological parameters. We find the conservative bound $G\mu/c^2 < 5.4 (\pm
1.1) \times 10^{-6}$ which is consistent with all other limits. We consider
variations of the standard cosmological scenario, finding that an under dense,
$\Omega_0 < 1$ universe has little effect on the spectrum, whereas the portion
of the spectrum probed by gravitational wave detectors is strongly sensitive to
the thermal history of the cosmological fluid. We discuss the opportunity for
the observation of this stochastic background by resonant mass and laser
interferometer gravitational wave detectors.
