*Comparisons of Solutions at R=8M*_{0}

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**The Quasi-Equilibrium solution compared to the exact integrated solution**

We compare the wavetrain from the QE approximation with the exact integrated solution. The initial radius of the shell is R=8M_{0}, corresponding to a moderately strong gravitational field.

The QE approximation shows remarkable agreement with the exact, integrated solution. Here we compare the two approaches by plotting the energy and gravitational luminosity as a function of apocenter radius.

A comparison of the actual gravitational wavetrains for the exact integrated solution and the QE approximation are very similar:

The above pictures compare blowups of the wavetrains at three different times.

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**The Monopole Approximation compared to the exact, integrated solution**

We also compare the exact integrated solution to the "monopole formula." The monopole approximation is given by the equation in the top panel.

At 8M_{0}, corresponding to a moderately strong field, the monopole approximation breaks down. Here we see that the monopole approximation keeps phase coherence, but amplitude and shape do not match the exact wavetrain.

Above we compare the monopole approximation with the exact, integrated solution at three different times.

*last updated 7 June 01 by rlc *