  We derive a nonlinear integral equation (NLIE) for some bulk excited states
of the sine-Gordon model on a finite interval with general integrable boundary
interactions, including boundary terms proportional to the first time
derivative of the field. We use this NLIE to compute numerically the dimensions
of these states as a function of scale, and check the UV and IR limits
analytically. We also find further support for the ground-state NLIE by
comparison with boundary conformal perturbation theory (BCPT), boundary
truncated conformal space approach (BTCSA) and the boundary analogue of the
Luscher formula.
