A differential vertical force applied by the external perturbation and the halo wake can warp the disk plane. HT describe a linearized solution for the dynamical evolution of the plane for an isolated disk. The bending analysis here uses the formalism developed by HT with several modifications. First, because our equilibrium disk models are embedded in an external halo, we must retain their equation (12) rather than simplify using relationships based on the specific form of the background model. Second, we solve the linearized equations of motion under a forced disturbance (their eq. 19) by Laplace transform for consistency with the approach in §3.2. The vertical force follows directly from the expansion coefficients describing the external perturbation, equation (9) and the halo wake, equation (10). The back-reaction to the in-plane distortion is included and is a relatively minor contribution to the total response. This calculation does not consider the back-reaction of the halo to the vertical distortion (Nelson & Tremaine 1995).
For reasons described in HT, their polynomial disk models are well suited to numerical analysis and adopted here as described in §2. I also tried exponential disks with different basis sets but could not find an alternative which allowed an accurate computation of the height alone, rather than the combining height times the surface density.