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We linearize the Einstein equations when the metric is Bondi-Sachs, when thebackground is Schwarzschild or Minkowski, and when there is a matter source inthe form of a thin shell whose density varies with time and angular position.By performing an eigenfunction decomposition, we reduce the problem to a systemof linear ordinary differential equations which we are able to solve. Thesolutions are relevant to the characteristic formulation of numericalrelativity: (a) as exact solutions against which computations of gravitationalradiation can be compared; and (b) in formulating boundary conditions on the$r=2M$ Schwarzschild horizon.Comment: Revised following referee comment

Linearized solutions of the Einstein equations within a Bondi–Sachs framework, and implications for boundary conditions in numerical simulations