Summation of the Born series for the normal modes of the Earth

We derive an explicit formula for the response of a laterally heterogeneous, self‐gravitating, rotating, dissipative, and physically dispersive earth to an earthquake. The long‐period spectrum at a single station is given by u(ω) =R^H₀ [V(ω) +ωW−ω²T]⁻¹·S₀(ω) where R₀ and S₀(ω) are the unperturbed source and receiver vectors and H denotes the Hermitian transpose. The quantities V(ω), W, and T are the potential energy, Coriolis, and relative kinetic energy matrices, and V(ω) +ωW−ω²T is the Lagrangian matrix whose eigenvalues are the perturbed complex eigenfrequencies. This simple result is obtained by formal summation of the Born series.