(ed.). Gravitational waves (IOP, 2001)(422s).

Testing theories of gravity 155• Toroidal modes: these are obtained by setting C 0 = C 2 = 0, and C 1 ̸= 0. Inthis case the displacements in (11.4) can be written in terms of the basis:⃗ψ T nlm (r,θ,ϕ)= T nl(r) ⃗LY lm (θ, ϕ) (11.6)with T nl (r) proportional to j l (k nl r). The eigenfrequencies are determined bythe boundary conditions (11.2) which read [10]wheref 1 (kR) = 0 (11.7)f 1 (z) ≡ d dz[jl (z)z]. (11.8)• Spheroidal modes: these are obtained by setting C 1 = 0, C 0 ̸= 0 andC 2 ̸= 0. The displacements of (11.4) can be expanded in the basis⃗ψ S nlm (⃗x) = A nl(r)Y lm (θ, ϕ)⃗n − B nl (r)⃗n × ⃗LY lm (θ, ϕ) (11.9)where A nl (r) and B nl (r) are dimensionless radial eigenfunctions [9], whichcan be expressed in terms of the spherical Bessel functions and theirderivatives. The eigenfrequencies are determined by the boundary conditions(11.2) which read [9](f2 (qR) −2µ λdetq2 R 2 )f 0 (qR) l(l + 1) f 1 (kR)1f 1 (qR)2f 2 (kR) + [ l(l+1)= 02− 1] f 0 (kR)(11.10)wheref 0 (z) ≡ j l(z)z 2 , f 2(z) ≡ d2dz 2 j l(z). (11.11)The eigenfrequencies can be determined numerically for both toroidal andspheroidal vibrations. Each mode of order l is (2l + 1)-fold degenerate. Theeigenfrequency values can be obtained from√ µω nl =ρ(kR) nlR . (11.12)11.2.2 Interaction of a metric GW with the sphere vibrational modesThe detector is assumed to be non-relativistic (with sound velocity v s ≪ c andradius R ≪ λ the GW wavelength) and endowed with a high quality factor(Q nl = ω nl τ nl ≫ 1, where τ nl is the decay time of the mode nl). Thedisplacement ⃗u of a point in the detector can be decomposed in normal modesas:⃗u(⃗x, t) = ∑ A N (t) ⃗ψ N (⃗x) (11.13)N