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

154 Detection of scalar gravitational waves11.2 Testing theories of gravity11.2.1 Free vibrations of an elastic sphereBefore discussing the interaction with an external GW field, let us considerthe basic equations governing the free vibrations of a perfectly homogeneous,isotropic sphere of radius R, made of a material having density ρ and Lamécoefficients λ and µ [8].Following the notation of [9], let x i , i = 1, 2, 3 be the equilibrium position ofthe element of the elastic sphere and xi ′ be the deformed position, then u i = xi ′ −x iis the displacement vector. Such vector is assumed small, so that the linear theoryof elasticity is applicable. The strain tensor is defined as u ij = (1/2)(u i, j + u j,i )and is related to the stress tensor by σ ij = δ ij λu ll + 2µu ij . The equations ofmotion of the free vibrating sphere are thus [8]ρ ∂2 u i∂t 2 = ∂∂x j (δ ijλu ll + 2µu ij ) (11.1)with the boundary condition:n j σ ij = 0 (11.2)at r = R where n i ≡ x i /r is the unit normal. These conditions simplystate that the surface of the sphere is free to vibrate. The displacement u i is atime-dependent vector, whose time dependence can be factorized as u i (⃗x, t) =u i (⃗x) exp(iωt), where ω is the frequency. The equations of motion then become:µ∇ 2 u i + (λ + µ)∇ i (∇ j u j ) =−ω 2 ρu i . (11.3)Their solutions can be expressed as a sum of a longitudinal and two transversevectors [10]:⃗u(⃗x) = C 0⃗∇φ(⃗x) + C 1⃗Lχ(⃗x) + C 2⃗∇×⃗Lχ(⃗x) (11.4)where C 0 , C 1 , C 2 are dimensioned constants and ⃗L ≡ ⃗x × ⃗∇ is the angularmomentum operator. Regularity at r = 0 restricts the scalar functions φ and χ tobe expressed as φ(r,θ,ϕ) ≡ j l (qr)Y lm (θ, ϕ) and χ(r,θ,ϕ) ≡ j l (kr)Y lm (θ, ϕ).Y lm (θ, ϕ) are the spherical harmonics and j l the spherical Bessel functions [11]:( ) 1 d l ( ) sin xj l (x) =x dx x(11.5)q 2 ≡ ρω 2 /(λ + 2µ) and k 2 ≡ ρω 2 /µ are the longitudinal and transversewavevectors, respectively.Imposing the boundary conditions (11.2) at r = R yields two families ofsolutions: