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

168 Detection of scalar gravitational waves11.4 The hollow sphereAn appealing variant of the massive sphere is a hollow sphere [25]. The latterhas the remarkable property that it enables the detector to monitor GW signals ina significantly lower frequency range—down to about 200 Hz—than its massivecounterpart for comparable sphere masses. This can be considered a positiveadvantage for a future worldwide network of GW detectors, as the sensitivityrange of such antenna overlaps with that of the large-scale interferometers, now ina rather advanced state of construction [6,7]. In this section we study the responseof such a detector to the GW energy emitted by a binary system constituted of starsof masses of the order of the solar mass. A hollow sphere obviously has the samesymmetry of the massive one, so the general structure of its normal modes ofvibration is very similar [25] to that of the solid sphere. In particular, the hollowsphere is very well adapted to sense and monitors the presence of scalar modes inthe incoming GW signal. The extension of the analysis of the previous sectionsto a hollow sphere is quite straightforward and in the following we will only givethe main results. Due to the different geometry, the vibrational modes of a hollowsphere differ from those studied in section 11.2. In the case of a hollow sphere,we have two boundaries given by the outer and the inner surfaces of the soliditself. We use the notation a for the inner radius and R for the outer radius. Theboundary conditions are thus expressed byσ ij n j = 0 atr = R and at r = a (R ≥ a ≥ 0), (11.86)(11.3) must now be solved subject to these boundary conditions. The solutionthat leads to spheroidal modes is still (11.9) where the radial functions A nl (r) andB nl (r) have rather complicated expressions:A nl (r) = C nl[ 1q S nl+ D nl1q S nlddr j l(qnl S r) − l(l + 1)K j l (knl S r)nlknl S rddr y l(qnl S r) − l(l + 1) y l (knl ˜D S r) ]nlknl S rB nl (r) = C nl[jl (q S nl r)q S nl r − K nl1k S nl r ddr {rj l(k S nl r)}(11.87)+ D nly l (q S nl r)q S nl r − ˜D nl1k S nl r ddr {ry l(k S nl r)} ]. (11.88)Here knl S R and qS nlR are dimensionless eigenvalues, and they are the solutionto a rather complicated algebraic equation for the frequencies ω = ω nl —see [25]for details. In (11.87) and (11.88) we have setK nl ≡ C tqnlSC l knlS , D nl ≡ qS nlknlS E, ˜D nl ≡ C t FqnlSC l knlS(11.89)