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

Sensitivity for various GW signals 97where V s is the maximum signal. For the thermal noise we haveVnb 2 = 1 ∫ ∞β122π −∞ β1 2 + S ω2 uu dω = S uuβ 12(8.31)where Vnb 2 = α2 kT emω02(8.5)) .Introducing the signal energy E svariable y =β ω 1)SNR = 1 Sg 2β ∫1 ∞2 2π S uuis the mean square narrow-band noise (see equations (8.4) and−∞= 1 2 mω2 0 ( V sα )2 we calculate (using thedy1 + Ɣ(1 + y 2 ) = V s2√ =Ɣ8V 2 nbE s4kT e√Ɣ. (8.32)The factor of 1 2has been introduced because we have supposed the signal to beall at a given phase, while we add to the noise in phase the noise in quadrature,thus reducing the SNR by a factor of two. The effective noise temperature is [9]T eff = 4T e√Ɣ. (8.33)For a continuous source we can directly apply equation (8.22). With a totalmeasuring time t m the continuous source with amplitude h 0 and angular frequencyω 0 appears as a wavepacket with its Fourier transform at ω 0S(ω 0 ) =( )h0 t 2 m(8.34)2and with bandwidthδ f = 2(8.35)t mwhich is very small for long observation times. Indicating with N(ω 0 ) the powerspectrum of the measured noise at the resonance, we obtain the amplitude of thewave that can be observed with SNR = 1:h 0 =√2N(ω 0 )t m. (8.36)A similar result has been obtained in the past [10] using a different procedure.In practical cases it is often not possible to calculate the Fourier spectrumN(ω 0 ) from experimental data over the entire period of measurement t m , eitherbecause the number of steps in the spectrum would be too large for a computeror because the physical conditions change as, for instance, a change in frequencydue to the Doppler effect. It is then necessary to divide the period t m in n subperiodsof length △t = t mn. In the search for a monochromatic wave we then haveto consider two cases: (a) The wave frequency is exactly known. In this case we