2.2 Long-wavelength approximation

Let L be the size of the detector and ¯λ := λ∕(2π ) be the reduced wavelength of the gravitational wave impinging on the detector. In the long-wavelength approximation the condition ¯λ ≫ L is fulfilled. The angular frequency of the wave equals ω = c∕¯λ. Time delays caused by the finite speed of the wave propagating across the detector are of order Δt ∼ L ∕c, but
ωΔt ∼ L-≪ 1, (14 ) ¯λ
so time delays across the detector are much shorter than the period of the gravitational wave and can be neglected. It means that with a good accuracy the gravitational-wave field can be treated as being uniform (but time-dependent) in the space region that covers the entire detector. To detect gravitational waves with some dominant angular frequency ω one must collect data over time intervals longer (sometimes much longer) than the gravitational-wave period. This implies that in Eq. (8View Equation) for the relative frequency shift, the typical value of the quantity ¯ t := t1 − z1∕c will be much larger than the retardation time Δt := L12∕c. Therefore, we can expand this equation with respect to Δt and keep terms only linear in Δt. After doing this one obtains (see Section 5.3 in [66Jump To The Next Citation Point] for more details):
L12-T ˙( z1) ( 2 2) y12 = − 2c n ⋅H t1 − c ⋅ n + 𝒪 h ,Δt , (15 )
where overdot denotes differentiation with respect to time.

For the configuration of particles shown in Figure 1View Image, the relative frequency shifts y12 and y21 given by Eqs. (12) can be written, by virtue of the formula (15View Equation), in the form

( T ) L3- T ˙ k--⋅ x1 ( 2 2) y12(t) = y21(t) = − 2cn3 ⋅H t + c ⋅ n3 + 𝒪 h ,Δt , (16 )
so that they are equal to each other up to terms ( 2 2) 𝒪 h ,Δt. The photon’s total round-trip frequency shift y121 [cf. Eq. (13View Equation)] is thus equal to
( T ) y (t) = − L3nT ⋅H˙ t + k--⋅ x1 ⋅ n + 𝒪 (h2,Δt2 ). (17 ) 121 c 3 c 3

There are important cases where the long-wavelength approximation is not valid. These include satellite Doppler tracking measurements and the space-borne LISA detector for gravitational-wave frequencies larger than a few mHz.


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