### 2.2 Long-wavelength approximation

Let be the size of the detector and be the reduced wavelength of the
gravitational wave impinging on the detector. In the long-wavelength approximation the condition
is fulfilled. The angular frequency of the wave equals . Time delays caused
by the finite speed of the wave propagating across the detector are of order , but
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 one must collect data over time intervals longer (sometimes much
longer) than the gravitational-wave period. This implies that in Eq. (8) for the relative frequency shift, the
typical value of the quantity will be much larger than the retardation time
. Therefore, we can expand this equation with respect to and keep terms
only linear in . After doing this one obtains (see Section 5.3 in [66] for more details):
where overdot denotes differentiation with respect to time.
For the configuration of particles shown in Figure 1, the relative frequency shifts and given
by Eqs. (12) can be written, by virtue of the formula (15), in the form

so that they are equal to each other up to terms . The photon’s total round-trip frequency
shift [cf. Eq. (13)] is thus equal to
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.