### 4.5 Gravitational wave detection

A gravitational wave detector can be modelled as a body of mass M at a distance L from a fiducial
laboratory point, connected to the point by a spring of resonant frequency and quality
factor Q. From the equation of geodesic deviation, the infinitesimal displacement of the
mass along the line of separation from its equilibrium position satisfies the equation of motion
where and are “beam-pattern” factors that depend on the direction of the source
and on a polarization angle , and and are gravitational waveforms
corresponding to the two polarizations of the gravitational wave (for a review, see [255]). In a source
coordinate system in which the x–y plane is the plane of the sky and the z-direction points toward the
detector, these two modes are given by
where represent transverse-traceless (TT) projections of the calculated waveform of Equation (68),
given by
where is a unit vector pointing toward the detector. The beam pattern factors depend on the
orientation and nature of the detector. For a wave approaching along the laboratory z-direction, and for a
mass whose location on the x–y plane makes an angle with the x-axis, the beam pattern
factors are given by and . For a resonant cylinder oriented along the
laboratory z-axis, and for source direction , they are given by ,
(the angle measures the relative orientation of the laboratory and
source x-axes). For a laser interferometer with one arm along the laboratory x-axis, the other
along the y-axis, and with defined as the differential displacement along the two arms,
the beam pattern functions are and
.
The waveforms and depend on the nature and evolution of the source. For example, for
a binary system in a circular orbit, with an inclination i relative to the plane of the sky, and the x-axis
oriented along the major axis of the projected orbit, the quadrupole approximation of Equation (70) gives

where is the orbital phase.