  Binary black hole (BBH) systems are usually located in the gravitational
potential well formed by a massive black hole (BH), which is mostly located in
the center of a galaxy. In most existing studies, the BBH systems are treated
as isolated systems, while the effect of the background is ignored. The
validity of the approximation is based on the belief that the background
gravitational field from other sources is extremely weak compared with the
strong gravitational field produced by the BBH itself during the evolution, and
can be neglected in gravitational wave detection. However, it is still
interesting to check how valid this approximation is. In this work, instead of
simulating the three-BH problem with a fully relativistic treatment, we use a
perturbational scheme to investigate the effect of the background gravitational
potential on the evolution of a BBH, especially on the waveform of its
gravitational radiation. Four scenarios are considered including the head-on
collision and the inspiral-to-merger process of a BBH which is either
freefalling towards or circularly orbiting around a third large BH. The head-on
collision and the circular inspiral are two limits of all possible
configurations. The existence of the background gravitational potential changes
the arrival time of the gravitational wavefront of a BH, prolongs the
wavelength, and increases the gravitational radiation energy. And most
interestingly, the background gravitational potential induces the higher-order
modes of the gravitational wave of a BBH. These interesting phenomena can be
explained by the gravitational redshift effect and the change of eccentricity
of a BBH's orbit from the background gravitational potential.
