  The adiabatic growth of a central massive black hole could compress the
surrounding dark matter halo, leading to a steeper profile of the dark matter
halo. This phenomenon is called adiabatic compression. We investigate the
adiabatic compression of wave dark matter - a light bosonic dark matter
candidate with its mass smaller than a few eV. Using the adiabatic theorem, we
show that the adiabatic compression leads to a much denser wave dark matter
halo similar to the particle dark matter halo in the semiclassical limit. The
compressed wave halo differs from that of the particle halo near the center
where the semiclassical approximation breaks down, and the central profile
depends on dark matter and the central black hole mass as they determine
whether the soliton and low angular momentum modes can survive over the
astrophysical time scale without being absorbed by the black hole. Such a
compressed profile has several astrophysical implications. As one example, we
study the gravitational waves from the inspiral between a central
intermediate-mass black hole and a compact solar-mass object within the wave
dark matter halo. Due to the enhanced mass density, the compressed wave dark
matter halo exerts stronger dynamical friction on the orbiting object, leading
to the dephasing of the gravitational waves. The pattern of dephasing is
distinctive from that of inspirals in the particle dark matter halo because of
the difference in density profile and because of the relatively suppressed
dynamical friction force, originating from the wave nature of dark matter. We
investigate the prospects of future gravitational wave detectors, such as Laser
Interferometer Space Antenna, and identify physics scenarios where the wave
dark matter halo can be reconstructed from gravitational wave observations.
