  Gravitational radiation is an excellent field for testing theories of gravity
in strong gravitational fields. The current observations on the
gravitational-wave (GW) bursts by LIGO have already placed various constraints
on the alternative theories of gravity. In this paper, we investigate the
possible bounds which could be placed on the Brans-Dicke gravity using GW
detection from inspiralling compact binaries with the proposed Einstein
Telescope, a third-generation GW detector. We first calculate in details the
waveforms of gravitational radiation in the lowest post-Newtonian
approximation, including the tensor and scalar fields, which can be divided
into the three polarization modes, i.e. "plus mode", "cross mode" and
"breathing mode". Applying the stationary phase approximation, we obtain their
Fourier transforms, and derive the correction terms in amplitude, phase and
polarization of GWs, relative to the corresponding results in General
Relativity. Imposing the noise level of Einstein Telescope, we find that the GW
detection from inspiralling compact binaries, composed of a neutron star and a
black hole, can place stringent constraints on the Brans-Dicke gravity. The
bound on the coupling constant $\omega_{\rm BD}$ depends on the mass,
sky-position, orbital angle, polarization angle, luminosity distance, redshift
distribution and total observed number $N_{\rm GW}$ of the binary systems.
Taking into account all the burst events up to redshift $z=5$, we find that the
bound could be $\omega_{\rm BD}\gtrsim 10^{6}\times(N_{\rm GW}/10^4)^{1/2}$.
Even for the conservative estimation with $10^{4}$ observed events, the bound
is still more than one order tighter than the current limit from Solar System
experiments. So, we conclude that Einstein Telescope will provide a powerful
platform to test alternative theories of gravity.
