  We study the gravitational wave (GW) signature in the doublet left-right
symmetric model (DLRSM) resulting from the strong first-order phase transition
(SFOPT) associated with $SU(2)_R\times U(1)_{B-L}$-breaking. For different
values of the symmetry-breaking scale $v_R =20,~30$, and $50$ TeV, we construct
the one-loop finite temperature effective potential to explore the parameter
space for regions showing SFOPT. We identify the region where the associated
stochastic GW background is strong enough to be detected at planned GW
observatories. A strong GW background favors a relatively light neutral CP-even
scalar $H_{3}$, arising from the $SU(2)_R$ doublet. The $SU(2)_L$ subgroup of
DLRSM is broken by three vevs: $\kappa_1,~\kappa_2$, and $v_L$. We observe a
preference for $\mathcal{O}(1)$ values of the ratio $w=v_L/\kappa_1$, but no
clear preference for the ratio $r=\kappa_2/\kappa_1$. A large number of points
with strong GW background can be ruled out from precise measurement of the
trilinear Higgs coupling and searches for $H_3$ at future colliders.
