  We explore the gravitational wave spectrum generated by string-wall
structures in an $SO(10)$ ($Spin(10)$) based scenario of pseudo-Goldstone boson
dark matter (pGDM) particle. This dark matter candidate is a linear combination
of the Standard Model (SM) singlets present in the 126 and 16 dimensional Higgs
fields. The Higgs $126$-plet vacuum expectation value (VEV)
$\left<126_H\right>$ leaves unbroken the $\mathbb{Z}_2$ subgroup of
$\mathbb{Z}_4$, the center of $SO(10)$. Among other things, this yields
topologically stable cosmic strings with a string tension $\mu \sim
\left<126_H\right>^2$. The subsequent (spontaneous) breaking of $\mathbb{Z}_2$
at a significantly lower scale by the $16$-plet VEV $\left<16_H\right>$ leads
to the appearance of domain walls bounded by the strings produced earlier. We
display the gravitational wave spectrum for $G \mu$ values varying between
$10^{-15}$ and $10^{-9}$ ($\left<126_H\right>\sim 10^{11}$ - $10^{14}$ GeV),
and $\left<16_H\right>\sim 0.1$ - $10^2$ TeV range ($G$ denotes Newton's
constant.) These predictions can be tested, as we show, by a variety of
(proposed) experiments including LISA, ET, CE and others.
