  The gravitational wave (GW) background produced at the cosmological chiral
phase transition in a conformal extension of the standard model is studied. To
obtain the bounce solution of coupled field equations we implement an iterative
method. We find that the corresponding $O(3)$ symmetric Euclidean action $S_3$
divided by the temperature $T$ has a simple behavior near the critical
temperature $T_C$: $S_3/T \propto (1-T/T_C)^{-\gamma}$, which is subsequently
used to determine the transition's inverse duration $\beta$ normalized to the
Hubble parameter $H$. It turns out that $\beta/H \gtrsim 10^3$, implying that
the sound wave period $\tau_\text{sw}$ as an active GW source, too, can be much
shorter than the Hubble time. We therefore compute $\tau_\text{sw} H$ and use
it as the reduction factor for the sound wave contribution. The signal-to-noise
ratio (SNR) for Deci-Hertz Interferometer Gravitational Wave Observatory
(DECIGO) and Big Bang Observer (BBO) is evaluated, with the result:
SNR$^\text{DECIGO} \lesssim 1.2$ and SNR$^\text{BBO} \lesssim 12.0$ for five
years observation, from which we conclude that the GW signal predicted by the
model in the optimistic case could be detected at BBO.
