  We revisit the calculation of the gravitational wave spectra generated in a
classically scale-invariant $SU(2)$ gauge sector with a scalar field in the
adjoint representation, as discussed by J.~Jaeckel, et al. The
finite-temperature potential at 1-loop level can induce a strong first-order
phase transition, during which gravitational waves can be generated. With the
accurate numerical computation of the on-shell Euclidean actions of the
nucleation bubbles, we find that the triangle approximation employed by
J.~Jaeckel, et al. strongly distorts the actual potential near its maximum and
thus greatly underestimates the action values. As a result, the gravitational
wave spectra predicted by J.~Jaeckel, et al deviate significantly from the
exact ones in peak frequencies and shapes.
