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\begin{document}
\begin{center}
\Large
\textbf{Supplemental Material to ``Low frequency tail of gravitational wave spectra from hydromagnetic turbulence''}\\
\normalsize
by Ramkishor Sharma $\&$ Axel Brandenburg
\end{center}
\begin{abstract}
\noindent
In Figs.~3 and 4 of the original version of our paper
\url{https://arxiv.org/abs/2206.00055}, we have shown the time evolution
of $|\tildeT(k)|$ and $|\dot{\tildeh}(k)|$ for five different $k$ values
of the wave vector $(k,0,0)$.
In this Supplemental Material, we provide the time evolution of the same
quantities for wave vectors $(0,k,0)$ and $(0,0,k)$.
The overall behavior of the phases is similar to that for $(k,0,0)$.
\end{abstract}

\section*{Time evolution of stress and the derivative of gravitational wave strain at selected wave vectors}

As discussed in Section~II.B of the paper, the time evolutions
of $|\tildeT(k)|$ and $|\dot{\tildeh}(k)|$ for a particular wave
vector can be very different than the collective effect of all the
wave vectors of the same length. Furthermore, $\arg(\tildeT)$ and
$\arg(\dot{\tildeh})$ remain approximately constant for some time and
start evolving more strongly after that. Here, we provide the time time
evolutions of $|\tildeT(k)|$, $|\dot{\tildeh}(k)|$, $\arg(\tildeT)$, and
$\arg(\dot{\tildeh})$ for two more wave vectors $(0,k,0)$ and $(0,0,k)$ in
\Figs{rslice_stress_plot_LowFreq_y}{rslice_stress_plot_LowFreq_z}, respectively.
\Figs{rslice_stress_plot_LowFreq_y}{rslice_stress_plot_LowFreq_z}
are for the helical case (HEL) and
\Figs{rslice_stress_plot_LowFreq_y}{rslice_stress_plot_LowFreq_z} are the
corresponding figures for the nonhelical case (NHEL).

\section*{Conclusion}
As is evident from these figures, the over behavior of $\arg(\tildeT)$
and $\arg(\dot{\tildeh})$ is similar as for the wave vector $(k,0,0)$
shown in Figure 3 and 4 of the paper for the helical and non helical
case, respectively.

\vskip 1cm

\begin{figure*}[ht]\begin{center}
\includegraphics[width=\textwidth]{rslice_stress_plot_LowFreq_y}
\end{center}\caption{
Modulus and phase of $\tilde{T}(k,t)$ and $\dot{\tilde{h}}(k,t)$
for the helical case for $\kk=(0,k,0)$ with $k=0.3$ (orange), 0.4 (red),
0.5 (green), 0.6 (blue), and 0.7 (black).
The inset shows the phase with a linear abscissa.
}\label{rslice_stress_plot_LowFreq_y}\end{figure*}

\begin{figure*}[ht]\begin{center}
\includegraphics[width=\textwidth]{rslice_stress_plot_LowFreq_z}
\end{center}\caption{
Same as \Fig{rslice_stress_plot_LowFreq_nohel_y}, but for the wave vector $\kk=(0,0,k)$.
}\label{rslice_stress_plot_LowFreq_z}\end{figure*}

\begin{figure*}[ht]\begin{center}
\includegraphics[width=\textwidth]{rslice_stress_plot_LowFreq_nohel_y}
\end{center}\caption{
Same as \Fig{rslice_stress_plot_LowFreq_y}, but for the nonhelical case
}\label{rslice_stress_plot_LowFreq_nohel_y}\end{figure*}

\begin{figure*}[ht]\begin{center}
\includegraphics[width=\textwidth]{rslice_stress_plot_LowFreq_nohel_z}
\end{center}\caption{
Same as \Fig{rslice_stress_plot_LowFreq_z}, but for the nonhelical case.
}\label{rslice_stress_plot_LowFreq_nohel_z}\end{figure*}

\end{document}
