= Solution
Fix $\omega=kc$. For a P-wave,
$$
m_P^2=\frac{\omega^2}{c_P^2}-k^2
=k^2\left(\frac{c^2}{c_P^2}-1\right).
$$
Set
$$
a=\frac{m_P}{k}
=\sqrt{\frac{c^2}{c_P^2}-1}.
$$
Equal-amplitude P-waves with vertical wavenumbers $\pm m_P$ then give, after an irrelevant common normalization,
$$
\boxed{
\mathbf u_P
=A e^{ik(x-ct)}
\bigl(\cos(akz),\,0,\,ia\sin(akz)\bigr)
}.
$$
Thus $f_1$ is even and both scalar functions multiplying $1$ and $i$ are real.
Similarly, put
$$
b=\frac{m_S}{k}
=\sqrt{\frac{c^2}{c_S^2}-1}.
$$
Using the in-plane transverse polarizations $(m_S,0,-k)$ and $(m_S,0,k)$ for the two vertical directions gives
$$
\boxed{
\mathbf u_{SV}
=B e^{ik(x-ct)}
\bigl(b\cos(bkz),\,0,\,-i\sin(bkz)\bigr)
}.
$$
This has the same required parity and reality properties.
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