= Solution
Use the usual minor and major symmetries of the <elastic stiffness tensor>, positive density, and <strong ellipticity in elasticity>. For the plane <elastic wave> $u=a e^{ik(n\cdot x-vt)}$, substitution into momentum balance gives the <acoustic tensor> problem
$$
\Gamma_{ip}(n)a_p=\rho v^2a_i,\qquad
\Gamma_{ip}(n)=c_{ijpq}n_jn_q,\qquad
\boxed{\det(\Gamma(n)-\rho v^2I)=0.}
$$
The major symmetry makes $\Gamma$ a real <symmetric matrix>. Moreover $a^{\mathsf T}\Gamma a=c_{ijpq}a_i n_j a_p n_q>0$ for nonzero $a$ under <strong ellipticity in elasticity>. Hence its three <eigenvalues> are positive and the three positive <wave speeds> are real. Symmetry alone proves only reality of $v^2$: an unstable <isotropic> tensor with negative <shear modulus> has negative transverse $v^2$. The requested reality therefore uses stability as part of the physical elastic-solid assumption.
The <elastic slowness surface> consists of three sheets, with $s=n/v_\alpha(n)$, satisfying
$$
\det(c_{ijpq}s_js_q-\rho\delta_{ip})=0.
$$
It is centrosymmetric. Sheets can meet at degenerate polarizations; they need not be spherical or convex in an <anisotropic> solid. The <group velocity> is normal to the corresponding smooth sheet: for a sheet tangent $ds$, differentiation of $a^{\mathsf T}[c_{ijpq}s_js_q-\rho I]a=0$ gives $a_i c_{ijpq}a_p s_q\,ds_j=0$. The coefficient vector is proportional to the <group velocity>, so its scalar product with every tangent vanishes.
Fix horizontal <slowness> $p=(n_1/c,n_2/c)$ and put $q=s_3$. Define
$$
T_{ip}=c_{i3p3},\qquad
R_{ip}=\sum_{\alpha=1}^2c_{i3p\alpha}p_\alpha,\qquad
Q_{ip}=\sum_{\alpha,\beta=1}^2c_{i\alpha p\beta}p_\alpha p_\beta.
$$
The symbol $p$ on a tensor index is an index, whereas $p_\alpha$ is a horizontal <slowness> component. The vertical <slowness> equation is the <vertical slowness sextic of an anisotropic solid>
$$
\boxed{\det\{Tq^2+(R+R^{\mathsf T})q+Q-\rho I\}=0.}
$$
The leading coefficient is $\det T>0$, so it is a degree-six polynomial with real coefficients. Nonreal roots occur in conjugate pairs, leaving between zero and six real roots counted with multiplicity. Generically the counts are $0,2,4,6$; a grazing repeated root can give an odd count of distinct real roots. Geometrically these are the intersections of a vertical line of fixed horizontal <slowness> with the three sheets.
All four generic counts are actually possible in a stable material. For example, take $\rho=1$ and orthotropic Voigt stiffnesses
$$
C_{11}=25,\ C_{22}=16,\ C_{33}=9,\ C_{13}=-1,\ C_{12}=C_{23}=0,\quad
C_{44}=4,\ C_{55}=1,\ C_{66}=2.
$$
The stiffness matrix is positive definite. For horizontal <slowness> $(p,0)$, the three diagonal acoustic equations are
$$
25p^2+q^2=1,\qquad 2p^2+4q^2=1,\qquad p^2+9q^2=1.
$$
The choices $p=0.1,0.3,0.8,1.2$ give respectively six, four, two and zero real vertical roots.
For the general depth-dependent amplitude, substitution of the horizontal harmonic factor yields
$$
T U''+i\omega(R+R^{\mathsf T})U'-\omega^2(Q-\rho I)U=0.
$$
For six simple roots $q_\ell$, choose nonzero null vectors $a_\ell$ of the corresponding matrix. The complete solution is
$$
\boxed{U(x_3)=\sum_{\ell=1}^6 A_\ell a_\ell e^{i\omega q_\ell x_3}.}
$$
Complex vertical roots give <evanescent waves>. At defective repeated roots the usual polynomial-in-depth factors multiplying exponentials must be included, or the solution can be defined by a limiting procedure.
To classify the modes, take $x_3$ positive downwards and $\omega>0$. The reduced <traction> for a mode is
$$
b_\ell=(Tq_\ell+R)a_\ell,\qquad \sigma_{i3}=i\omega b_{\ell,i}.
$$
Mechanical <energy> crosses a plane at rate $-\sigma_{i3}u_{i,t}$. Averaging the real harmonic fields uses $\langle\operatorname{Re}(f e^{-i\omega t})\operatorname{Re}(g e^{-i\omega t})\rangle=\operatorname{Re}(f g^*)/2$. With velocity amplitude $-i\omega a_\ell$ and <traction> amplitude $i\omega b_\ell$, the time-averaged vertical <acoustic energy flux> is
$$
J_{3,\ell}=\frac{\omega^2}{2}\operatorname{Re}(a_\ell^*b_\ell).
$$
For a real propagating mode, write its physical wavevector as $k=\omega(p_1,p_2,q)$. Differentiate $a^{\mathsf T}\Gamma(k)a=\rho\omega(k)^2a^{\mathsf T}a$ in $k_3$. The derivatives of $a$ cancel by the <eigenvalue> equation, while $a^{\mathsf T}(\partial_{k_3}\Gamma)a=2\omega a^{\mathsf T}(Tq+R)a$. Thus its vertical <group velocity> and flux are
$$
g_{3,\ell}=\frac{a_\ell^{\mathsf T}(Tq_\ell+R)a_\ell}
{\rho\,a_\ell^{\mathsf T}a_\ell},\qquad
J_{3,\ell}=\frac12\rho\omega^2|a_\ell|^2g_{3,\ell}.
$$
Thus the <elastic mode classification by vertical energy flux> is \b[upgoing if $J_3<0$, downgoing if $J_3>0$]. In <anisotropy>, the sign of vertical phase <slowness> alone need not classify energy propagation. For evanescent modes, select $\operatorname{Im}q>0$ for decay into a downward half-space and $\operatorname{Im}q<0$ for decay into an upward half-space. Away from grazing and degeneracy, continuation from zero horizontal <slowness>, with a small radiation damping if needed, separates three outgoing modes on each side. Superpose only the appropriate three to construct the most general upgoing or downgoing field; limiting modes handle exceptional cases.
For a free surface at $x_3=0$ with solid occupying $x_3>0$, let $(q_0,a_0)$ be a unit-amplitude incident upgoing mode. The reflected field consists of the three downgoing propagating or depth-decaying modes:
$$
u=\left[a_0e^{i\omega q_0x_3}+
\sum_{r=1}^3 A_ra_re^{i\omega q_rx_3}\right]
e^{i\omega(p_1x_1+p_2x_2-t)}.
$$
Frequency and horizontal <slowness> are common to all modes, by translation invariance of the surface. Zero surface <traction> imposes the three equations
$$
b_0+\sum_{r=1}^3A_rb_r=0,\qquad
B=(b_1,b_2,b_3).
$$
Hence the <free-surface anisotropic reflection amplitudes> are
$$
\boxed{(A_1,A_2,A_3)^{\mathsf T}=-B^{-1}b_0}
$$
whenever $B$ is invertible. This specifies mode conversion as well as reflection. If $B$ is singular, a nonzero outgoing homogeneous surface mode exists; these are surface-wave poles, including <Rayleigh waves> in the <isotropic> case. At such parameters one must use the appropriate limiting radiation solution and check solvability instead of assuming a unique finite inverse.
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