An alternative, non-algebraic way to fix the ordering is to use the physical requirement that a surface wave must be evanescent (its amplitude must decay with depth away from the free surface) with respect to BOTH the P and S wavefields it is built from.
A wave of horizontal phase velocity \(c\) has a vertical wavenumber \(k_z = k\sqrt{(V/c)^2 - 1}\) for a body wave of velocity \(V\). For this vertical wavenumber to be imaginary (giving exponential decay with depth, which is what makes it a surface wave rather than a wave that radiates energy into the half-space), we need \(c < V\) for that wave type. Since the Rayleigh wave must decay evanescently in terms of both its P-wave and SV-wave potentials, its phase velocity must be less than the SLOWER of the two body-wave velocities, i.e. less than \(V_s\) (because \(V_s < V_p\) always). Hence \(V_r < V_s\) is not a coincidence of the Poisson-solid root -- it is a structural requirement for the wave to exist as a surface wave at all.
Combined with the universal elastic-solid inequality \(V_s < V_p\) (shear modulus alone is always less than the P-wave modulus \(K + 4\mu/3\)), the only consistent ordering is
\[ V_r < V_s < V_p \]which is option (A). Options (B), (C) and (D) all place \(V_r\) at or above \(V_s\), which would make the Rayleigh wave a radiating (non-evanescent, non-surface) wave -- physically inconsistent with it being a surface wave at all.
\(\boxed{\text{Answer: (A)}}\)