= Solution
Use geometric rays for body-wave arrivals and a traction-free upper surface. A compact <isotropic explosion source> has moment tensor proportional to $\delta_{ij}$ and produces direct P body radiation in a homogeneous <isotropic> medium. Thus the basic near-offset arrival is the direct upper-layer <P wave>, with time $X/\alpha_1$. A pure explosion is not a shear fault: it does not automatically emit a direct <S wave> of comparable strength. Interfaces can nevertheless convert its P <energy> into SV motion, and the free surface can support <Rayleigh waves>.
Downward P <energy> reflects from the interface to give a PP <reflection> returning to the receiver; for a horizontal interface its ray is symmetric and its time is $\sqrt{X^2+4H^2}/\alpha_1$. Other transmitted P <energy> enters the faster half-space. At $\sin i_c=\alpha_1/\alpha_2$, a critical ray travels along the interface and feeds an upgoing <seismic head wave>, often labeled Pn in this simplified crustal model. Its ideal travel-time branch is
$$
t_{\rm head}=\frac{X}{\alpha_2}+\frac{2H\cos i_c}{\alpha_1}.
$$
It requires enough offset for the two inclined upper-layer legs, $X\ge2H\tan i_c$. Once $X$ exceeds the larger <first-arrival crossover distance> $X_c=2H\sqrt{(\alpha_2+\alpha_1)/(\alpha_2-\alpha_1)}$, this branch arrives before the direct upper-layer wave because it spends most of the path in faster material. A transmitted ray descending into an otherwise uniform infinite half-space does not return as a separate <reflection> without an additional reflector.
\Image[/past-exam-of-the-physics-of-the-earth-as-a-planet-course-of-the-university-of-cambridge/2016/paper-1-seismic-phases.png]
{title=Original two-layer P-ray sketches and illustrative direct, reflected and critically refracted travel-time branches}
At oblique incidence, the bonded-interface conditions couple P and SV <polarizations>. A downgoing P ray can return as a converted SV <reflection> in the upper layer, with common horizontal slowness $p=\sin i_P/\alpha_1=\sin i_S/\beta_1$. There are transmitted P/SV branches and, after further interaction with the free surface or interface, mixed-mode <reflections> and <layer reverberations>. Conversion coefficients depend on <density> and impedance as well as wave <speeds>, and some vanish at normal incidence. The supplied inequalities do not order $\beta_2$ against $\alpha_1$, so one cannot universally draw a P-incident critical head ray travelling at the lower-layer S <speed>: that direct conversion construction requires $\beta_2>\alpha_1$. A shear head-wave phase can be present if the appropriate SV <energy> and ray conditions are available, but is not an inevitable direct explosion arrival from the stated data alone.
At later times a shallow explosion can excite dispersive Rayleigh-wave trains, with coupled P-SV motion decaying with depth. Short wavelengths mainly sample the slow upper layer, whereas long wavelengths penetrate the lower half-space; group and phase <speeds> need not be equal. Multiple reflected/converted body waves can overlap this coda. The shear-speed contrast also permits Love-wave modes structurally, but SH motion is decoupled from P-SV in this flat <isotropic> model. \b[An ideal <isotropic> explosion does not excite <Love waves> here; appreciable SH/Love <energy> would require source asymmetry, scattering or a broken symmetry of the model.]
Exact amplitudes, resolved arrival separation and attenuation cannot be calculated without <densities>, source spectrum, damping and instrument response. The qualitative robust features are a near-offset direct P arrival, reflected and converted phases, a faster-lower-layer P head wave that can become first at long range, reverberations and P-SV surface-wave <energy>. The ray diagram uses declared illustrative <speed> ratios solely to show the geometry; they are not additional supplied material constants.
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