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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 isIt requires enough offset for the two inclined upper-layer legs, $X\ge 2H\tan i_c$. 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.

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. 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.

The normal-incidence acoustic impedance is $Z=\rho V_p$, with units $\mathrm{kg}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. ThusDefine the pressure-amplitude reflection coefficient and pressure-amplitude transmission coefficient for incidence downwards from layer $i$ into layer $j$ byHere $p$ denotes compressional normal-stress amplitude; in the water it is the acoustic pressure. Continuity of normal traction and normal particle velocity gives these expressions. They describe each interface separately, before any layer layer reverberation.

For a lossless normal-incidence plane wave, intensity is proportional to $p^2/Z$, soA transmitted pressure amplitude exceeding unity is therefore compatible with conservation of energy. If instead the coefficient is defined for particle velocity measured along the same downward coordinate, $r_v=-r_p$ and $t_v=2Z_i/(Z_i+Z_j)=1-r_p$: the two transmitted velocity coefficients are $0.8000$ and $18/29=0.62069$. Upward incidence reverses the sign of $r_p$, interchanges the transmitted pressure and velocity coefficients, and leaves $R_E,T_E$ unchanged. At normal incidence between these ideal isotropic layers there is no P-to-S seismic mode conversion.

Take horizontal interfaces, water depth $h_w=50\mathrm{m}$, and sediment thickness $h_s=20\mathrm{m}$. Both source and receiver are just below the sea surface. Every angle below is measured from the vertical interface normal. Snell's law preserves the horizontal seismic ray parameterA basement seismic head wave of speed $V_h$ has horizontal propagation at the basement interface, so $p=1/V_h$. The two possibilities are:

The water supports only P waves. At oblique incidence the seabed can convert a P wave into an SV-wave in the sediment, and the return SV-wave can convert back to a P wave in the water. Further seismic mode conversion can occur at the basement. Thus the down-going and up-going sediment legs can independently be P or S. The S entries refer to SV-waves; a fluid P wave source in this two-dimensional isotropic geometry does not excite the decoupled SH-wave sector.

Let $L_h\ge 0$ be the distance travelled along the basement interface. For sediment legs $a,b\in \{P,S\}$,The first-appearance distance is the limiting value at $L_h=0$:

For example, the unconverted P branch gives $2(50+20)\tan 36.870^{\circ }=105\mathrm{m}$. The earliest geometrically permitted basement P seismic head wave is the doubly converted path with sediment S legs, at approximately $90\mathrm{m}$. The corresponding basement S family first appears at approximately $297\mathrm{m}$. These limiting distances describe existence of ray branches; a finite detectable wave amplitude also depends on conversion efficiency and acquisition noise.

There is no separate finite-distance P head wave at the water–sediment interface: $V_{p,w}=V_{p,s}=1500\mathrm{m}{\mathrm{s}}^{-1}$, so the nominal critical angle is $90^{\circ }$, and a ray from a source 50 m above the interface cannot reach it at finite horizontal distance. Nor can the slower sediment S speed, $900\mathrm{m}{\mathrm{s}}^{-1}$, support a seismic head wave launched from water: it would require $\sin {\theta }_w=1500/900>1$. A seismic reflection still exists at the seabed because the densities, hence the acoustic impedances, differ.

First appearance and first arrival differ. For each permitted branch the travel time isThe all-P basement P branch has $t_h=x/2500+0.0746667\mathrm{s}$. It overtakes the direct water P wave, $t_d=x/1500$, only at $x=280\mathrm{m}$. The converted P-head branches have larger seismic intercept times, so even where they exist at shorter offsets they are later arrivals. The S-head branches are likewise later than the all-P P-head branch wherever they exist in this model. Reflections and layer reverberations can supply additional later phases.