= Solution
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 \b[vertical interface normal]. <Snell's law> preserves the horizontal <seismic ray parameter>
$$
p=\frac{\sin\theta_w}{1500}=\frac{\sin\theta_{sP}}{1500}=\frac{\sin\theta_{sS}}{900}.
$$
A basement <seismic head wave> of speed $V_h$ has horizontal propagation at the basement interface, so $p=1/V_h$. The two possibilities are:
|| Basement head mode
|| $p$ / $\mathrm{s\,m^{-1}}$
|| Water P angle
|| Sediment P angle
|| Sediment S angle
| P, $V_h=2500\ \mathrm{m\,s^{-1}}$
| $0.000400$
| $36.870^\circ$
| $36.870^\circ$
| $21.100^\circ$
| S, $V_h=1600\ \mathrm{m\,s^{-1}}$
| $0.000625$
| $69.636^\circ$
| $69.636^\circ$
| $34.229^\circ$
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 $L_h\geq0$ be the distance travelled along the basement interface. For sediment legs $a,b\in\{P,S\}$,
$$
x=2h_w\tan\theta_w+h_s(\tan\theta_{sa}+\tan\theta_{sb})+L_h.
$$
The \b[first-appearance distance] is the limiting value at $L_h=0$:
|| Basement head mode
|| Sediment down/up modes
|| $x_{\min}$ / m
| P
| P/P
| $105.00$
| P
| P/S or S/P
| $97.72$
| P
| S/S
| $90.43$
| S
| P/P
| $377.17$
| S
| P/S or S/P
| $336.90$
| S
| S/S
| $296.62$
For example, the unconverted P branch gives $2(50+20)\tan36.870^\circ=105\ \mathrm m$. 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.
\Image[/past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2024/paper-3-q1-head-wave-rays.png]
{title=Surface-to-surface P and converted-S head-wave paths through the water and sediment layers}
{description=Each panel adds an illustrative 40 m basement head segment; the tabulated first-appearance limit removes that segment. Angles are measured from the vertical normal. Reversing the mixed sediment path gives its reciprocal S/P case.}
There is \b[no separate finite-distance P head wave at the water–sediment interface]: $V_{p,w}=V_{p,s}=1500\ \mathrm{m\,s^{-1}}$, 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, $900\ \mathrm{m\,s^{-1}}$, 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.
\b[First appearance and first arrival differ.] For each permitted branch the <travel time> is
$$
t_h(x)=\frac{x}{V_h}+2h_w\sqrt{1500^{-2}-V_h^{-2}}+h_s\left(\sqrt{V_{sa}^{-2}-V_h^{-2}}+\sqrt{V_{sb}^{-2}-V_h^{-2}}\right),\qquad x\geq x_{\min}.
$$
The 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.
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