Past exam of the physics of the Earth as a Planet course of the University of Cambridge / 2012 / Paper 1 / 3

Hawaii: short-wavelength volcanic loading and long-wavelength mantle support must be distinguished. A dense volcanic edifice replacing seawater adds mass and gives a positive free-air gravity anomaly. Its load bends the lithosphere down, producing a surrounding moat and an outer flexural arch. The associated displaced density interfaces contribute negative and positive gravity signals at neighbouring distances. Observed gravity near the Hawaiian ridge includes strong positive anomalies over the ridge, negative flanks, and a broader positive belt. At much longer wavelengths, the Hawaiian swell also correlates with a positive gravity anomaly, with a smaller gravity-to-topography ratio that requires compensation; the precise source of support is not determined by that correlation alone.^[ref]

The mechanical distinction can be expressed with a simple elastic-plate model. For load height $h$, downward deflection $w$, plate rigidity $D$, loading density contrast $\Delta {\rho }_l$ relative to water, and restoring density contrast $\Delta {\rho }_b$ at an interface,For a Fourier component of horizontal wavenumber $q$, this gives $w_q=\Delta {\rho }_{l}g{h}_q/(D{q}^4+\Delta {\rho }_{b}g)$. The load's direct gravity contribution competes with the negative contribution of a depressed interface at reference depth $t$. In a linear thin-interface approximation,At short wavelengths rigidity suppresses compensation, leaving relatively high positive gravity admittance. At long wavelengths the plate can deflect and compensate more completely, reducing admittance. Fitting the wavelength dependence constrains effective elastic thickness through $D=E_Y{T}_e^3/[12(1-{\nu }^2)]$, subject to density structure, load distribution and subsurface loading assumptions. This explains why local Airy compensation alone misses the spatially extended flexural response.

The broad swell needs a deeper explanation than the narrow volcanic pile: hot, buoyant mantle and mantle flow can support regional topography, while deep density anomalies and surface elevation jointly determine its gravity. A free-air anomaly keeps the attracting mass of the topography; a Bouguer correction removes an assumed direct topographic attraction and can reveal its compensating mass deficit. Gravity plus topography constrain support as a function of wavelength; neither alone uniquely fixes the mantle density distribution.

For the OR alternative, triple junctions are points where three plate boundaries meet. Each boundary can be a spreading ridge, subduction trench, or transform fault. The three plate velocities satisfyThis velocity triangle is necessary but not enough: a single junction velocity ${\mathbf{u}}_J$ must also keep the meeting point on all three moving boundaries. For symmetric spreading between A and B, the ridge constraint is $[{\mathbf{u}}_J-({\mathbf{v}}_A+{\mathbf{v}}_B)/2]\cdot \mathbf{n}=0$. For a transform, ${\mathbf{v}}_A\cdot \mathbf{n}={\mathbf{v}}_B\cdot \mathbf{n}$, and the junction must have that same normal component. For a trench, its normal migration must match the adopted subduction/rollback model; a trench fixed relative to the overriding plate shares that plate's normal velocity.

In velocity space each boundary supplies a line of possible junction velocities. A common intersection means the junction can migrate while preserving its local configuration; incompatible constraints force boundary reorganization. This is kinematic stability, not the claim that the junction is stationary. A symmetric ridge–ridge–ridge example has three arms meeting at $120^{\circ }$ and a common junction velocity; unequal spreading rates do not require those equal angles. Transform and trench junctions depend on the allowed relative motions and boundary geometry. Stable and unstable evolution were formulated through plate-velocity geometry in the original triple-junction analysis.^[ref]

These original diagrams show an illustrative flexural response and a symmetric ridge-junction construction; they are not maps or measured Hawaiian profiles.

For the ray sketches in (i) and (ii), take a horizontally layered isotropic medium, a surface source and receiver, depth $z$ positive downwards, and wave speed $v(z)$. Snell's law conserves $p=\sin i/v$, where $i$ is measured from the vertical. A returning diving ray turns where $pv(z_t)=1$, withFor a uniform positive gradient $v=v_s+\beta z$, rays are circular arcs andThus ordinary travel time increases with distance but has a decreasing slope as deeper rays sample faster material. The following two leaves discuss the labelled changes to this reference structure.

For the OR alternative, the supplied fault-plane diagrams, assume the usual convention that shaded quadrants are compressional first motions on a lower-hemisphere projection. Each pair of nodal planes represents a fault plane and an auxiliary plane; first-motion polarity alone cannot choose which slipped. Geological mapping, aftershock locations or rupture observations can resolve that ambiguity.^[ref]

Diagram (i): normal faulting. The central unshaded band indicates the extensional dip-slip pattern. It is compatible with crustal extension, such as a continental rift or a spreading environment, with a hanging wall moving down relative to the footwall on the actual fault. Its two curved nodal traces are the possible fault and auxiliary planes.

Diagram (ii): strike-slip faulting. The two straight nodal diameters divide four alternating quadrants, as expected for two vertical candidate planes. It is compatible with a transform plate boundary or continental shear zone. One candidate is right lateral and the conjugate is left lateral; polarity alone does not select the actual plane. Without a north arrow, geographical strikes cannot be assigned.

Diagram (iii): oblique-slip faulting with a reverse component. The off-centre nodal intersection and curved traces distinguish it from pure vertical strike slip, while shading through the centre indicates the reverse component under the stated convention. It is compatible with transpression, such as oblique convergence at a plate margin or a compressional strike-slip bend. Both strike-slip and dip-slip components are required; it is not merely a rotated pure strike-slip pattern. Exact strike, dip, rake, and slip-plane selection cannot be recovered numerically from this uncalibrated printed sketch. Reversing the shading convention would reverse the normal/reverse interpretation, so the convention is essential.

A thin high-positive-gradient zone bends rays back rapidly, bunches their return distances, and can produce a triplicated travel-time curve. Rays turning above it retain the ordinary branch. Rays entering and turning within it have a stronger curvature and can return at smaller distances than slightly shallower rays; deeper-turning rays below it eventually regain increasing range. The mapping from turning depth to distance can therefore fold twice. In the overlapping distance interval, three rays with different turning depths have distinct travel times.

The travel-time curve has two fold points, with branches joined at caustics. Its first-arrival curve is the lower envelope, changing branch where travel times cross; later branches remain physically distinct arrivals. The apparent surface slowness on each branch is $dT/dX=p$, with the deeper, faster-sampling ray generally supplying a smaller slope. A finite high-gradient zone does not require a velocity jump or a separate reflected ray. Whether the branches actually triplicate, and the ranges of the caustics, depend on gradient contrast and thickness, which are not numerically supplied.

The original upper-row example uses a specified illustrative piecewise-linear speed profile to show the folds. The lower row is the low-velocity example in (ii). All distances and times in this illustration come from those chosen profiles, not from extra exam data.

In a zone with $dv/dz<0$, a descending ray bends towards the vertical instead of turning back towards the surface. A ray cannot have its ordinary smooth diving-ray turning point within that decreasing-speed interval. There are rays turning above the zone and rays that penetrate it and turn deeper, after speed has increased sufficiently. The latter spend extra time traversing slower material, so their branch is delayed relative to a simple extrapolation of the original uniform-gradient curve.

Turning depths within the low-velocity zone are missing. In a piecewise-gradient example this can leave a range interval without a returning diving ray: a geometrical shadow zone between the shallow and deeper branches. The lower row of the original plot shows that situation, with the interrupted shallow branch and the delayed deeper branch. The boundaries are profile-dependent; reflected, refracted or diffracted energy can still arrive, and smoothing an idealized velocity maximum can change the branch limits. Thus the robust signature is the interruption/reorganization of the ordinary turning-ray family, not a universal claim of zero recorded signal. The supplied description alone does not specify unique numerical travel times or shadow-zone widths.