= Solution
\b[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.<Gravity and bathymetry in the central Pacific Ocean by A. B. Watts>[Abstract, ridge, flanking, outer-belt and long-wavelength swell anomalies]{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,
$$
D\nabla^4w+\Delta\rho_b g w=\Delta\rho_l g h.
$$
For a Fourier component of horizontal <wavenumber> $q$, this gives $w_q=\Delta\rho_l g h_q/(Dq^4+\Delta\rho_b g)$. 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,
$$
Z(q)\equiv\frac{\delta g_q}{h_q}=2\pi G\Delta\rho_l\left[1-\frac{e^{-qt}}{1+Dq^4/(\Delta\rho_b g)}\right].
$$
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. \b[<Gravity> plus topography constrain support as a function of wavelength; neither alone uniquely fixes the <mantle> <density> distribution.]
For the \b[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 satisfy
$$
\mathbf v_{AB}+\mathbf v_{BC}+\mathbf v_{CA}=0.
$$
This 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.<Evolution of Triple Junctions by Dan McKenzie and W. J. Morgan>[Abstract, stable versus unstable junction configurations]{ref}
\Image[/past-exam-of-the-physics-of-the-earth-as-a-planet-course-of-the-university-of-cambridge/2012/paper-1-plate-support.png]
{title=Plate support and triple junction}
These original diagrams show an illustrative flexural response and a symmetric ridge-junction construction; they are not maps or measured Hawaiian profiles.
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