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With $ϵ={\beta }^{\prime }/\beta \ll 1$, $\eta ={\mu }^{\prime }/\mu =O({ϵ}^2)$ and a fixed mode index, put $y=qh$, $W=\omega h/{\beta }^{\prime }$ and $K=kh$. The exact Love wave branch has $W=y\sqrt{1+{\eta }^2{\tan }^{2}y}/\sqrt{1-{ϵ}^2}$ and $K=y\sqrt{{ϵ}^2+{\eta }^2{\tan }^{2}y}/\sqrt{1-{ϵ}^2}$. Its minimum group velocity occurs near $y=(n+1/2)\pi$, with transition width $W-(n+1/2)\pi$ of order $({\eta }^2(n+1/2)\pi {)}^{1/3}$. Well above that width, the rigid-base approximation to Love waves captures both velocities. Density ratios and the mode index must be controlled in this estimate.

For subcritical incidence, a reflected SH-wave loses energy into the faster substrate. Postcritical incidence has an evanescent substrate tail and no outgoing normal energy flux in the ideal lossless model. Repeated reflections can then interfere to form Love waves. Individual reflected rays are contributions to the field; normal-mode frequencies require the reflection-phase condition and are not supplied by a geometrical travel-time curve alone.

For a slow uniform layer over a faster half-space, the nth Love wave branch reaches the half-space shear wave speed at the displayed threshold. Its half-space decay exponent then vanishes, so the threshold is a limiting nonlocalized state, not a trapped finite-energy mode. The fundamental has zero threshold and exists at every positive frequency; overtones require frequency strictly above their positive threshold. At high frequency each fixed branch approaches the layer shear wave speed from above.

For a slower surface layer supporting propagating SH-waves with vertical wavenumber $q>0$ above an evanescent half-space with decay exponent $p>0$, continuity of displacement field and traction gives the displayed reflection coefficient for a downward incident displacement field wave. Its modulus is one and its wave phase is $-2\arctan (\mu p/({\mu }_{0}q))$. A traction-free top reflects displacement field with coefficient $+1$. The round-trip condition $R_h{e}^{2iqh}=1$ then gives $\tan (qh)=\mu p/({\mu }_{0}q)$, equivalent to direct Love wave mode matching.

The full Love wave problem replaces the rigid base by a decaying substrate displacement. With $Y=A\cos (qz)$ in the layer and $Y=B{e}^{-\kappa (z-h)}$ below it,where ${\mu }^{\prime }={\rho }^{\prime }{\beta }^{\prime 2}$ and $\mu =\rho {\beta }^2$. Displacement continuity gives $B=A\cos (qh)$ and traction continuity gives ${\mu }^{\prime }q\sin (qh)=\mu \kappa \cos (qh)$. The rigid base is therefore a good approximation only when $\mu \kappa \gg {\mu }^{\prime }q$: then the substrate displacement is small and $qh$ lies close to ${\chi }_n=(n+1/2)\pi$.

Put $ϵ={\beta }^{\prime }/\beta$, $\eta ={\mu }^{\prime }/\mu$, $y=qh$, $W=\omega h/{\beta }^{\prime }$ and $K=kh$. The complete dispersion branch can be parameterized without root finding:Then $c={\beta }^{\prime }W/K$ and $U={\beta }^{\prime }{W}^{\prime }/K^{\prime }$. At the lower end, $c\to \beta$ and $U\to \beta$. The Love-wave cutoff frequencies are ${\omega }_{c,n}=n\pi {\beta }^{\prime }/[h\sqrt{1-{ϵ}^2}]$, with zero cutoff for the fundamental. Thus the rigid-layer formula cannot describe the fundamental's low-frequency limit or the low-frequency part of an overtone: it would either give no propagating mode or predict $c>\beta$, outside the trapping interval ${\beta }^{\prime }<c<\beta$.

For comparable densities and a fixed mode index, $\eta =({\rho }^{\prime }/\rho ){ϵ}^2\ll 1$. Write $y={\chi }_n-\delta$ near the strong-dispersion transition. In the region controlling the group-velocity minimum of a high-contrast Love wave, $\delta \ll 1$ and $\eta /\delta \gg ϵ$. The leading derivatives giveTheir minimum is atThe frequency estimate follows by inserting ${\delta }_{\min }$ into $W\simeq ({\chi }_n-\delta )[1+{\eta }^2/(2{\delta }^2)+{ϵ}^2/2]$: the two leading terms of size $\delta$ cancel. The real group velocity has a finite minimum and turns upwards near the rigid-layer threshold, whereas the rigid formula has zero group speed at its artificial threshold.

For a useful estimate of the validity range, put $\Delta =W-{\chi }_n>0$. The rigid prediction has $K_R^2=W^2-{\chi }_n^2\simeq 2{\chi }_n\Delta$. The finite-load phase correction is $\delta \simeq \eta {\chi }_n/(\kappa h)$; its changes to both $K$ and $dK/dW$ are small ifThe first condition keeps $K_R$ away from the substrate propagation boundary; the second controls the rapidly changing interface phase. Equivalently, the rigid formulas are reliable sufficiently above the actual group-velocity minimum, beyond its narrow transition interval. At fixed $\omega /{\omega }_n>1$ separated from one, they become asymptotically accurate as the contrast grows. They are particularly simple at $\omega \gg {\omega }_n$, where both velocities approach ${\beta }^{\prime }$. Within the transition and below it the full dispersion relation is required.

The small speed ratio alone is insufficient if the layer density grows so strongly that ${\mu }^{\prime }/\mu$ is not small. For such materials the exact load criterion $\mu \kappa \gg {\mu }^{\prime }q$ and its frequency derivatives should be used; the preceding contrast estimates assume a bounded order-one density ratio and fixed $n$.

Choose depth $z$ positive downwards, the free surface at $z=0$ and the interface at $z=h$. A Love wave has only transverse displacement field $u_y=U(z)e^{i(\kappa x-\omega t)}$. Take $\omega >0$, $\kappa =\omega /c_L$, and defineA trapped mode requires ${\beta }_0<c_L<\beta$, so both $q,p$ are positive. The SH equations are $U_0^{\prime \prime }+q^2{U}_0=0$ in the layer and $U^{\prime \prime }-p^{2}U=0$ in the half-space. Zero shear traction at the free surface and decay at large depth selectContinuity of displacement field and shear traction at $h$ gives $C=A\cos (qh)$ and $-{\mu }_{0}qA\sin (qh)=-\mu pC$. ConsequentlySubstituting the definitions of $q,p$ yields the requested speed-frequency dispersion relation. For negative frequency use $|\omega |$ in both vertical wavenumbers; the real displacement field has conjugate frequency partners. Neither equality in the trapped-speed interval gives a strictly localized finite-frequency Love mode.

Take a stable isotropic elastic material with mass density $\rho >0$ and Lamé parameters $\lambda ,\mu$, where $\mu >0$ and $3\lambda +2\mu >0$. Its elastic stiffness tensor isThe P wave is longitudinal and the S wave transverse. The wavefront-transport alternative and all three steps of the Love wave alternative are given below.

Take a horizontally polarized shear displacement $u(y)e^{i(kx-\omega t)}$, with $k>0$ and $c=\omega /k$. In the layer put $q=\sqrt{c^2/{\bar{c}}_s^2-1}$ and in the half-space put $\gamma =\sqrt{1-c^2/c_s^2}$. The required inequality makes both positive. The shear-wave equations givewhere the first form satisfies zero traction at $y=h$ and the second decays as $y\to -\infty$. Displacement and traction continuity at zero give $A\cos (kqh)=B$ and $\bar{\mu }kqA\sin (kqh)=\mu k\gamma B$. Hence the Love wave dispersion relation isThe left-hand square root uses the layer speed ${\bar{c}}_s$, as in the original PDF; the TeX aid incorrectly omits its bar there.

Let $K=kh>0$ and $q_{max}=\sqrt{c_s^2/{\bar{c}}_s^2-1}$. The allowed interval is $0<q<q_{max}$, and the right side becomesa positive strictly decreasing function from infinity to zero. On each positive branch of $\tan (Kq)$ beginning at $Kq=m\pi$, the left side is strictly increasing from zero to infinity, or to a positive endpoint value if $q_{max}$ comes first. There is exactly one intersection on that branch whenever $m\pi <K{q}_{max}$. The fundamental branch $m=0$ thus supplies a real solution for every positive $kh$.

The second branch exists precisely when $K{q}_{max}>\pi$. Thus the strict cutoff of the second Love-wave mode isAt equality the second solution would have $c=c_s$, zero half-space decay exponent, and fails the required disturbance decay. Consequently the cutoff is the infimum, rather than an attained smallest value, under the printed strict speed inequality. Also $kh=0$ itself is a limiting case, not a trapped propagating mode; the “any value” assertion is valid for positive $kh$.

Seismic surface-wave dispersion. Rayleigh waves combine vertical and longitudinal motion in the vertical plane of propagation; Love waves have horizontal transverse motion and can be guided by a low-shear-speed surface layer. A homogeneous elastic half-space supports a Rayleigh wave with a frequency-independent speed in the ideal elastic limit, but depth-dependent Earth structure makes the surface-wave eigenfunctions and their propagation speeds depend on frequency. This structural dispersion can occur even when each constituent material is itself nondispersive.

Distinguish phase velocity $c=\omega /k$ from group velocity $U=d\omega /dk$, which controls a narrow-band envelope in the appropriate weakly attenuating limit. Longer periods generally have deeper sensitivity, while shorter periods emphasize shallower structure. In a profile whose shear speed increases with depth this often makes long-period waves faster, but low-velocity zones, anisotropy and higher modes can change that trend. Frequency dependence need not be monotonic in every Earth model.

Measure phase delays or wave-packet arrival times across known source–receiver paths for many periods, accounting for source phase, cycle ambiguity, multipathing and mode identification. Compute predicted dispersion from trial elastic profiles and compare with observations. Linearization expresses the sensitivity schematically aswhere $\alpha ,\beta$ are compressional and shear speeds. A regularized inverse problem then estimates depth structure; many crossing paths additionally provide surface-wave tomography. Joint Love wave and Rayleigh wave measurements help distinguish shear-wave polarizations and constrain radial seismic anisotropy. Finite bandwidth, depth averaging and correlated sensitivities mean that one dispersion curve does not uniquely determine a detailed Earth profile.

Seismic ray theory, the eikonal equation and Fermat's principle. In a smoothly varying isotropic medium, insert a high-frequency ansatz $u=A(\mathbf{x})e^{i\omega [T(\mathbf{x})-t]}$ into the scalar wave equation. The leading powers of frequency giveThe travel-time surfaces are wavefronts and rays are their normal trajectories, with slowness vector $\mathbf{p}=\nabla T$. The next-order equation transports amplitude, including geometrical spreading; attenuation is an additional material effect.

Fermat's principle says that the ray makes $\int ds/c$ stationary under path variations with fixed endpoints. Stationarity, rather than always the global minimum, permits multiple arrivals. Its Euler–Lagrange equation, with arclength $s$ and unit tangent $\hat{\mathbf{t}}$, isFor horizontal layering, the horizontal slowness $p_h=\sin i/c$ is conserved, giving Snell's law at interfaces and smooth refraction between them. Here $i$ is measured from vertical. A turning point satisfies $p_{h}c(z)=1$; increasing speed with depth bends a downgoing ray back upwards. Travel-time tomography uses these ray integrals to infer spatial slowness variations.

The approximation requires wavelength to be small compared with structural variation scales. Caustics, turning-point amplitudes, diffraction, scattering and finite-frequency sensitivity require wave-based corrections. Thus seismic ray theory gives an economical geometric description and useful travel times, while surface-wave dispersion retains essential interference and boundary-mode physics.

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 asthenosphere exists because the mantle's ability to deform is strongly dependent on temperature, pressure, composition, water and deformation mechanism. Cooling from the surface creates a strong lithosphere. Beneath it, hotter mantle deforms by material creep at much lower stress on geological timescales. The asthenosphere is predominantly solid weak rock, not a global ocean of magma. On the much shorter timescale of a seismic wave, it still supports S waves.

The hot shallow mantle can be close to its solidus. Water weakening of rocks and, in some settings, a small connected melt fraction can further lower effective viscosity. At greater depths, changes in pressure, homologous temperature, mineral structure and water storage modify the deformation laws; increasing depth does not imply indefinitely decreasing viscosity. Explaining a weak upper-mantle interval does not require every location to contain the same amount of melt.

There are several physically distinct boundaries to test. The thermal lithosphere-asthenosphere boundary describes a thermal transition, the mechanical boundary a contrast in long-term strength or viscosity, and the seismic lithosphere-asthenosphere boundary a change in seismic velocity or other wave properties. They need not have identical depths. A smooth geothermal gradient can explain broad thermal weakening, but a very sharp decrease of seismic velocity may require changes in melt fraction, hydration, composition or seismic anisotropy. A sharp shallow reflector within a craton might instead be an internal lithospheric boundary.^[ref]

Begin with a thermal null model. A half-space cooling model givesIf a thermal boundary corresponds to a fixed temperature $T_{\ast }$, its depth isso it deepens as the square root of plate age. Test that prediction against independently determined seafloor ages, bathymetry and heat flux. An oceanic plate cooling model adds finite thickness and basal heat supply, allowing the old-age depth to approach a finite value. For continents, include radiogenic heating, crustal thickness, the history of continental rifting or tectonic thickening, and advective heating by magma or mantle flow. A cold, dry craton and a recently extended volcanic region should not share the same thermal profile.

Then calculate the observable consequences of competing mechanisms rather than merely assigning a boundary to a map:

Compare models against the observed boundary depth and its sharpness, including uncertainty and resolution. An age–depth trend with matching heat flux favours thermal control; a sharp discontinuity that fails the thermal prediction but coincides with an electrical anomaly can support a melt- or hydration-related transition. Neither observation alone proves a unique cause. The spatially varying boundary is best understood as the result of thermal evolution interacting with composition, fluids and deformation history.

A layer with shear modulus ${\mu }_1$ and wave speed $c_1$ lies between a rigid support and a half-space with shear modulus ${\mu }_2$ and wave speed $c_2>c_1$. A trapped antiplane shear normal mode has layer amplitude $A\sin (my)$ and exterior amplitude $A\sin (mh)e^{-\ell (y-h)}$, where $m=k\sqrt{c^2/c_1^2-1}$ and $\ell =k\sqrt{1-c^2/c_2^2}$. Continuity of displacement field and traction gives the displayed dispersion relation. It requires $c_1<c<c_2$ and differs from a Love wave with a free boundary.

A free top and rigid base impose transverse wavenumbers $(n+1/2)\pi /h$ on guided SH modes between rigid and free planes. Their phase velocity and group velocity are $c={\beta }^{\prime }/\sqrt{1-{\omega }_n^2/{\omega }^2}$ and $U={\beta }^{\prime }\sqrt{1-{\omega }_n^2/{\omega }^2}$ for $\omega >{\omega }_n$. A real Love wave replaces the rigid base by an evanescent elastic load; the approximation becomes reliable sufficiently above the corresponding strong-dispersion transition, not at the actual mode cutoff.

Seismic surface-wave dispersion is the frequency dependence of Rayleigh wave or Love wave propagation caused by depth-dependent elastic properties, mode structure and possibly intrinsic anelasticity. Phase velocity is $\omega /k$; group velocity is $d\omega /dk$. Different periods and branches have different depth sensitivities, enabling surface-wave tomography.

Surface-wave tomography combines path-dependent measurements of seismic surface-wave dispersion to infer lateral and depth variation in elastic properties. Joint Rayleigh wave and Love wave measurements can help constrain seismic anisotropy.

A toroidal normal mode of the Earth is a free oscillation with tangential, divergence-free displacement. It is the spherical counterpart of the transverse shear motion in a Love wave. A bounded spherical Earth has discrete angular orders and eigenfrequencies, so the arbitrarily low-frequency continuum of a flat infinite layer is not a literal statement about global nonrigid oscillations.