The adiabatic bulk modulus divided by density, $\Phi =K_S/\rho ={\alpha }^2-4{\beta }^2/3$ for an isotropic solid. It has units of squared speed and is a key input to the Adams-Williamson density relation.

In a chemically homogeneous adiabatic spherical region, $d\rho /dr=-\rho g/\Phi$, coupled to $dm/dr=4\pi r^2\rho$ and $g=Gm/r^2$. Seismic velocities supply $\Phi$; density normalization and jumps require further data and assumptions.

Repeated reflection and conversion between a layer surface and its lower interface produces additional delayed phases. Their travel times and amplitude depend on layer thickness, wave speeds, impedance contrasts, attenuation and source spectrum.

A seismogram records ground motion through an instrument response. Comparing it with a synthetic seismogram requires consistent component orientation, sampling, timing, filtering and instrument calibration.

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.

Seismic ray theory approximates high-frequency propagation through smoothly varying media. The travel-time field satisfies the eikonal equation, and its characteristics are paths of stationary travel time according to Fermat's principle. Diffraction and caustics require refinements beyond a simple geometrical-ray approximation.

Seismic anisotropy is direction dependence of elastic-wave propagation. Aligned crystals or layering can produce it; using it to infer deformation requires a model connecting fabric to flow.

Arrival-time location estimates a hypocenter and origin time by comparing picked phase arrivals with predicted travel times. Linearization gives residual changes $\delta t_i=\delta t_0+\nabla T_i\cdot \delta \mathbf{x}$. Poorly distinct derivative columns cause tradeoffs, while incorrect wave speeds or phase labels cause systematic bias even with many stations.

A low-velocity zone has a wave-speed decrease with depth over some interval. Surface rays do not turn within its negative-gradient interior; rays penetrating it can return only after encountering sufficient speed recovery below. This can produce shadow ranges, multiple travel-time branches and poor turning-depth coverage. A wave-specific low-velocity zone is not identical to a mechanical layer boundary.

A seismic phase labels a propagation type and path, such as direct P, direct S, a reflection or a converted wave. Misidentifying the phase assigns the wrong travel-time function even if its arrival is picked precisely.

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.

An odd-only segmented sieve represents only odd integers in each bounded interval, handling the prime two separately. For an odd first represented value $L$, array index $i$ stands for $L+2i$. To mark multiples of an odd prime $p$, begin at the smallest odd multiple at least $\max (L,p^2)$ and advance by $2p$, corresponding to index stride $p$. Base primes up to the square root of the upper endpoint suffice. The segmentation limits working storage while omitting even candidates roughly halves it again. The number one remains nonprime and must be excluded explicitly.

An observer at fixed exterior Schwarzschild spatial coordinates. Its proper time is $dT=\sqrt{f}dt$, with $f=1-2GM/(r{c}^2)>0$. It is accelerated, not freely falling; no such static timelike worldline exists on or inside the event horizon.

multi-angle light scattering measures scattered light at multiple angles to infer molecular size and, with appropriate concentration and optical-contrast information, molar mass. Coupling it to size-exclusion chromatography helps distinguish eluting assemblies, although unresolved mixtures complicate inference.

Cloudiness measured through attenuation or scattering of light. actin filaments, bundles and aggregates can contribute differently, so the signal is not automatically proportional to polymer mass.

Measurement of time-averaged scattering intensity as a function of angle and concentration. For suitable dilute polymer solutions its small-wave-number behavior gives a radius of gyration; intensity weighting and interparticle correlations require attention.