Source: https://jmrplens.github.io/phonometry/simulation/elastic-waves/ # Elastic waves and fluid-solid coupling A fluid carries one wave; a solid carries a family of them. This guide covers the elastic companion of the [2D FDTD wave simulation](https://jmrplens.github.io/phonometry/simulation/fdtd-simulation/): `elastic_fdtd_simulation` integrates the plane-strain velocity-stress system of Virieux (1986) on the same staggered grid as the acoustic solver, and with it the compressional **P wave**, the shear **S wave**, the **Rayleigh wave** of a free surface and the **Scholte wave** of a fluid-solid contact all emerge from the same update. Because a fluid is just the shear-free limit of the elastic equations, one pair of material maps couples water, air, sediment and steel in a single run: oblique reflection with **mode conversion**, seabed interface waves and the transmission of immersed plates come out of the maps alone, each pinned to its exact closed form by the validation suite. The guide assumes the acoustic solver's vocabulary (sources, probes, obstacle masks, sponges and the Courant bound); read [2D FDTD wave simulation](https://jmrplens.github.io/phonometry/simulation/fdtd-simulation/) first if those are new. The closed forms this solver cross-checks live in the building, vibration and underwater guides: the mass law of [panel sound insulation](https://jmrplens.github.io/phonometry/buildings/design/panel-sound-insulation/), the flexural waves of [bending-wave transmission at plate junctions](https://jmrplens.github.io/phonometry/vibration/structural/junction-transmission/), and the shear-supporting seabed that the fluid-bottom reflection model of [underwater sound propagation](https://jmrplens.github.io/phonometry/underwater/underwater-propagation/) cannot represent. ## 1. The P-SV velocity-stress scheme A solid carries more than one wave. Alongside the compressional **P wave** at $c_\mathrm{P} = \sqrt{(\lambda + 2\mu)/\rho}$ a shear **S wave** propagates at $c_\mathrm{S} = \sqrt{\mu/\rho}$, and every traction-free surface guides a **Rayleigh wave** just below $c_\mathrm{S}$. `elastic_fdtd_simulation` integrates the 2D plane-strain velocity-stress system of Virieux (1986, Eq. 2), $$ \rho\,\frac{\partial \mathbf{v}}{\partial t} = \nabla\!\cdot\!\boldsymbol{\tau}, \qquad \frac{\partial \boldsymbol{\tau}}{\partial t} = \lambda\,(\nabla\!\cdot\!\mathbf{v})\,\mathbf{I} + \mu\left(\nabla\mathbf{v} + \nabla\mathbf{v}^{\mathsf T}\right), $$ on the same staggered layout as the acoustic solver: the normal stresses ``txx``/``tyy`` share the cell centres where the acoustic pressure lives, the velocities sit on the faces and the shear stress ``txy`` on the corners (Virieux's fully staggered cell, shifted half a cell). The API mirrors `fdtd_simulation` piece by piece: wave-speed maps `c_p` and `c_s` (plus `rho`) instead of `c`, `ExplosionSource` (an isotropic stress injection) and `ForceSource` (a directional body force) as sources, probes that record ``p``, ``vx`` or ``vy``, the same sponge and obstacle machinery, and a frozen `ElasticFDTDResult` with the same `.plot()`. What the run writes down travels as one `ElasticRecording` (the probe cells and their fields, the snapshot cadence and its field) and how the domain ends as one `ElasticBoundaries` (the sides and the sponge thickness). The Courant bound depends only on the fastest $c_\mathrm{P}$ in the map (Virieux Eqs. 6-7), while the resolution rule uses the slowest wave speed anywhere in the domain -- the slowest non-zero $c_\mathrm{S}$ of the solid cells or, if a fluid region carries a still slower $c_\mathrm{P}$, that fluid speed: $\Delta x \le c_\text{min} / (10 f)$ (in a wholly solid map the S wavelength is always the shortest; in mixed domains the fluid P wavelength can be shorter, as in water at 1480 m/s over a 2000 m/s-shear sediment). ```python import numpy as np from phonometry import simulation # An aluminium block hit by a tiny explosion: the P front reaches two # probes 100 and 220 cells away; c_P falls out of the differential delay. w = 8e-6 res = simulation.elastic_fdtd_simulation( 6320.0, 3130.0, 0.002, 1.1e-4, rho=2700.0, shape=(501, 501), sources=[simulation.ExplosionSource( ix=250, iy=250, waveform=simulation.GaussianPulse(0, 0, width=w).value)], recording=simulation.ElasticRecording( probes=[(350, 250), (470, 250)], probe_fields=("p",)), ) t = res.times[1:] t1, t2 = (t[np.abs(res.signals[k, 0, 1:]).argmax()] for k in range(2)) print(round(0.24 / (t2 - t1))) # 6316 c_P = sqrt((lambda+2mu)/rho) = 6320 ``` ## 2. Fluids inside the elastic solver Setting ``c_s = 0`` marks a **fluid** cell: the shear modulus vanishes, the system degenerates to the acoustic equations, and with a uniform ``c_s = 0`` map the elastic solver reproduces the acoustic one **bit for bit** (the regression suite asserts exact equality). A water column over a steel half-space is therefore just two bands of the material maps: with the density averaged arithmetically onto the faces and the shear modulus harmonically onto the corners (Moczo et al. 2007, Eqs. 7.37-7.39), the traction continuity of every internal interface, fluid-solid contacts included, emerges from the maps alone. The validation suite measures the normal-incidence reflection of a water-steel interface within 2 % of $(Z_2 - Z_1)/(Z_2 + Z_1)$ (typically a fraction of a percent) and recovers the normal-incidence mass law of a 3 mm immersed steel plate within 0.3 dB of `mass_law_transmission_loss`. ## 3. Free surfaces and Rayleigh waves A side declared ``"free"`` becomes a traction-free surface through **stress imaging** (Moczo et al. Eq. 9.9): the normal stress is pinned to zero on the surface plane and the shear stress above it is its antisymmetric image. That single boundary condition is what makes surfaces wave-bearing: strike the free surface of an aluminium block vertically (Lamb's problem) and, besides the P and S body fronts, a **Rayleigh wave** rolls along the surface at $c_\mathrm{R} \approx 0.93\,c_\mathrm{S}$, the root of the exact Rayleigh characteristic equation (Cremer, Heckl & Petersson Eq. 3.149). The tests pin the measured Rayleigh speed to that root within 2 % and the flexural wave of a thin free-free strip to the Kirchhoff plate dispersion $c_\mathrm{B} \propto \sqrt{\omega}$ in its thin-plate domain. Free surfaces ride the Rayleigh sampling rule: allow 15-20 cells per wavelength there, the second-order imaging surface being the most dispersive part of the scheme. *One snapshot, three speeds: the P front has covered twice the distance of the S front (dotted arcs at the exact $c_\mathrm{P} t$ and $c_\mathrm{S} t$ radii), and the strongest motion travels along the free surface as the Rayleigh wave, a whisker slower than S.*

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```python import numpy as np import matplotlib.pyplot as plt from phonometry import simulation # A 0.6 x 0.3 m aluminium block, struck vertically at its free surface. dx, w = 0.001, 8e-6 dt = 0.6 * dx / (6320.0 * np.sqrt(2.0)) steps = round(7.3e-5 / dt) res = simulation.elastic_fdtd_simulation( 6320.0, 3130.0, dx, 7.3e-5, rho=2700.0, shape=(300, 600), sources=[simulation.ForceSource( ix=300, iy=0, direction="y", amplitude=1e6, waveform=simulation.GaussianPulse(0, 0, width=w).value)], boundaries=simulation.ElasticBoundaries({"top": "free"}), recording=simulation.ElasticRecording(snapshot_every=steps, snapshot_field="vy"), ) res.plot(kind="snapshot") plt.show() ```

One frame is enough for the two body speeds — the dotted arcs are drawn at the exact $c_\mathrm{P} t$ and $c_\mathrm{S} t$ radii, so they measure rather than label. It is not enough for the third wave, whose speed differs from the shear front's by 7 % and whose defining property is that it stays on the surface while the body fronts leave. Run the same hit for 163 µs, with sponges on the other three sides and the corner between a sponge and the free surface pushed outside the frame, and run it twice: top declared `"free"`, then top left at the solver default, the clamped rigid wall. Same aluminium, same hit, same two body fronts, no surface train at all in the second. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_elastic_halfspace_waves.webm) Both body arcs run out of the frame and keep going. The surface lobes do not go anywhere but along the surface, and they are still there when the body waves have left — which is the whole of "a Rayleigh wave". The inset traces two surface probes 0.15 m apart, and the delay between their peaks is the measurement: the simulated $c_\mathrm{R}$ against 2921 m/s, the root of the exact characteristic equation, which the tests pin to within 2 %. The two field panels carry their own colour scales, and say so: a vertical force applied *on* a clamped surface nearly cancels against its own image, so that run radiates far less than the free one. The probe traces share a single scale, which is where the comparison stays quantitative — the missing surface arrival is a missing peak, not a rescaled one. The other wave a free surface makes possible — the **flexural** wave of a thin free-free strip, the one pinned to the Kirchhoff dispersion above — is worth watching in motion too. The clip below is this same solver launching a 4 kHz bending packet along a 10 mm steel plate: on the control panel the plate runs straight and the packet simply leaves, and on the junction panel a perpendicular plate of the same thickness turns the corner into a scatterer. The packet splits there into the reflected and transmitted bending waves the closed form prices at $\tau_{12}(0°) = 0.5$, plus the fast in-plane precursor that races ahead down the receiving plate — the mode conversion the pinned-junction model deliberately leaves out, and the reason this page needs an elastic solver rather than a flexural one. The [bending-wave transmission guide](https://jmrplens.github.io/phonometry/vibration/structural/junction-transmission/) takes the same run apart against the EN 12354 vibration reduction index. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_elastic_plate_junction.webm) ## 4. Fluid-solid coupling at normal incidence Because the fluid is just the ``c_s = 0`` limit of the elastic scheme, a water-steel contact is **one solver, two bands of the material maps**, and the whole physics of fluid-solid coupling comes out of the same update. The `Material` constants and `ElasticFDTD2D.from_regions` build those layered maps without manual array surgery: a background material and a list of painted regions. ```python import numpy as np from phonometry.simulation import ElasticFDTD2D, STEEL, WATER # A 12 m water column over a steel half-space, as a 1D-like strip. dx = 0.005 sim = ElasticFDTD2D.from_regions( (2400, 3), dx, background=WATER, regions=[(np.s_[1200:, :], STEEL)]) y = (np.arange(2400) + 0.5) * dx p0 = np.exp(-(((y - 3.0) / 0.15) ** 2))[:, None] # plane pulse sim.txx[:] = -p0 sim.tyy[:] = -p0 y_face = np.arange(1, 2400) * dx + 0.5 * WATER.c_p * sim.dt sim.vy += (np.exp(-(((y_face - 3.0) / 0.15) ** 2)) / (WATER.rho * WATER.c_p))[:, None] # one-way, leapfrog-consistent trace = [] for _ in range(round(3.4e-3 / sim.dt)): sim.step() trace.append(sim.p[900, 1]) trace = np.asarray(trace) t = (np.arange(trace.size) + 1) * sim.dt incident = trace[t < 1.8e-3].max() echo = trace[t > 2.6e-3] print(round(float(echo[np.abs(echo).argmax()] / incident), 3)) # 0.938 # (Z2 - Z1)/(Z2 + Z1) = 0.938: at normal incidence no shear is excited # and the steel behaves as a liquid of its rho and c_P (B&G Eq. 4.2.27) ``` ## 5. Oblique incidence: mode conversion Away from normal incidence the solid stops being a liquid. The incident sound refracts into **two** transmitted waves, P and SV (**mode conversion**), each with its own critical angle: for water over steel, 14.5° (P) and 27.5° (S). Between them the transmitted P is evanescent and the shear wave carries the transmitted power, so the reflection dips to $|V| = 0.918$ at 20° instead of reflecting totally as a fluid bottom of the same $c_\mathrm{P}$ would; beyond the S critical angle the reflection is total, $|V| = 1$ with a phase. The validation suite launches oblique carrier beams at 10° and 20° and matches the exact reflection coefficient of Brekhovskikh & Godin (Eqs. 4.2.22-4.2.26) within a fraction of a percent, mode conversion included. The regimes are worth watching rather than tabulating, because "the transmitted P is evanescent" is not a shallower beam but the absence of one. The clip runs the scene at 10°, 20° and 35°, one incidence from each of the first three regimes, driving a sustained phase-graded beam onto the contact. Read it as a picture of the regimes, not as a second measurement: each panel carries the closed-form $|V|$, because the beam a 90 mm panel can hold is six wavelengths wide and at that width a probe sits in the source's near field rather than in a formed beam. The measured column above comes from the validation suite, which has the room to do it properly. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_elastic_mode_conversion.webm) At 10° two beams leave along the dashed Snell directions, at different angles because they travel at different speeds: the slower shear wave at 22.1° from the normal, the faster compressional wave bent much further out, to 43.8°. At 20° nothing departs at the compressional angle; a skin clings to the contact instead, and the shear beam alone carries what crosses. At 35° both are evanescent and nothing propagates *away* into the steel — what fills the solid there runs along the contact and dies with depth — while the water above settles into the interference of the incident beam with a reflection that is total in amplitude and shifted in phase. Each panel draws its steel half with its own annotated display gain: at 10° the transmitted energy leaves in two beams and the amplitude anywhere is small, while at 35° nothing leaves and the evanescent field piles up against the contact, bright and going nowhere. Brightness in the steel is not power — read the printed $|V|$ and the gain factor for that. This is exactly the shear physics that the fluid-bottom `reflection_coefficient` of the [underwater module](https://jmrplens.github.io/phonometry/underwater/underwater-propagation/) cannot represent, and the reason a shear-supporting seabed loses more energy than its equivalent-fluid model. ## 6. The Scholte interface wave Total reflection beyond the S critical angle leaves both media carrying only evanescent fields, and those fields can lock together into a true interface wave: the **Scholte wave**, the fluid-solid analogue of the Rayleigh wave. It travels slower than both the fluid sound speed and the solid shear speed, has no low-frequency cut-off and does not disperse over homogeneous half-spaces; `scholte_speed` solves its exact characteristic equation (B&G Eq. 4.4.20). ```python from phonometry.simulation import Material, STEEL, WATER, scholte_speed print(round(scholte_speed(WATER, STEEL), 1)) # 1479.6, 0.03 % below water seabed = Material(c_p=3500.0, c_s=2000.0, rho=2500.0) print(round(scholte_speed(Material(1500.0, 0.0, 1000.0), seabed), 1)) # 1436.0 ``` Those two numbers tell the whole story. Over a **stiff** bed (steel) the Scholte wave hugs the water speed (1480 m/s) to within 0.03 % and its evanescent tail reaches ~7 wavelengths up into the water: it is essentially a grazing water wave and cannot be separated by time of flight in any reasonable domain (over air-solid contacts the deficit collapses to ~$10^{-12}$ of $c$ and the wave is unobservable outright). Over a **soft** sediment the speed drops well below the water speed and the wave squeezes to within half a wavelength of the contact, which is why measured seabed interface waves are a standard probe of sediment shear speed. The snapshot below runs that soft case: an explosive shot just above the bottom, and the strongest late feature is the interface-hugging Scholte train, timed by the test suite at 1436 m/s within 2 % between two contact probes. *The van Vossen benchmark media (water over a 3500/2000/2500 sediment): the Scholte train crawls along the contact at 1436 m/s, evanescent into both media, while the direct water wave runs ahead at 1500 m/s.*

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```python import numpy as np import matplotlib.pyplot as plt from phonometry import simulation # Water over a soft seabed; explosive 50 Hz Ricker 10 m above the contact. ny, nx, dx = 200, 500, 1.0 c_p = np.full((ny, nx), 1500.0) c_s = np.zeros((ny, nx)) rho = np.full((ny, nx), 1000.0) c_p[100:], c_s[100:], rho[100:] = 3500.0, 2000.0, 2500.0 f0, t0 = 50.0, 0.030 dt = 0.6 * dx / (3500.0 * np.sqrt(2.0)) steps = round(0.232 / dt) def ricker(t): a = (np.pi * f0 * (t - t0)) ** 2 return (1.0 - 2.0 * a) * np.exp(-a) res = simulation.elastic_fdtd_simulation( c_p, c_s, dx, 0.232, rho=rho, sources=[simulation.ExplosionSource(ix=60, iy=89, waveform=ricker, amplitude=1e3)], boundaries=simulation.ElasticBoundaries("absorbing", absorbing_layer_cells=20), recording=simulation.ElasticRecording(snapshot_every=steps, snapshot_field="vy"), ) res.plot(kind="snapshot") plt.show() ```

## 7. Immersed plates and the mass law An **immersed plate** closes the loop with the building-acoustics module. At normal incidence no shear is excited, so an elastic plate in water is exactly the three-media fluid layer of B&G §2.4: its transmission follows the closed form of Eq. 2.4.14, which reduces to the familiar mass law for thin plates and low frequencies and predicts **total transmission** at the half-wave thickness resonances $f_n = n c_\mathrm{P} / (2 h)$ (Eq. 2.4.19). For a 10 mm steel plate that first resonance sits at 295 kHz, and one broadband FDTD run reproduces the whole curve: 5.8 dB at 10 kHz (where `mass_law_transmission_loss` with water as the ambient fluid agrees with the exact form to 0.02 dB), 18.1 dB at 50 kHz, and a transmission-loss dip within 0.1 % of the 295 kHz resonance. The same suite stress-tests the extreme contrast of an air-steel contact (impedance ratio ~$10^5$:1): stable over 10 000 steps with the reflected amplitude conserved to 0.5 %. At oblique incidence the plate physics gets richer, and the clip below is this solver driving the same 10 mm steel plate, now lying in air, with a sustained 45° plane wave arriving on it. The two panels differ in **one number only** — the drive frequency, $f_\mathrm{c}/2 = 603$ Hz on the left and $2 f_\mathrm{c} = 2413$ Hz on the right, either side of the 1206 Hz coincidence frequency the library computes from the same $m''$ and $B'$ used above. Everything else, the plate, the angle, the mesh and the colour scale, is held fixed. Below $f_\mathrm{c}$ the plate reflects almost everything and the transmitted level lands on the oblique mass law; above it the acoustic trace wavelength matches the free bending wavelength, the plate re-radiates a 45° beam that grows along the lit span, and the transmitted level holds at the low-frequency figure where the mass law demanded 12 dB more blocking. The air below the plate is drawn on both panels with the display gain measured off the settled field of the two runs together (×150, that is +44 dB) and printed on the canvas: read the *annotations* for levels, not the brightness. The [panel sound insulation guide](https://jmrplens.github.io/phonometry/buildings/design/panel-sound-insulation/) takes the same run apart against the plateau method and the mass law. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_elastic_coincidence.webm) ## What this solver does not do Everything the 2D acoustic solver cannot do, this one cannot either: the domain is a cross-section, so a source is a line source with cylindrical $1/\sqrt{r}$ spreading rather than the $1/r$ of a 3D point source, and absolute levels and decay rates are not those of a 3D problem. The solid is isotropic and purely elastic: no anisotropy, and no viscoelastic damping beyond the bulk `damping` decay rate, so material loss factors are not modelled. There is no elastic perfectly matched layer — the sponge absorbs grazing Rayleigh waves less effectively than body waves. And guided-wave dispersion beyond the thin-plate Kirchhoff regime (full Lamb modes) is observable in the fields but has no dedicated closed-form oracle here. ## Quick answers ### When do I need the elastic solver instead of the acoustic one? Whenever any part of the domain supports shear: mode conversion at oblique incidence, Rayleigh waves on free surfaces, Scholte waves at fluid-solid contacts and plate transmission all need the elastic equations. If every cell is fluid, use the acoustic solver: with a uniform ``c_s = 0`` map the elastic solver reproduces it bit for bit, at roughly twice the memory and stepping cost. ### What grid resolution does an elastic simulation need? Resolve at least 10 cells per shortest wavelength using the slowest wave speed in the domain -- the slowest non-zero shear speed or a slower fluid sound speed if one is present -- $\Delta x \le c_\text{min} / (10 f)$. Allow 15-20 cells per wavelength along free surfaces (the stress-imaging boundary is the most dispersive part of the scheme), and at least 15 points per wavelength when an interface wave has to be timed (van Vossen et al. 2002). ### What is a Scholte wave and how fast does it travel? The Scholte wave is the true interface wave of a fluid-solid contact: evanescent into both media, elliptical particle motion, no low-frequency cut-off and non-dispersive over homogeneous half-spaces. Its speed solves the exact characteristic equation (Brekhovskikh & Godin Eq. 4.4.20, `scholte_speed`) and always lies below both the fluid sound speed and the solid shear speed: 0.03 % below the water speed (1480 m/s) over steel (1479.6 m/s), but 4 % below the 1500 m/s ocean water of the sediment example (1436 m/s for that water over a 3500/2000/2500 bed), which is why seabed interface waves probe the sediment shear stiffness. ## See also - [2D FDTD wave simulation](https://jmrplens.github.io/phonometry/simulation/fdtd-simulation/): the acoustic solver this guide extends, with the staggered grid, the Courant bound, the sources, probes and boundary machinery shared by both. - [Predicting Panel Sound Insulation](https://jmrplens.github.io/phonometry/buildings/design/panel-sound-insulation/): the mass law and coincidence closed forms; the immersed-plate run of section 7 is their full-wave cross-check. - [Bending-wave transmission at plate junctions](https://jmrplens.github.io/phonometry/vibration/structural/junction-transmission/): the flexural-wave closed forms of the building model; the Kirchhoff dispersion validated in section 3 is the wave they transport. - [Underwater sound propagation](https://jmrplens.github.io/phonometry/underwater/underwater-propagation/): the fluid-bottom Rayleigh reflection model whose missing shear physics section 5 quantifies. - API reference: [`simulation.elastic-fdtd`](https://jmrplens.github.io/phonometry/reference/api/simulation/elastic-fdtd/). ## References - Virieux, J. (1986). P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method. *Geophysics*, 51(4), 889-901. [doi:10.1190/1.1442147](https://doi.org/10.1190/1.1442147). The elastic solver of section 1: the velocity-stress system (Eq. 2), the fully staggered cell and update (Fig. 1, Eq. 5), the P-only Courant bound (Eqs. 6-7), the dispersion relations (Eqs. 13-14) and the liquid as the shear-free limit. - Moczo, P., Kristek, J., Galis, M., Pazak, P., & Balazovjech, M. (2007). The finite-difference and finite-element modeling of seismic wave propagation and earthquake motion. *Acta Physica Slovaca*, 57(2), 177-406. The heterogeneous effective parameters of section 2 (harmonic shear modulus, arithmetic density; Eqs. 7.37-7.39) and the stress-imaging free surface of section 3 (Eq. 9.9). - Cremer, L., Heckl, M., & Petersson, B. A. T. (2005). *Structure-borne sound* (3rd ed.). Springer. [doi:10.1007/b137728](https://doi.org/10.1007/b137728). The analytic oracles of the elastic solver: the exact Rayleigh characteristic equation (Eq. 3.149) and the Kirchhoff flexural dispersion with its thickness correction (Eqs. 3.83-3.89, 3.196b). - Brekhovskikh, L. M., & Godin, O. A. (1990). *Acoustics of layered media I: Plane and quasi-plane waves*. Springer. [doi:10.1007/978-3-642-52369-4](https://doi.org/10.1007/978-3-642-52369-4). The fluid-solid oracles of sections 4-7: the oblique reflection coefficient with mode conversion (Eqs. 4.2.22-4.2.26) and its critical-angle limits (Eqs. 4.2.27-4.2.31), the exact Scholte characteristic equation (Eqs. 4.4.18-4.4.20) with the light-fluid asymptotics (Eqs. 4.4.21-4.4.24), and the three-media layer transmission (Eqs. 2.4.10-2.4.19). - van Vossen, R., Robertsson, J. O. A., & Chapman, C. H. (2002). Finite-difference modeling of wave propagation in a fluid-solid configuration. *Geophysics*, 67(2), 618-624. [doi:10.1190/1.1468623](https://doi.org/10.1190/1.1468623). The fluid-solid benchmark of the same staggered scheme: the harmonic shear modulus and arithmetic density averages (Eqs. 9-10) that satisfy the interface conditions implicitly, the soft-bed configuration of the Scholte test of section 6 and the >= 15 points-per-wavelength rule for the O(2,2) scheme with an interface wave. ## Standards and sources No measurement standard governs the elastic solver: it implements the textbook velocity-stress method of Virieux (1986) with the heterogeneous-medium and free-surface treatments of Moczo et al. (2007). Validation is anchored to closed forms: the P and S body-wave speeds, the exact Rayleigh root and the Kirchhoff flexural dispersion (Cremer, Heckl & Petersson), the bit-exact acoustic limit, the fluid-solid reflection at normal and oblique incidence with mode conversion, the exact Scholte-wave speed and the immersed-plate transmission with its thickness resonance (Brekhovskikh & Godin), the immersed-panel mass law, and the van Vossen (2002) soft-bed interface-wave benchmark.