Solitary Wave Induced Surface Dilation

• Solitary Wave Induced Surface Dilation is the lofting of a granular assembly’s free surface by a laterally propagating compressive wave under near-zero overburden pressure.
• The phenomenon is modeled by extending 1D Hertzian contact laws into 3D granular channels, using calibrated scaling laws to relate impact speed, elastic properties, and force redirection.
• Numerical simulations and analytical models link SID to Lunar Cold Spots by predicting surface relief, bulk-density drops, and dilation depths consistent with observed regolith anomalies.

Solitary Wave Induced Surface Dilation (SID) denotes the lofting or dilation of the upper layers of a granular assembly by a laterally propagating solitary wave at a mechanically free surface. In the granular-channel studies that explicitly formulate the mechanism, the wave is generated by lateral or floor-driven impact loading, propagates as a compressive front only slightly above the sound speed, and redirects part of its force upward near the surface, where the absence of significant overburden allows a detached or dilated band to form (Frizzell et al., 3 Sep 2025). SID is quantified by a lofting depth $z_{loft}$, by the resulting at-rest surface relief $\Delta z$, and by the associated bulk-density reduction; the phenomenon has been proposed as a mechanism for Lunar Cold Spots in vacuum-exposed, low-gravity regolith (Frizzell et al., 4 Sep 2025).

1. Physical setting and defining characteristics

In the 2025 granular literature, SID is studied in channels of monodisperse, cohesive spherical particles modeled as Hertzian contacts and exposed to vacuum, with a free upper surface and a hard or rough lower boundary. One formulation uses a randomly packed monolayer of depth $\sim 160\,R$ in which solitary waves are driven by “frozen-in” impacts along the bottom layer; another uses a compact bed in which a lateral impulse generates an initial shock that rapidly decays and then propagates as a nearly nondispersive solitary-wave front (Frizzell et al., 3 Sep 2025).

The defining physical condition is that the free surface experiences little or no confining pressure. In this setting, a lateral compressive–shear solitary wave traveling just above the weakly precompressed sound speed arrives at the surface with a curved front, slower at the surface than at depth, so that a normal component of the wavefront force is redirected upward. The 2025 SSDEM study states that surface dilation requires zero overburden pressure, is precipitated by waves traveling barely above the sound speed $(> \mathrm{Mach}\ 1.05)$, is induced by compressive solitary waves, is insensitive to channel length, and requires a hard subsurface floor to maintain the wave over the entire channel (Frizzell et al., 4 Sep 2025).

Within this usage, SID is not simply generic shear dilation. The relevant mechanism is inertial and wave-mediated: the passing solitary front temporarily overcomes the local gravitational loading of the upper bed and produces a lofted or dilated band whose depth can be predicted from force balance.

2. Force scaling from 1D Hertzian chains to 3D granular channels

The central constitutive step in the 2025 SID model is the extension of a known 1D solitary-wave force law to a laterally driven 3D random packing. For an uncompressed 1D chain of Hertzian spheres of mass $m$, radius $R$, effective elastic modulus $E_{eq}$, and impact speed $v_0$, the peak force in the leading solitary wave is written as

$F_p \approx 0.719\,(m^3 E_{eq}^2 R v_0^6)^{1/5},$

with

$$m=\frac{4}{3}\pi R^3\rho, \qquad E_{eq}=\frac{E}{1-\nu^2}.$$

The 3D formulation introduces a single dimensionless scaling constant $\Delta z$0 and writes the driven-wavefront force as

$\Delta z$1

where

$\Delta z$2

The same source reports that simulations in the 3D channel confirm

$\Delta z$3

over several decades in each parameter, with correlation coefficients $\Delta z$4 (Frizzell et al., 3 Sep 2025).

This scaling is the key reduction that makes SID analytically tractable. The model assumes that the 1D-to-3D transition can be captured by a single fit constant and that finer 3D effects, including surface-layer “anomalies,” are averaged out. In that sense, the 3D SID formulation is not a first-principles derivation of every contact-network detail; it is a calibrated force law whose exponents follow the 1D Hertzian-chain result.

3. Overburden, pre-compression, and the lofting criterion

The lofting model couples the peak solitary-wave force to gravity-induced pre-compression. In static equilibrium under gravity $\Delta z$5, the overburden force at depth $\Delta z$6 is

$\Delta z$7

where $\Delta z$8 is the packing fraction. Equating this to the Hertz spring force yields a static overlap

$\Delta z$9

Lofting is assumed to begin when the vertical component of the wavefront force exceeds the overburden. In the 3D SID formulation, the near-surface curvature introduces a small tilt $\sim 160\,R$0, so the effective vertical solitary-wave force is $\sim 160\,R$1, where $\sim 160\,R$2 is the inter-particle friction. Cohesion enters through a difference in normal contact area between the maximum overlap $\sim 160\,R$3 and the initial overlap $\sim 160\,R$4, but for typical lunar-like grains $\sim 160\,R$5 is reported to be small and often neglected. The resulting force balance at the maximum lofting depth is

$\sim 160\,R$6

with $\sim 160\,R$7 for a $\sim 160\,R$8-deep channel (Frizzell et al., 3 Sep 2025).

The companion SSDEM study expresses the same idea in scaled form using a wavefront force $\sim 160\,R$9:

$(> \mathrm{Mach}\ 1.05)$0

so that

$(> \mathrm{Mach}\ 1.05)$1

In the authors’ interpretation, $(> \mathrm{Mach}\ 1.05)$2 tends to zero at depth and to a few degrees near the surface, so the free surface is the region where vertical force redirection is dynamically significant (Frizzell et al., 4 Sep 2025).

The principal modeling assumptions are explicit. All solitary-wave energy is taken up in the leading wavefront; damping and plastic losses are neglected in the 1D model used as the basis for scaling; the static compression $(> \mathrm{Mach}\ 1.05)$3 is assumed small compared with $(> \mathrm{Mach}\ 1.05)$4; only the small-angle component $(> \mathrm{Mach}\ 1.05)$5 acts vertically; cohesion is generally negligible for lunar-like grains; and grain restitution $(> \mathrm{Mach}\ 1.05)$6, Poisson’s ratio $(> \mathrm{Mach}\ 1.05)$7, and rolling parameters are reported to have negligible influence on $(> \mathrm{Mach}\ 1.05)$8 (Frizzell et al., 3 Sep 2025).

4. Numerical modeling and parameter dependence

The SID literature uses soft-sphere DEM or SSDEM with Hertzian normal elasticity, tangential elasticity, viscous damping, JKR cohesion, Coulomb friction, and rolling resistance. In the detailed SSDEM formulation, the particle equations of motion are integrated with $(> \mathrm{Mach}\ 1.05)$9; solitary-wave speed is measured by tracking the first peak in bin-averaged contact force versus position; and the Mach number is defined as $m$0, where $m$1 is the weak-impact acoustic speed and $m$2 is the solitary-front speed (Frizzell et al., 4 Sep 2025).

The solitary-wave dynamics display threshold behavior. The SSDEM study reports that there is no dilation for $m$3, small dilation for $m$4, and saturation at approximately $m$5 mm to a few mm for $m$6. Barely supersonic waves with $m$7 are sufficient to trigger surface dilation. Strong impacts initially produce shocks with speed scaling as $m$8; after rapid decay, the shock leaves a nearly nondispersive solitary-wave front of speed $m$9 (Frizzell et al., 4 Sep 2025).

Quantitatively, in a $R$0 compact bed shocked at $R$1 $R$2, the SSDEM simulations find a uniform $R$3 mm for $R$4 m, a corresponding bulk-density drop $R$5, and a surface-band thickness of approximately $R$6–$R$7 m, with a fully detached band extending to $R$8 cm. Spatially, only the initial shock-decay region in the first $R$9 m and the end-wall reflection zone in the last $E_{eq}$0 m depart from the uniform $E_{eq}$1 profile. Channel length between $E_{eq}$2 and $E_{eq}$3 m produces identical dilation within $E_{eq}$4 mm random-seed error, provided the bed is long enough for the solitary pulse to detach from the initial shock and reach a steady shape (Frizzell et al., 4 Sep 2025).

The material-parameter study based on LIGGGHTS reports complementary scaling trends. In a $E_{eq}$5 channel filled with $E_{eq}$6 monodisperse spheres under lunar gravity $E_{eq}$7, vacuum, and $E_{eq}$8–$E_{eq}$9, both $v_0$0 and $v_0$1 follow the predicted power laws with $v_0$2. From the force law and lofting criterion, $v_0$3 increases with $v_0$4 and $v_0$5 and decreases with $v_0$6 and $v_0$7 (Frizzell et al., 3 Sep 2025).

Packing and bed geometry matter strongly. At $v_0$8, increasing $v_0$9 from $F_p \approx 0.719\,(m^3 E_{eq}^2 R v_0^6)^{1/5},$0 to $F_p \approx 0.719\,(m^3 E_{eq}^2 R v_0^6)^{1/5},$1 increases $F_p \approx 0.719\,(m^3 E_{eq}^2 R v_0^6)^{1/5},$2 from $F_p \approx 0.719\,(m^3 E_{eq}^2 R v_0^6)^{1/5},$3 to $F_p \approx 0.719\,(m^3 E_{eq}^2 R v_0^6)^{1/5},$4 mm, corresponding to $F_p \approx 0.719\,(m^3 E_{eq}^2 R v_0^6)^{1/5},$5–$F_p \approx 0.719\,(m^3 E_{eq}^2 R v_0^6)^{1/5},$6. The SSDEM abstract states that dilation increases with bed height, whereas the detailed sensitivity summary states that, for constant $F_p \approx 0.719\,(m^3 E_{eq}^2 R v_0^6)^{1/5},$7 and $F_p \approx 0.719\,(m^3 E_{eq}^2 R v_0^6)^{1/5},$8–$F_p \approx 0.719\,(m^3 E_{eq}^2 R v_0^6)^{1/5},$9 cm, $$m=\frac{4}{3}\pi R^3\rho, \qquad E_{eq}=\frac{E}{1-\nu^2}.$$0–$$m=\frac{4}{3}\pi R^3\rho, \qquad E_{eq}=\frac{E}{1-\nu^2}.$$1 mm decreases linearly with $$m=\frac{4}{3}\pi R^3\rho, \qquad E_{eq}=\frac{E}{1-\nu^2}.$$2; for constant $$m=\frac{4}{3}\pi R^3\rho, \qquad E_{eq}=\frac{E}{1-\nu^2}.$$3, the height change is nearly constant. A hard floor is consistently reported as necessary: without floor support, the wave rapidly decays (Frizzell et al., 4 Sep 2025).

5. Lunar-regolith interpretation and Lunar Cold Spots

Both 2025 granular studies connect SID to Lunar Cold Spots (LCS), but they report the inferred regolith anomaly with slightly different numerical summaries. One characterizes LCS as crater halos of reduced thermal inertia, with a $$m=\frac{4}{3}\pi R^3\rho, \qquad E_{eq}=\frac{E}{1-\nu^2}.$$4–$$m=\frac{4}{3}\pi R^3\rho, \qquad E_{eq}=\frac{E}{1-\nu^2}.$$5 bulk-density deficit to approximately $$m=\frac{4}{3}\pi R^3\rho, \qquad E_{eq}=\frac{E}{1-\nu^2}.$$6 cm depth and radial extent greater than $$m=\frac{4}{3}\pi R^3\rho, \qquad E_{eq}=\frac{E}{1-\nu^2}.$$7 crater radii, but with no obvious ejecta. The other describes LCS as distal low-thermal-inertia halos, approximately $$m=\frac{4}{3}\pi R^3\rho, \qquad E_{eq}=\frac{E}{1-\nu^2}.$$8–$$m=\frac{4}{3}\pi R^3\rho, \qquad E_{eq}=\frac{E}{1-\nu^2}.$$9 km from young craters, with bulk density inferred to drop by approximately $\Delta z$00 in the top $\Delta z$01 cm of regolith (Frizzell et al., 3 Sep 2025).

The granular SID calculations are intended to show that these observed scales are dynamically plausible. For particles with $\Delta z$02 MPa, $\Delta z$03, and $\Delta z$04, described as “lunar” grains, the LIGGGHTS simulations report $\Delta z$05 at $\Delta z$06, corresponding to approximately $\Delta z$07 height relief, and a predicted $\Delta z$08 cm. The same source states that the simulated dilation extends to a depth comparable to the channel bottom, namely $\Delta z$09 cm in the laboratory configuration, and argues that because actual LCS are approximately $\Delta z$10 cm deep and regolith extends approximately $\Delta z$11 m above megaregolith, the laboratory channel samples only a fraction of the relevant scale; longer channels would reduce $\Delta z$12 but still yield percent-level dilation. By contrast, rubbery soft grains with $\Delta z$13 MPa modulus and millimeter size show only approximately $\Delta z$14 height change and less than $\Delta z$15 near-surface dilation (Frizzell et al., 3 Sep 2025).

The SSDEM scaling analysis reaches a related conclusion from a different parameter set. Scaling from $\Delta z$16 mm steel-simulant grains to lunar regolith, under the assumptions of grain modulus approximately $\Delta z$17 MPa, $\Delta z$18, and $\Delta z$19, gives an approximately $\Delta z$20 deeper lofting band for the same wave strength, that is, approximately $\Delta z$21 cm rather than $\Delta z$22 cm. That study further notes that a $\Delta z$23 density drop in the top $\Delta z$24 cm corresponds to a thermal-inertia drop of order $\Delta z$25–$\Delta z$26, using $\Delta z$27, consistent with observed LCS thermal anomalies of approximately $\Delta z$28 K at sunrise (Frizzell et al., 4 Sep 2025).

The causal interpretation is explicitly partly speculative. One source states that a driven solitary wave in buried megaregolith could produce a continuous solitary wave in the regolith above over approximately $\Delta z$29 crater radii, lofting it gently and producing a uniform low-density shell; farther out, decayed solitary waves would arrive as discrete outbursts that loft only patches of surface, consistent with the “fuzzier” outer halo. This is presented as speculation rather than as a demonstrated field-scale inference (Frizzell et al., 3 Sep 2025).

6. Related fluid-mechanical usage and terminological scope

A distinct use of surface-dilation language appears in weakly nonlinear two-layer fluid theory. Jiang, Kovačič, and Zhou develop a model in which an underlying interfacial solitary wave modulates the surface signature in a two-layer fluid system, capturing three asymmetric behaviors: surface waves become short in wavelength at the leading edge and long at the trailing edge; they propagate toward the trailing edge with a relatively small group velocity and toward the leading edge with a relatively large group velocity; and they become high in amplitude at the leading edge and low at the trailing edge (Jiang et al., 2019).

In the detailed formulation associated with that work, the interfacial solitary wave is described by a KdV-type approximation, while the surface response is treated by ray theory. The local surface wavelength is written as

$\Delta z$30

with the local wavenumber determined implicitly by a Doppler-shifted dispersion relation. The slowly varying amplitude satisfies a wave-action conservation law, and the surface “dilation” is defined as a horizontal strain measured by

$\Delta z$31

The maximum dilation scales as

$\Delta z$32

This fluid-mechanical usage concerns wavelength, amplitude, and horizontal strain of surface-wave packets above an interfacial solitary wave, not lofting of grains under vacuum and low gravity (Jiang et al., 2019).

This suggests that “SID” is context-dependent. In the granular-vacuum literature, it refers to free-surface lofting or inertial bulk dilation induced by a laterally propagating compressive solitary wave. In the two-layer-fluid setting, it refers to a ray-theoretic surface strain induced by an underlying interfacial solitary wave. The common thread is solitary-wave forcing of a surface manifestation; the operative mechanics and observables are different.