Metallic Wavy Surfaces (MWS): Morphology and Applications

• Metallic Wavy Surfaces (MWS) are metallic interfaces with periodic corrugation used to control plastic accommodation, turbulence, and electromagnetic scattering.
• They are defined by geometric parameters—such as amplitude, wavelength, and cusp height—with distinct profiles for explosive welding, turbulent flows, and THz reflectors.
• MWS applications span explosive welding, drag modulation in turbulent flows, and THz non-line-of-sight links, enabling precise process control and improved signal security.

Metallic wavy surfaces (MWS) are metallic interfaces or exposed metallic surfaces with ordered or quasi-ordered waviness, used or observed in several technically distinct settings. In the cited literature, the term encompasses self-organized wavy interfaces formed during explosive welding, shallow sinusoidal walls introduced to modify turbulent boundary layers, and one-dimensional corrugated aluminum reflectors used to create non-line-of-sight terahertz links through Bragg-scattering lobes (Greenberg et al., 2016, Ghebali et al., 2017, Li et al., 2024, Li et al., 6 Aug 2025). Across these settings, MWS are characterized not by a single function but by a shared reliance on surface corrugation, interfacial excess area, or periodic geometry to control plastic accommodation, near-wall turbulence, or electromagnetic scattering.

1. Geometric forms and defining parameters

The literature uses closely related but not identical geometric descriptions for MWS. In explosive welding, the relevant quantities are the wave amplitude $A$, wavelength $\lambda$, and splash height $h_s$ defined on the bonded interface. In turbulent-flow studies, the wall is prescribed as a sinusoidal height field with amplitude $A_w$ and projected wavelengths $\lambda_x$ and $\lambda_z$. In terahertz studies, the reflector is a one-dimensional sinusoidal or cosinoidal corrugation with amplitude $A$ and spatial period $\Lambda$ (Greenberg et al., 2016, Ghebali et al., 2017, Li et al., 2024, Li et al., 6 Aug 2025).

Context	Surface description	Key parameters
Explosive welding	Transition from cusps to quasi-waves to periodic waves	$A$, $\lambda$, $\lambda$0
Turbulent channel flow	$\lambda$1	$\lambda$2, $\lambda$3, $\lambda$4, $\lambda$5
THz NLoS reflector	$\lambda$6 or $\lambda$7	$\lambda$8 mm, $\lambda$9 mm

For explosive-welded interfaces, typical values from SEM are $h_s$0–$h_s$1, $h_s$2–$h_s$3, and $h_s$4–$h_s$5, with dimensionless roughness $h_s$6 and, for Cu–Ta, $h_s$7–$h_s$8 (Greenberg et al., 2016). In skewed-wavy-wall DNS, the mean flow makes an angle $h_s$9 with the wave crests, the streamwise-projected wavelength is $A_w$0, the spanwise-projected wavelength is $A_w$1, and the wave slope is $A_w$2 (Ghebali et al., 2017). In the THz work under rain, the aluminum MWS has $A_w$3 mm, $A_w$4 mm, and root-mean-square roughness $A_w$5 from the un-corrugated aluminum surface; in the covered-surface study, the corrugation is machined into a $A_w$6 high-purity aluminum plate with conductivity $A_w$7 and surface roughness $A_w$8 (Li et al., 2024, Li et al., 6 Aug 2025).

A plausible implication is that “MWS” is best treated as a morphology class rather than a single device class: the common structural motif is periodic or quasi-periodic metallic waviness, but the operative physics differ substantially across welding, wall-bounded turbulence, and THz scattering.

2. Explosive-welding MWS as a sequence of interfacial transition states

In explosive welding, MWS arise through a sequence of morphological transition states. Below the lower boundary of the weldability window, isolated cusps (“wedges”) of the harder metal penetrate the softer partner; these cusps are solid-phase, diffusionless, self-similar, and act as topological anchors. Near the lower boundary, the cusps evolve into “splashes,” described as solid-state protrusions resembling water splashes, with typical height $A_w$9–$\lambda_x$0 and lateral size $\lambda_x$1. Slightly above the lower boundary, these splashes coalesce into patches or rows, creating a “patchwork quilt” of quasi-waves with local wavelengths $\lambda_x$2 varying from $\lambda_x$3 to $\lambda_x$4 and amplitudes $\lambda_x$5–$\lambda_x$6. Fully into the weldability window, a more uniform, periodic wavy interface forms; for Cu–Ta, $\lambda_x$7–$\lambda_x$8 and $\lambda_x$9–$\lambda_z$0, whereas for Cu–Ti with intermetallics, waves contain embedded cusps (Greenberg et al., 2016).

Electron-microscope observations organize this sequence in detail. SEM of the Ta interface in a Cu–Ta joint shows isolated cusps after etching Cu away, then regularly spaced splashes at the lower boundary, and then splash groups whose contact regions seed larger-scale organization. As collision intensity increases, splashes align into rows and bands; the bands coalesce, and boundary splashes act as “zip-fastener” seeds, leading to sinusoidal relief. The role of mutual solubility is explicit: in insoluble systems such as Cu–Ta, waves are clean curves without cusps, whereas in soluble systems such as Cu–Ti, intermetallic particles “lock” cusps in the wave bodies, producing intermittent waves (Greenberg et al., 2016).

The proposed mechanism is plastic and self-organizing rather than hydrodynamically liquid in the ordinary sense. Collision pressures far exceed yield strengths, producing intense plastic deflection and an irreversible increase of the contact area. Upon pressure release, the non-equilibrium interface seeks to relax but must conserve an excess surface area; plastic flow at the collision front creates initial cusps, and neighboring splashes merge under lateral stresses and oscillating shock-wave interactions. Material properties, especially hardness, mutual solubility, and intermetallic precipitation, determine whether cusps lock into waves or unzip into a clean sinusoid (Greenberg et al., 2016).

The same source states that, although no detailed analytic model is presented in the paper, a generic energetic framework applies:

$\lambda_z$1

with bending contribution

$\lambda_z$2

and a critical wavelength analogous to wrinkling,

$\lambda_z$3

The details further give a perturbation-growth form

$\lambda_z$4

These relations are presented as a generic framework, not as a closed derivation of the explosive-welding process itself (Greenberg et al., 2016).

3. Imitation experiments, interface heterogeneity, and process control

Simulation and imitation experiments were devised to reproduce key geometric features of explosive-welding MWS. The setup used two plates—a ruler and a thin metal sheet—clamped at the ends, with ruler thickness $\lambda_z$5 mm and plate thickness $\lambda_z$6 mm. The ruler was bent to radius $\lambda_z$7–$\lambda_z$8 mm and then released. Cases included the plate glued to the tensile side, compressive side, or mid-line of the ruler, with different metals including Al, Cu, Pb, and Ta (Greenberg et al., 2016).

These experiments reproduced several motifs seen on welded interfaces. Plates on the tensile side formed discrete “roof” bends, typically $\lambda_z$9–$A$0 arcs, interpreted as analogous to splash clusters and quasi-waves. Plates on the compressive side developed wrinkles akin to cusp arrays, and unbonded plates formed dome-shapes illustrating plastic accommodation of excess area. The curvature is $A$1, so that for $A$2 mm, $A$3, and for $A$4 mm, $A$5. Observed roof-bend spacing obeyed $A$6–$A$7 with number of bends $A$8–$A$9, summarized by $\Lambda$0. The relaxation time scale was minutes to hours at room temperature (Greenberg et al., 2016).

The same report treats interface heterogeneity and bending as primary selectors of morphology. Initial heterogeneities, including surface roughness and cusps, serve as nucleation sites for splashes and control local $\Lambda$1 and $\Lambda$2. Bending-induced plastic strains impose periodic stress fields and favor formation of waves aligned with principal curvature. Material heterogeneity, such as grain size and hardness contrasts, leads to patchwise wave parameters, described as a “patchwork quilt” (Greenberg et al., 2016).

Industrial implications are stated directly. By engineering the starting roughness and imposing pre-bending, one can bias the direction and uniformity of the wavy interface. Materials with moderate mutual solubility, forming small intermetallics, yield intermittent or embedded-cusp waves, whereas insoluble pairs yield cleaner sinusoidal waves. Controlled heterogeneity enables tailored mechanical interlocking without large intermetallic embrittlement. For desired wave patterns, collision velocity $\Lambda$3–$\Lambda$4 minimal weldability velocity $\Lambda$5, just above the lower boundary, produces controlled quasi-waves with $\Lambda$6 and $\Lambda$7; larger $\Lambda$8 toward the center of the weldability window and collision angle $\Lambda$9–$A$0 yield fully periodic waves with $A$1 and $A$2. A stand-off distance $A$3 in the range $A$4–$A$5 mm and pre-texturing of the flyer plate to roughness $A$6 with slight pre-bending $A$7 are likewise given as practical guidance (Greenberg et al., 2016).

A common misconception is that the wavy morphology in explosive welding is simply a frozen liquid-like instability. The cited study instead emphasizes solid-phase, diffusionless cusps, plastic deflection, and excess-area relaxation, even though some morphological features are described by analogy to splashes (Greenberg et al., 2016).

4. Near-wall turbulence over metallic waviness: drag reduction and drag increase

In wall-bounded turbulence, MWS are used in two distinct ways in the cited literature. One line of work investigates skewed wavy walls as a passive means of emulating a Spatial Stokes Layer (SSL) and reducing turbulent skin-friction drag. Another examines two-dimensional sinusoidal waviness in the “waviness” regime, where increasing slope enhances near-surface turbulent mixing and raises total drag (Ghebali et al., 2017, Jayaraman et al., 2019).

For skewed shallow waviness, the lower wall is

$A$8

and the mean spanwise velocity profile above the wall creates a transverse shear strain $A$9. This is compared with the analytic SSL field

$\lambda$0

whose shear is

$\lambda$1

The skin-friction coefficient is defined as

$\lambda$2

and the total drag change is

$\lambda$3

with empirically $\lambda$4. For small wave slopes $\lambda$5, the friction-drag reduction grows quadratically, $\lambda$6; for moderate slopes $\lambda$7, the trend becomes approximately linear, while the pressure-drag term remains quadratic. The reported optimum lies at $\lambda$8–$\lambda$9, corresponding to $\lambda$00–$\lambda$01, $\lambda$02–$\lambda$03, and skew angle $\lambda$04–$\lambda$05. Under these conditions, $\lambda$06, $\lambda$07, and the net drag reduction $\lambda$08; on the finest grids, the asymptotic value falls to $\lambda$09 (Ghebali et al., 2017).

The DNS methodology for this passive-drag-reduction problem is also specified. The domain is a pressure-driven turbulent channel of height $\lambda$10, periodic in streamwise and spanwise directions, with the lower wall or both walls conformally meshed to the sinusoidal geometry. The flow is driven at an angle $\lambda$11 to the mesh, enabling statistical averaging. The solver is a fully-explicit fractional-step finite-volume method with collocated variables, second-order spatial differencing, third-order gear-type time stepping, Rhie–Chow interpolation, and a Poisson solver via successive-line over-relaxation with multigrid acceleration. A representative case at $\lambda$12 used $\lambda$13, $\lambda$14, wall-normal spacing from $\lambda$15 at the wall to $\lambda$16 in the core, domain size $\lambda$17, $\lambda$18, and $\lambda$19, with statistics averaged for $\lambda$20 (Ghebali et al., 2017).

A separate DNS study of two-dimensional sinusoidal waviness,

$\lambda$21

holds $\lambda$22–$\lambda$23 fixed while varying mean surface slope $\lambda$24 from $\lambda$25 to $\lambda$26 at $\lambda$27. In this regime, the total drag is split into viscous and pressure contributions,

$\lambda$28

Increasing $\lambda$29 produces an $\lambda$30–$\lambda$31 growth in $\lambda$32 and an $\lambda$33–$\lambda$34 non-zero $\lambda$35, rising from zero at $\lambda$36 to $\lambda$37 at $\lambda$38. The form-drag fraction grows monotonically with $\lambda$39. The increase in wave slope enhances near-surface turbulent mixing, yields higher fraction of form drag, accelerates return to isotropy, and modulates the buffer layer. Direct shear-production of $\lambda$40 is zero; spanwise variance is produced by the pressure-rate-of-strain mechanism, concentrated primarily on the windward side for flows with very little flow separation (Jayaraman et al., 2019).

Taken together, these results show that “wavy surface” is not synonymous with drag reduction. Skewed three-dimensional waviness can passively reproduce an SSL-like shear-strain pattern and deliver a modest overall-drag reduction, whereas two-dimensional sinusoidal waviness at comparable inner-scaled heights can increase drag while enhancing mixing (Ghebali et al., 2017, Jayaraman et al., 2019).

5. THz non-line-of-sight links, Bragg scattering, and rain-conditioned vulnerability

In THz communications, MWS are used as passive non-line-of-sight reflectors. The surface studied under rainfall conditions is a one-dimensional sinusoidal corrugation machined into an aluminum plate,

$\lambda$41

with $\lambda$42 mm and $\lambda$43 mm. For a single-period deterministic profile, the autocorrelation function is

$\lambda$44

and the one-dimensional power spectral density is

$\lambda$45

Under the Kirchhoff approximation, the bistatic differential scattering cross section per unit length in $\lambda$46 is

$\lambda$47

and substituting the sinusoidal profile yields discrete Bragg-scattering lobes indexed by integer $\lambda$48:

$\lambda$49

For each lobe,

$\lambda$50

with the specular component $\lambda$51 and first-order Bragg lobes $\lambda$52 dominating for $\lambda$53 (Li et al., 2024).

Rain modifies the scattering in two ways. First, the effective reflection coefficient becomes

$\lambda$54

using $\lambda$55 and a rain attenuation law

$\lambda$56

Second, total bistatic scattering is reduced along the free-space paths according to

$\lambda$57

Surface wetting is modeled by an effective RMS height

$\lambda$58

with droplet height $\lambda$59 mm, introducing an additional small-perturbation attenuation factor

$\lambda$60

At $\lambda$61 GHz and extreme rain intensity $\lambda$62 mm/hr, $\lambda$63 Np, so that $\lambda$64 drops by $\lambda$65 dB relative to dry conditions; over the full path, $\lambda$66 Np, giving a uniform $\lambda$67–$\lambda$68 dB reduction of scattering peaks (Li et al., 2024).

The security analysis uses three metrics. The normalized secrecy capacity is

$\lambda$69

with $\lambda$70 indicating that Eve’s SNR approaches Bob’s. The variation parameter is

$\lambda$71

and the backscatter parameter is

$\lambda$72

Eavesdropping is declared successful when $\lambda$73, $\lambda$74, and $\lambda$75 simultaneously (Li et al., 2024).

The reported measurements used a controlled-rain chamber, frequencies from $\lambda$76 to $\lambda$77 GHz with analysis focused on continuous waves at $\lambda$78 and $\lambda$79 GHz and $\lambda$80 dBm, Tx–MWS distance $\lambda$81 m, MWS–Rx distance $\lambda$82 m, horn antennas with $\lambda$83 cm dielectric lenses and $\lambda$84 dBi gain, and angular scans over $\lambda$85 in $\lambda$86 steps. Even under heavy rain, the MWS exhibited distinct Bragg lobes at angles satisfying $\lambda$87. The sinusoidal grooves fostered uniform film formation but prevented large isolated droplets, so the main peaks remained within $\lambda$88 dB of dry values, whereas a smooth surface showed up to $\lambda$89 dB fluctuations when wet. Peaks in side-scattered power corresponded to $\lambda$90, typically at $\lambda$91 for $\lambda$92 and $\lambda$93 at $\lambda$94 GHz, and at $\lambda$95 and $\lambda$96 at $\lambda$97 GHz for $\lambda$98; at these angles, $\lambda$99 and $h_s$00 as well (Li et al., 2024).

6. Covered MWS in indoor THz relays: redistribution rather than elimination of risk

The covered-surface literature extends the THz use of MWS to indoor passive relays concealed beneath common materials. The underlying MWS is a one-dimensional cosinoidal corrugation machined into a high-purity aluminum plate,

$h_s$01

with $h_s$02 mm and $h_s$03 mm, selected to satisfy Bragg’s condition for angular redirection in the $h_s$04–$h_s$05 GHz band. The channel model combines free-space path loss

$h_s$06

with an angular-dependent reflection coefficient

$h_s$07

where $h_s$08 satisfies Bragg’s condition, $h_s$09 is the loss constant of the covering, $h_s$10 its thickness, and $h_s$11 is proportional to $h_s$12. The end-to-end gain is

$h_s$13

with $h_s$14 and $h_s$15 (Li et al., 6 Aug 2025).

The experimental testbed used a CW source at $h_s$16 dBm, up-converted to $h_s$17, $h_s$18, and $h_s$19 GHz, horn antennas with $h_s$20 cm dielectric lenses producing a $h_s$21 mm beam, and power readings via a calibrated sensor with $h_s$22 dB accuracy. The MWS was fixed at $h_s$23 cm from Tx and $h_s$24 cm from Rx on a vibration-isolated optical bench. Coverings included wallpaper samples P1–P3 with thickness $h_s$25–$h_s$26 mm mounted via adhesive with micro-airgaps, a polyester curtain with $h_s$27 mm, and titanium-dioxide plaster with $h_s$28–$h_s$29 mm, troweled to fill corrugations and then dried and sanded flat. Permittivity was extracted using T-SPEC 800 THz-TDS over $h_s$30–$h_s$31 THz, and two angular measurement modes were used: receiver rotation of $h_s$32 around the MWS, and MWS rotation of $h_s$33 with Tx/Rx fixed to simulate clandestine reorientation (Li et al., 6 Aug 2025).

Quantitative effects depend on the covering. For wallpaper at $h_s$34 GHz, Mode 1 patterns show an approximately $h_s$35 dB rise in the specular direction versus the bare MWS due to constructive interference between surface reflection and substrate Bragg lobe, while off-specular lobes drop by $h_s$36–$h_s$37 dB. The secrecy-capacity metric

$h_s$38

satisfies $h_s$39 over multiple angles, and under Mode 2 the variation parameter

$h_s$40

remains below $h_s$41 at eavesdropping-optimal rotations of $h_s$42 and $h_s$43; the backscatter metric $h_s$44 also stays below $h_s$45. For curtain, angular gain patterns deviate by no more than $h_s$46 dB from the bare case, $h_s$47 persists over a broad set of angles, and covert eavesdropping remains possible around $h_s$48 rotation with simultaneous $h_s$49 and $h_s$50. For wall plaster, the specular gain is $h_s$51 dB over bare, side lobes smear and weaken, but $h_s$52 still occurs near specular and $h_s$53, and a $h_s$54 rotation again satisfies both $h_s$55 and $h_s$56. Across all coverings, BER predictions for 16-QAM fall below the forward error-correction threshold $h_s$57 in some angular regions, indicating recoverable data at the eavesdropper (Li et al., 6 Aug 2025).

The physical interpretation given in the source is that coverings redistribute rather than remove the scattering risk. Bragg-based scattering lobes are fixed by $h_s$58 and $h_s$59, so coverings with thickness $h_s$60 cannot shift the angular positions of strong beams. Dual-path constructive interference between covering reflections and substrate Bragg scattering can enhance eavesdropping angles without alerting Bob or Alice, and conventional backscatter monitoring can miss these changes because covered corrugations mask them below detection thresholds. The wavelength-scale thickness and dielectric constants of wallpaper and curtain still transmit sufficient power to the aluminum substrate for the underlying scattering to remain dominant (Li et al., 6 Aug 2025).

The same work proposes five hardening directions: active perturbation of corrugation phase via embedded piezoelectric actuators introducing pseudo-random sub-wavelength deformations $h_s$61; frequency hopping combined with corrugation segment switching; integrated absorptive cladding using a thin lossy polymer layer with $h_s$62 over non-specular lobes; multi-angle backscatter sensing using an array of low-gain receivers around the transmitter; and cryptographic beam-null coding through small amplitude-phase modulation patterns on the CW carrier (Li et al., 6 Aug 2025). This suggests that, in THz relaying, passive geometry alone is insufficient as a security measure even when the reflector is hidden beneath ordinary indoor materials.