Surface wakes on ultra-soft solids

Abstract

We explore the dynamical response of the free surface of an ultra-soft solid driven by a localized moving pressure disturbance. Experiments reveal a steady V-shaped wake analogous to a surface Mach wedge. A simple geometric argument provides a qualitative explanation consistent with observations. A theoretical framework combining elastodynamic, capillary, and gravitational effects yields a generalized dispersion relation that smoothly interpolates between Kelvin's theory of liquid interface wakes and Rayleigh's theory of elastic surface waves. Together, our experiments and theory reveal the existence of a soft wake regime that bridges fluid and solid surface wave physics, offering new routes for probing the dynamics of soft surfaces.

Figures (6)

• Figure 1: (a) Schematic of the experimental setup, where a pressure source of compressed air moving at speed $U$ impinges on the surface of an elastic solid, generating a surface wake. (b) Top-view images of wakes for two representative speeds, $U=0.55 ~m/s$ (left) and $U=1.28~ m/s$ (right) leads to different opening angles ($2\alpha$) on an ultra-soft solid ($\mu \approx 6.5 Pa$). The wake angle decreases with the increasing speed of the pressure source. A projected grid on the surface of the gel slab from the top enables visualization of the surface deformation. (c) Experimental data showing Mach-like scaling, where $\sin\alpha$ varies linearly with $c_s/U$, and $c_s = \sqrt{\mu/\rho}$ is the Rayleigh (shear) wave speed. (d) Schematic of the geometry of wave fronts on a soft solid surface due to a moving pressure source.
• Figure 2: Dimensionless phase speed $C_p$ of free waves on the surface of an infinitely deep ultra-soft solid with weak elasticity $(C_s \ll 1)$, shown as a function of the scaled wavenumber $K$ following (\ref{['redDis_phasespeed']}). The minimum occurs at $K^*=(\rho g L^2/\gamma)^{1/2}$, with a value $C_{p,min} = (2/(\mathrm{Fr}^2 \mathrm{Bo}) + 4 \varphi_2(\lambda^2)C_s^2)^{1/2}$. The dashed line corresponds to the capillary-gravity waves in infinitely deep inviscid liquid without elasticity $(C_s=0)$.
• Figure 3: Wake pattern on ultra-soft solids showing the normalized surface elevation, computed using (\ref{['ND-disp']}) at $\text{Bo} \approx 3.7$, for different scaled shear-wave speeds $C_s=c_s/U$ and Froude numbers $\mathrm{Fr}$ as functions of the scaled coordinates $X/\Lambda_g$ and $Y/\Lambda_g$, where $\Lambda_g =2\pi \text{Fr}^2$ is the dimensionless wavelength of the gravity wave corresponding the source size $L$. The opening half-angle $\alpha$ of the leading dominant ridge (dashed black line) is shown for increasing (a) $C_s$ at $\mathrm{Fr}=1.1$ and (b) $\mathrm{Fr}$ at $C_s=0.02$. The computational domain is approximately $0.75~ \mathrm{Fr}^2$ times the length of the experimental tank.
• Figure 4: Plot of $\sin \alpha$ as a function of $c_s/U$ for different Froude numbers $\mathrm{Fr}$, obtained from simulations following (\ref{['ND-disp']}). Blue curves: iso-$\mathrm{Fr}$ contours from simulations, with $\mathrm{Fr}$ indicated above each curve. Yellow squares: simulations run at the experimental conditions. Red circles: experimental observations for comparison. The experimental data points that correspond to different pulse speeds $U$ move across iso-$\mathrm{Fr}$ curves, such that $\mathrm{Fr} ~C_s \sqrt{\mathrm{Bo}} = \frac{U}{\sqrt{gL}}\sqrt{\frac{\mu}{\rho U^2}}(\frac{\rho gL^2}{\gamma})^{1/4} =[(\mu^2/(\rho g \gamma)]^{1/4}$ is constant. The inset shows the wake angle scales with the inverse of $\mathrm{Fr}$, for both experiments and simulations.
• Figure 5: (a) Dimensionless phase and group speeds as functions of wavenumber from gravity-capillary elastodynamics ($C_s>0)$ following (\ref{['redDis_phasespeed']}) and (\ref{['eq:group_speed']}), showing the elasto-gravity ($C_g<C_p$) and elasto-capillary branches ($C_g>C_p$). (b) Schematic of the stationary wake pattern from interference of monochromatic waves generated by a source moving from $S_{-t}$ to $S_0$, adapted from crawford1984elementaryrabaud2014gravcap. (c) Radiation angle $\beta$ for allowed wavenumbers at fixed $\mathrm{Bo}, ~\mathrm{Fr}$, with varying shear-wave speeds $C_s$, following (\ref{['eq:wake-angle-of-K']}). The dashed curves in (b) and (c) correspond to $C_s=0$. (d) Schematic of the resultant wake pattern formed by superposition of broadband waves, defining the characteristic angles.