Linear waves at viscoelastic interfaces between viscoelastic media

We derive the general dispersion relation for interfacial waves along a planar viscoelastic boundary that separates two viscoelastic bulk media, including the effect of gravity. Our unified theory contains Rayleigh waves, capillary-gravity-flexural waves, Lucassen waves, bending waves in elastic plates, and the standard dispersion-free sound waves, as limiting cases. To illustrate our results, we consider waves at a viscoelastic interface immersed in water and at an air-water interface. We furthermore investigate waves at a viscoelastic interface separating two identical viscoelastic bulk media, for which we consider both Kelvin-Voigt and Maxwell materials, as applicable to polymer gels and solutions. For all cases, we study how material properties determine the crossovers, scaling, and existence regimes of the various interfacial waves. Since we include viscoelastic effects for all media involved, our theory allows to model waveguiding phenomena in biology, such as pressure pulses in axon membranes, which are possibly relevant for acoustic nerve pulse propagation phenomena.

Figure 1

We consider a planar viscoelastic interface (medium II), which is located at $z=0$ and separates two viscoelastic bulk media (media I and III), which are infinitely extended in the half-spaces $z<0$, $z>0$. All viscoelastic media are modeled as linear, isotropic, and homogeneous. We include gravitational acceleration, which acts in the negative $z$ direction. We consider wave solutions that travel in the $x$ direction, and are translationally invariant in $y$, and decay exponentially away from the interface at $z=0$.

Figure 2

(a, c) Properties of waves at an interface separating two Newtonian fluids with identical parameters, as discussed in Sec. 4a2. Parameters are given in Sec. 4a, and correspond to a DPPC membrane (interface) and water (bulk). The full dispersion relation (30) (solid blue and green lines) and the equations that correspond to the individual factors of the factorized symmetric dispersion relation (31), i.e., Eqs. (44) (dashed purple line) and (45) (dashed red line), are solved numerically to obtain the wave number $k$ as a function of $\omega$. The corresponding (a) phase velocity and (c) propagation distance are then calculated using Eqs. (15) and (16). (b, d) Phase velocities and propagation distances for an air-water interface, as discussed in Sec. 4a3. Both the full dispersion relation (12) (solid blue and green lines) and the equations corresponding to the individual factors of the factorized asymmetric dispersion relation (35) (dashed red and purple lines) are solved numerically, and the resulting wave number $k\left(\omega \right)$ is used to calculate $c$, ${\beta }^{-1}$ via Eqs. (15) and (16). Additionally, the solution of the asymmetric vacuum-water dispersion relation (32) is shown for comparison (dotted orange line). For all subplots, vertical dashed lines denote the various crossover frequencies discussed for (a) and (c) in Sec. 4a2 and for (b) and (d) in Sec. 4a3; green dash-dotted lines denote crossovers in the capillary-gravity-flexural wave (CGW), and blue dashed lines indicate crossovers in the Lucassen wave. Red dotted lines highlight the frequencies at which solutions of the full dispersion relation disappear. The power-law scalings of $c$ and ${\beta }^{-1}$ within each scaling regime are indicated by black bars.

Figure 3

Plots of the displacement field $\mathit{u}(\mathit{r},t)$ of Lucassen and capillary-gravity-flexural waves (CGW) in the elastic and surface-tension-dominated regime, respectively. The scaling is chosen such that the respective dominant decay lengths into the bulk media, $1/\mathrm{Re}\left({\lambda }_{M,\mathrm{t}}^{-1}\right)$ and $1/\mathrm{Re}\left({\lambda }_{M,\mathrm{l}}^{-1}\right)$, as well as the propagation distance ${\beta }^{-1}=1/\mathrm{Im}\left(k\right)$ are well visible in the plot. The displacement of the interface is shown as a green line. Bulk water displacement is shown as a blue grid, whereas bulk air displacement is shown as a red grid. Decay lengths and volume element trajectories are shown in black. (a) Lucassen wave at water-water interface with $\omega =1.02\times {10}^3{\mathrm{s}}^{-1}$. (b) Lucassen wave at air-water interface with $\omega =1.04\times {10}^3{\mathrm{s}}^{-1}$. (c) Capillary-gravity-flexural wave at water-water interface with $\omega =1.02\times {10}^3{\mathrm{s}}^{-1}$. (d) Capillary-gravity-flexural wave at air-water interface with $\omega =1.04\times {10}^3{\mathrm{s}}^{-1}$.

Figure 4

Pressure maps and velocity fields corresponding to the waves shown in Fig. 3. Velocity fields are shown as black arrows, where arrow lengths correspond to the magnitude of the velocity. The pressure is shown as a heatmap, where a red-blue colormap is used for water pressure. The air pressure is indicated by a purple-green colormap. The pressure in the bulk fluids is always shown relative to the pressure at the interface, i.e., setting $P_0=0$ in Eq. (48). Scalings have been chosen equal to the plots shown in Fig. 3. (a) Lucassen wave at water-water interface with $\omega =1.02\times {10}^3{\mathrm{s}}^{-1}$. (b) Lucassen wave at air-water interface with $\omega =1.04\times {10}^3{\mathrm{s}}^{-1}$. (c) Capillary-gravity-flexural wave at water-water interface with $\omega =1.02\times {10}^3{\mathrm{s}}^{-1}$. (d) Capillary-gravity-flexural wave at air-water interface with $\omega =1.04\times {10}^3{\mathrm{s}}^{-1}$.

Figure 5

Phase velocities $c$ and propagation distances ${\beta }^{-1}$ for Lucassen waves at an interface separating viscoelastic materials. Phase velocities and propagation distances are obtained from $k\left(\omega \right)$ via Eqs. (15) and (16). For (a) and (c), the viscoelastic bulk materials are two Kelvin-Voigt materials with identical properties, as discussed in Sec. 4b1. To obtain $k\left(\omega \right)$, the full symmetric dispersion relation (30) (solid blue line), the Kelvin-Voigt Lucassen dispersion relation (52) (dashed red line), and the Newtonian-fluid Lucassen dispersion relation (46) (dotted green line) are evaluated numerically. For (b) and (d), two Maxwell fluids with identical parameters are considered as bulk materials; cf. Sec. 4b2. To obtain $k\left(\omega \right)$, each of the dispersion relations (30) (solid blue line), (39) (dotted orange line), (46) (dotted green line), (55) (dashed red line) is evaluated numerically. For all subplots, vertical dashed lines denote crossover frequencies as discussed in Sec. 4b1 for (a), (c), and in Sec. 4b2 for (b), (d). Blue dashed lines denote crossovers in the Lucassen wave. Red dashed lines highlight the frequencies at which solutions of the full dispersion relation disappear. The crossover to the free membrane limit is colored orange for better distinguishability. The power-law scalings of $c$ and ${\beta }^{-1}$ within each scaling regime are indicated by black bars.

Figure 6

Phase diagram showing the crossover frequencies found from the analytical solution of the Lucassen wave on the polymer solution-polymer solution interface Eq. (55). The crossover from Newtonian to Maxwell-fluid is denoted by ${\omega }^{\mathrm{MW}}$ (blue line), whereas the crossover to the free membrane limit is denoted by ${\omega }_{2\mathrm{D}}^{\rho }$ (orange line). The red dotted line shows the crossover from elastic to viscous response ${\omega }_{2\mathrm{D}}^{\mathrm{elastic}}$. The black dashed line shows our choice of $\tau ={10}^{-3}$ s as an example.

Figure 7

Traditional Rayleigh wave solutions at the vacuum-water interface. The dispersion relation for the traditional Rayleigh wave equation (34) is solved numerically to obtain the wave number $k$ as a function of $\omega$. Phase velocities and propagation distances are obtained from $k\left(\omega \right)$ via Eqs. (15) and (16).

Figure 8

Displacement plots of Lucassen and capillary-gravity-flexural waves (CGW) in the dissipative regime. The scaling is chosen such that the wave-type-specific dominant decay lengths into the bulk media $1/\mathrm{Re}\left({\lambda }_{M,\mathrm{t}}^{-1}\right)$ and $1/\mathrm{Re}\left({\lambda }_{M,\mathrm{l}}^{-1}\right)$ as well as the propagation distance ${\beta }^{-1}=1/\mathrm{Im}\left(k\right)$ are of the same order of magnitude and visible in the plot. The displacement of the interface is shown as a green line. Bulk water displacement is shown as a blue grid, whereas bulk air displacement is shown as a red grid. Decay length and volume element trajectories are shown in black. (a) Lucassen wave at water-water interface with $\omega =1.01\times {10}^8{\mathrm{s}}^{-1}$. (b) Lucassen wave at air-water interface with $\omega =1.04\times {10}^8{\mathrm{s}}^{-1}$. Note that the vertical decay length into air $1/\mathrm{Re}\left({\lambda }_{\mathrm{I},\mathrm{t}}^{-1}\right)$ is too large to include in the diagram without rendering the displacements too small to observe. (c) Capillary-gravity-flexural wave at water-water interface with $\omega =1.01\times {10}^8{\mathrm{s}}^{-1}$. (d) Capillary-gravity-flexural wave at air-water interface with $\omega =1.04\times {10}^8{\mathrm{s}}^{-1}$.

Figure 9

Pressure maps and velocity fields corresponding to the waves shown in Fig. 8. Velocity fields are shown as black arrows, where arrow lengths correspond to the magnitude of the velocity. The pressure is shown as a heatmap, where a red-blue colormap is used for water pressure. The air pressure is indicated by a purple-green colormap. The pressure in the bulk fluids is always meant to be the difference to the pressure at the interface, i.e., $P_0=0$ in Eq. (48). Scalings have been chosen equal to the plots shown in Fig. 8. (a) Lucassen wave at water-water interface with $\omega =1.01\times {10}^8{\mathrm{s}}^{-1}$. (b) Lucassen wave at air-water interface with $\omega =1.04\times {10}^8{\mathrm{s}}^{-1}$. (c) Capillary-gravity-flexural wave at water-water interface with $\omega =1.01\times {10}^8{\mathrm{s}}^{-1}$. (d) Capillary-gravity-flexural wave at air-water interface with $\omega =1.04\times {10}^8{\mathrm{s}}^{-1}$.

Figure 10

Plot of real and imaginary parts of the inverse transversal damping length ${\lambda }_{\mathrm{t}}^{-1}(k,\omega )$ in the frequency region of the breakdown of the Lucassen wave for a purely elastic membrane bounded by half-spaces of polymer gels modelled as Kelvin-Voigt fluids, obtained by inserting the numerical solution $k\left(\omega \right)$ of Eq. (30) into Eq. (9). Dashed lines show negative values.