Coupled Scholte modes supported by soft elastic plates in water

Localized acoustic surface waves supported by a “soft” elastic plate in water are explored. Unlike many materials, such as aluminum, for soft interfaces the Scholte wave, a localized interface wave, has a speed well below that of sound in water, and the energy of the Scholte wave is no longer mainly localized to the water. We note that the Scholte velocity is largely independent of Poisson's ratio in the solid, and rather than the bulk speeds of sound, the ratio between the Young's modulus and the density of the solid may better indicate whether an interface is soft. The behavior of the coupled Scholte modes along a thin plate with soft interfaces are investigated. It is demonstrated, and experimentally verified using acrylic plates underwater, that for soft interfaces, the symmetric coupled Scholte mode exhibits dispersive behavior, and deviates from the Scholte and the fluid velocities at low frequencies.

Figure 1

Variation in the Scholte velocity $c_{\mathrm{sch}}$, and the transverse and longitudinal bulk speeds of sound $c_{\mathrm{T}}$ and $c_{\mathrm{L}}$ over the allowed range of positive Poisson's ratio $\nu$, while the Young's modulus $E$, solid density $\rho$, and fluid parameters remain constant. The region over which $c_{\mathrm{T}}$ and $c_{\mathrm{sch}}$ vary is expanded in the inset.

Figure 2

Color plot showing the variation of the Scholte velocity $c_{\mathrm{sch}}$ with respect to a fixed fluid sound velocity $c_{\mathrm{F}}$ over a range of the Young's modulus $E$ and the density $\rho$ of the solid. The arbitrary fixed Poisson's ratio is $\nu =0.33$, and fluid properties are that of water ($c_{\mathrm{F}}=1480 {\mathrm{ms}}^{-1}, {\rho }_{\mathrm{F}}=1000\text{kg}{\text{m}}^{-3}$). The red dashed line approximately indicated $c_{\mathrm{sch}}=0.95c_{\mathrm{F}}$. The Young's modulus and density of aluminum is given by the red cross (above the red dashed line, hard example), and of acrylic by the red dot (below the red dashed line, soft example).

Figure 3

Total energy density of the Scholte wave across the interface between water and (a) aluminum (hard interface), (b) acrylic (soft interface), calculated by summing the kinetic and strain energy densities [36, 37].

Figure 4

The dispersion curves for coupled Scholte modes in 10-mm-thick plates underwater are shown, together with the bulk speeds of sound in the system. (a) For an aluminum plate the antisymmetric coupled Scholte mode is shown, and the symmetric mode that follows $c_{\mathrm{F}}$ is omitted. This is an example of a hard plate. (b) For an acrylic (soft) plate, an additional symmetric coupled Scholte mode is clearly present, as well as the lower frequency antisymmetric mode.

Figure 5

Total energy density for the coupled Scholte mode across a 10-mm-thick acrylic plate in water, calculated by summing the kinetic and strain energy densities [36, 37]. (a) shows the distribution for a weakly coupled, high frequency Scholte mode, while (b) and (c) show the distribution for strongly coupled Scholte modes at $f=20$ kHz for the slower antisymmetric and faster symmetric mode, respectively (indicated by the inset displacement grids).

Figure 6

A schematic of the experimental setup used to excite and detect coupled Scholte modes in acrylic plates of thicknesses $d=5$, 10, and 20 mm underwater.

Figure 7

Experimental data obtained for acrylic plates of thickness $d$ = (a) 5 mm, (b) 10 mm, and (c) 20 mm are presented as color maps of the Fourier transformed data. The peaks in Fourier amplitude, shown in blue, indicate the dispersion of the modes, in agreement with the overlaid theoretical predictions.