Boundary-Layer Effects on Acoustic Transmission Through Narrow Slit Cavities

We explore the slit-width dependence of the resonant transmission of sound in air through both a slit array formed of aluminum slats and a single open-ended slit cavity in an aluminum plate. Our experimental results accord well with Lord Rayleigh’s theory concerning how thin viscous and thermal boundary layers at a slit’s walls affect the acoustic wave across the whole slit cavity. By measuring accurately the frequencies of the Fabry-Perot-like cavity resonances, we find a significant 5% reduction in the effective speed of sound through the slits when an individual viscous boundary layer occupies only 5% of the total slit width. Importantly, this effect is true for any airborne slit cavity, with the reduction being achieved despite the slit width being on a far larger scale than an individual boundary layer’s thickness. This work demonstrates that the recent prevalent loss-free treatment of narrow slit cavities within acoustic metamaterials is unrealistic.

Figure 1

(a) Schematic of the slit-array experimental configuration. The aluminum slats (hatched) were stacked vertically between two mirrors 3 m apart, with a speaker and microphone placed at their focal lengths 1 m away. The sample stand was covered in acoustic absorber (dotted fill) and the beam path is indicated by a dashed arrow. (b) Schematic of the slit-array sample itself, with dimensions labeled (not to scale). Here, $L=19.8\pm 0.12\mathrm{mm}$, $d=2.91\pm 0.03\mathrm{mm}$, and $\mathrm{\Lambda }=d+w$.

Figure 2

Schematic of the single slit experimental configuration. The dashed blocks represent the aluminum sample, and the dotted blocks represent an acoustic baffle. The microphone and speaker are $\sim 20°$ off normal in the $xz$ plane, covered in acoustic absorber, and separated from the sample faces by 220 and 400 mm, respectively. Here, $L=35.0\pm 0.1\mathrm{mm}$.

Figure 3

The fundamental resonant frequency of each slit cavity $f_{FP}^{\prime }$, normalized to that predicted by the Fabry-Perot condition $f_{FP}$ as a function of the ratio of viscous boundary-layer thickness ${\delta }_{\nu }$ to slit width $w$. The solid circles are the experimentally recorded mean frequencies, with the error bars representing their standard deviation. The dashed lines represent a lossless FEM numerical prediction, and the solid lines a more complete numerical prediction that includes the viscous and thermal properties of each system. The top and bottom panels represent the slit array and single slit samples, respectively. Insets: examples of the experimentally measured transmitted frequency spectra for the relevant sample. Four different slit widths of 2.0, 1.0, 0.5, and 0.2 mm are represented by the dot-dashed, dashed, long-dashed, and solid lines, respectively.

Figure 4

Ratio of the calculated effective speed of sound $v_p$ to adiabatic speed $c_0$ in each slit cavity, as a function of the ratio of viscous boundary-layer thickness ${\delta }_{\nu }$ to slit width $w$. Triangular points and accompanying error bars are experimental data converted from Fig. 3, with black and gray representing the single slit and slit-array data, respectively. The dashed line represents the prediction of Stinson’s [20] analytic theory.