Boundaries of negligible reflection coefficients achieved by single compliant layers.

Abstract

The basic boundary is assumed to possess a (specular) reflection coefficient that is not negligible, i.e., [Zp(ω)/Zf(Θ)] does not approach unity. Here, Zp(ω) is the surface impedance of the basic boundary at the frequency ω and Zf(Θ) is the surface impedance of the fluid that occupies the semi-infinite space above the boundary. It is noted that Zf(Θ)=[ρc/cos(Θ)], where ρ and c are the density and speed of sound in the fluid, respectively, and Θ is the angle of incidence from the normal to the plane of the boundary. For a basic boundary that possesses a surface impedance Zp(ω) that is mass controlled, namely, Zp(ω)=iωMp, where Mp is the surface mass, a compliant layer of surface impedance Zc(ω)=(K/iω)(1+iηc), placed atop the basic boundary, will render the reflection coefficient negligible if: ηc=[ωMp/Zf(Θ)]; ω2=(K/Mp)(1+ηc2). In another example, the surface impedance ratio ‖Zc(ω)/Zp(ω)‖ at the frequency ω of concern is small compared with unity, then one may render the reflection coefficient of the boundary negligible by adding a panel of surface impedance Z1(ω) that is mass controlled atop the compliant layer: Z1(ω)=iωM1. In this case the reflection coefficient will be negligible if: ηc=[Zf(Θ)/ωM1]; ω2=(K/M1). In both examples the reflection coefficient is rendered negligible by providing the reflecting boundary with a resonant dynamic system that is adjusted so that the resonance frequency coincides with the frequency of concern. To achieve negligible reflection coefficient, the loss factor ηc of the compliant layer needs to be appropriately adjusted. The need for the later adjustment will be discussed.