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transmission of an isolated element at a given frequency (Figure 2). It is then standard procedure to combine the re- flection/transmission coefficients to determine the 2 × 2 ma- trix that describes how waves propagate across a single peri- od of the transmission line. The Bloch-Floquet condition for determining the effective wavenumber (ke ) is then equiva- lent to finding the matrix eigenvalues. This gives sinke b= ωb √ ρe Ce , where ω is the radial frequency, b is one-half the period length, and the effective density and compressibility are ρe = ρ + 2SZM / (– iωb) and Ce = C + 1 / (– iωb2SZH), respec- tively, where ρ and C are the fluid density and compressibil- ity, respectively, and S is the duct cross section.
A simple example is given by the limiting case of a hole in the duct which reduces the Helmholtz resonance frequency to zero and yields negative effective compressibility Ce be- low a finite cutoff. Conversely, Ce is positive above the cutoff, which is a result well known in musical acoustics: tone holes in a flute are a high-pass filter.
Figure 2. An acoustic duct with alternating membrane masses and Helmholtz resonators. The resonances of the two elements produce negative effective density and effective compressibility in certain frequency ranges. Simultaneously negative density and compressibility, a “double-negative material,” can be ob- tained by matching the resonances. See text for details.
A membrane attached to the sides of the duct behaves as a mass (m) constrained by a spring (Figure 2) with ZM = –iωm [1 – (ω0 /ω ) 2 ]/(2S) 2 and therefore acts as a high-pass filter. In this way, Lee et al. (2009) demonstrated negative effective inertia in a periodic array of membranes in air by observ- ing transparency above 735 Hz and very low transmission below that. By contrast, the low-frequency absorber of Liu et al. (2000), with impedance ZM = –iωm [1 – (ω /ω0 ) 2 ]–1/ (2S) 2 is a low-pass device whose cutoff frequency ω0/2π can made very low (400 Hz) by increasing the mass of the inter- nal oscillator. By selecting the resonance frequencies prop- erly, it is clear that it is possible to simultaneously achieve negative density and negative compressibility over a finite range of frequencies. As mentioned above, the phase and
energy propagation directions are then opposite; a situation known as negative group velocity (group and energy propa- gation velocities are almost always the same, except in the case of large absorption that is not applicable here). To see this, note that energy travels in the direction of the acoustic energy flux (pv), where v is particle velocity. A plane wave with dependence cos ω ( _xc – t) has phase velocity c, which is real-valued because both the effective density and compress- ibility are negative. The pressure and particle velocity are re- lated by the dynamic impedance (ρc) and therefore, because ρ is negative, the energy flux direction must be opposite to the phase velocity.
AMMs take advantage of negative properties in a variety of ways, including radiation/sensing in leaky wave antennae (LWAs) and focusing in phononic crystals. LWAs are one- dimensional transmission line devices, like the duct consid- ered above, designed to radiate sound into a surrounding fluid much like a long flutelike instrument with the sound emanating from the tone holes. The coupling to the exterior fluid is particularly strong if the compressibility element is an open hole. LWAs are of interest because of the possibil- ity to refract acoustic waves at different angles using the frequency-dependent phase speed within the waveguide. Positive and negative effective properties allow the angle to range over 180°, from the forward direction with phase and energy velocities aligned to the negative direction where the energy in one direction in the fluid couples with the LWA wave with the phase velocity in the opposite direction. Naify et al. (2013) demonstrated radiation depending on excita- tion frequency with positive, negative, and zero refractive indexes, using a structure made of periodically arrayed subwavelength alternating membranes and open vents. The LWA performs just as well sensing incident acoustic waves as it does radiating, which is a consequence of acoustic reci- procity under time reversal. The LWA property to discrimi- nate direction using frequency therefore allows directional- ity detection of incoming sound using a single microphone in place of beamforming using multiple sensors (Esfahlani et al., 2016).
If both effective density and compressibility become zero at the same frequency, then the phase speed remains real as this frequency is crossed because the properties are double positive on one side and double negative on the other. Such zero index materials are of interest because the phase speed becomes infinite (index is zero), which allows for devices that can steer acoustic energy in unusual ways, such as unidi- rectional transmission and cloaking (Ma and Sheng, 2016).
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