Transmission loss of one-dimensional systems with frequency-dependent complex masses

The mass law is a convenient and useful benchmark for assessing transmission-loss behavior, not only for single-leaf partitions, but also for more complicated structures. Commonly, the mass is treated as a frequency-independent positive quantity. However, the mass law is easily generalized for the case of a complex mass depending on frequency ("dynamic mass"). It is not really a surprise that a negative mass leads to the same transmission loss as a positive one and that a complex mass implies dissipative effects. For a long time complex mass densities are used in the description of porous absorbers as equivalent fluids. By contrast the use of negative or complex masses in multi-layered structures appears to be novel. In particular, the familiar and often highly annoying transmission-loss minimum of mass-springmass systems can completely disappear when at least one mass is negative. An excellent performance is achieved with both masses negative and equal. Unfortunately this enticing solution is only fictitious. Nevertheless, with suitably designed metamaterials it might be possible to approach that fiction to some extent. As a simple realization of such a metamaterial a rigid frame with an internal resonator is tested. Its dynamic mass can indeed become nearly negative in a certain frequency range. In the considered examples, however, the desired transmission-loss improvement is accompanied by substantial reductions, in particular at higher frequencies. Thus the said transmission problem is shifted to higher frequencies. A more comprehensive parameter study could lead to more favorable configurations. Otherwise different metamaterial realizations including "dynamic springs" might be pursued.