Under investigation in this paper is the Whitham–Broer–Kaup (WBK) model for the dispersive long wave in shallow water. Connection between the WBK model and a second-order Ablowitz–Kaup–Newell–Segur (AKNS) system is revealed. By means of the Darboux transformation for the second-order AKNS system, the multisoliton solutions in terms of the double Wronskian determinant for the WBK model are derived. Inelastic interactions are graphically discussed. Our results could be helpful for interpreting certain nonlinear wave phenomena in shallow water.

A membrane absorber usually requires a large back cavity to achieve low-frequency sound absorption. This paper describes the design of a membrane acoustic metamaterial absorber in which magnetic negative stiffness is employed to reduce the size of the back cavity. As a baseline for the present research, analysis of a typical membrane sound absorber based on an equivalent circuit model is presented first. Then, a theoretical model is established by introducing negative stiffness into a standard absorber. It is demonstrated that a small cavity with negative stiffness can achieve the acoustic impedance of a large cavity and that the absorption peak is shifted to lower frequencies. Experimental results from an impedance tube test are also presented to validate this idea and show that negative stiffness can be employed to design compact low-frequency membrane absorbers.

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