## 1 Introduction  
In a two-dimensional cavity-duct system, an acoustic wave spreads in the tube with the existence of the cavity, and sound pressure is amplified around the cavity because of the interaction between the acoustic wave and the unstable shear layer. Trapped modes exist below or above the cut-off frequency and bring considerable damage to the system, such as noise, stability, and security issues. The trapped mode is a kind of free oscillation that often occurs in mathematical theory, and the embedded trapped modes emerge above the cut-off frequency [1]. [PERSON] found that trapped modes occur when a mass of fluid is bounded by fixed surfaces and by a free surface of infinite extent. Both a mode on a sloping beach and a mode near submerged circular cylinder were constructed mathematically, and the existence of trapped modes was discovered in a water environment [2]. On the basis of a battery of experiments, [PERSON] observed that a sequence of resonances, which are caused almost entirely by acoustic effects, are formed by pressure fluctuations [3]. [PERSON] also used relaxation methods to calculate the resonant frequencies and the pressure amplitude distribution, and the numerical results coincided well with those of his previous experiments. [PERSON] and [PERSON] applied the mathematical technique of [PERSON], and discovered that the trapped mode appears in the vicinity of a circular cylinder. [PERSON] presented an approach based on complex residue theory in order to simplify the transcendental equation, and found that it was appropriate for various configurations [4]. In a long waveguide including a number of circular cylinders in the center, [PERSON] also combined multipole expansions methods and addition theorems for Bessel functions, discovering that the number of trapped modes that depends on the geometric configuration is less than or equal to the number of cylinders [5]. [PERSON] introduced an open system with a perfectly matched layer (PML) as the absorbing boundary condition, and numerically computed the resonances for a few of obstacles in the one- and two-dimensional waveguides. He also observed that the value of imaginary part of the resonance influenced the trapped mode, and the frequency of trapped modesincreased as the imaginary part increased [6]. [PERSON] developed a two-step numerical code to investigate the influence of mean flow on the trapped modes. The mean flow was first solved, and the next step was the calculation of the acoustic field by using linearized acoustic perturbation equations (APEs). The effect of mean flow Mach number on the frequency of the acoustic mode was also researched. He observed experimentally that the trapped mode also depended on the size of the cavity [7]. More recently, [PERSON], [PERSON] investigated the characteristics of trapped modes numerically and experimentally [8]. The existence of trapped modes with an asymmetric radial component was shown in the numerical simulation. There was a complex interaction between particle velocity fluctuations and the region near the edge of the cavity. [PERSON] investigated the characteristics of trapped modes in a rectangular cavity attached to a cylindrical duct. The numerical part indicated the existence of trapped acoustic modes with the nonuniform acoustic velocity along the shear layer. The flow test part showed that the excitation of shear layer is not necessary for the acoustic resonances [9]. [PERSON], [PERSON] et al. considered the reflection-transmission problem in a waveguide with an obstacle, showing that the reflection-less modes could be characterized as eigenfunctions [10]. Acoustic wave transmission in a nonaxisymmetric waveguide, which consists of a cylindrical resonator and two cylindrical waveguides, was considered. The trapped mode was found because of full destructive interference [11]. [PERSON], [PERSON] proposed a method to solve the aeroacoustic problem in the time domain with nonuniform steady mean flow.