The parameters and appearing in Eq are given by The

The parameters and appearing in Eq. (7) are given by:
The potential functions corresponding to the scattered waves are expressed as:where stands for the Hankel function of the first kind of order n, and , and are unknown coefficients. Considering the discussed potential functions for the scattered and incident waves in the matrix, the displacement field is given by:
Now, applying the continuity condition for stress and displacement at the cylinder/matrix interface, the unknown coefficients are identified. Continuity condition for stress and displacement are:
To study the scattered waves, the form function is considered. For the case of an incident everolimus wave, is obtained as:and for an incident shear wave, it is:
In the case of scattered compression waves, is equal to and in the cases of scattered SH and SV waves, it is equal to and , respectively.
In this paper, for the purpose of analysis of multiple scattering, an infinite plane wave of frequency incident on a grating of N infinitely long transversely isotropic cylinders, with outer radius aligned in one row and embedded in an isotropic matrix is considered. The wave vector makes an angle with respect to any of the cylinder axes. Corresponding to each cylinder, a local cylindrical coordinate system is considered where each local z-axis coincides with the cylinder axis while the local x-axes (corresponding to ) are along the incident wave direction. To properly account for multiple scattering, by employing the Graf’s Addition Theorem, the scattered wave from any cylinder is considered as an incident wave for the other cylinders. Therefore, considering the type of the incident wave, the incident potential functions take a more general form as follows:
The parameters , and for the case in which the incident wave is scattered from the cylinder-number c to cylinder number j (for ) are calculated as [18]:and for , we get:
Now, with the aid of Eqs. (17) and (18), together with the equations presented for the scattering from a single cylinder, analysis of multiple scattering is possible.
To take into account the viscoelastic behavior of the matrix material, the Havreliak–Negami model is employed. Use of this model yields complex Lame constants which are functions of the frequency. In this regard, the frequency dependence of the complex shear modulus is expressed as [16]:where (relaxed modulus) and (unrelaxed modulus) are the limiting values of the shear modulus at low and high frequencies , respectively is the relaxation time and are dimensionless material parameters. is the loss factor expressed as:where and:
Therefore, is dependent on the ratio of to . Adopting the hypothesis of a real Poisson ratio, the Lame constants are given as [16]:

In order to validate the numerical results of the analytical model, a number of experiments are conducted. These experiments are performed by using the short-pulse Method of Isolation and Identification of Resonances (MIIR). In this method, a broad-band short-pulse is incident on the target by an ultrasonic transmitter probe. The interaction of the incident wave with the cylinder results in a backscattered field that is picked up by the same ultrasonic probe which acts as both transmitter and receiver. The schematic of the experimental setup everolimus is illustrated in Fig. 3.
In the measured signal, the first echo is the specular echo and the echoes succeeding it are due to resonances of the system. The frequency spectrum of the backscattered signal comprises of the effects of the waves scattered from the cylinder and the frequency characteristics of the transducer. After filtering the frequency characteristics of the transducer, the frequency spectrum of the remaining signal would represent the form function of the cylinder versus the dimensionless frequency ka. The range of variations of ka for each experiment depends on the probe bandwidth, the cylinder radius and the mechanical properties of both cylinders and surrounding media. Therefore, in order to make changes to the frequency range, either the cylinder radius or the probe bandwidth should be changed.