**4. Model specification**

Figure 1 shows a schematic of the cylindrical double shell of infinite length subjected to a plane wave with an incidence angle . The radii and the thicknesses of the shells are *Ri e*, and *i e*, *h* in which the subscripts *i* and *e* represent the inner and outer shells. A concentric layer of porous material is installed between the shells. The acoustic media in the outside and the inside of the shell are represented by density and speed of sound: 1 1 *s c*, outside and 3 3 *s c*, inside.

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 39

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/ *c* , 1*c* is the speed of sound in incident Medium,

**5. Applying full method to two-dimensional problem** 

 

incident wave can be expressed as [1]:

 

is the angle of incidence.

 , 1 1 cos *<sup>y</sup>* 

**Figure 2.** Illustration of wave propagation in the porous layer

*t*

[1]. The displacements in the solid phase are:

*x x*

where 1 sin *<sup>x</sup>* 

For a two-dimensional problem as shown in the *x y* plane of Fig. 2, the potential of the

*<sup>i</sup> e*

 , 1 1 

<sup>1</sup> ( ) *x y jx y*

 

Three kinds of the waves propagate in porous material, therefore six traveling waves, which have the same trace wave numbers, are induced by an oblique incident wave in a finite depth layer of porous material, as shown in Fig. 2. The *x* and *y* direction components of the displacements and stresses of the solid and fluid phases were derived by Bolton et. al.

2 22 2 <sup>ˆ</sup> *yy yy <sup>x</sup> j x <sup>j</sup> y jy jy jy*

*DD D <sup>D</sup> u je e e e e* 

2 5 6 *ty ty <sup>x</sup> ty j x jy jy*

*j e De De*

1 2 3 4

 

   

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**Figure 1.** Schematic diagram of the double-walled cylindrical composite shell lined with porous materials
