## Investigation of Characteristics of the Reflection of Stationary and Pulse Sound from Visco-Elastic Cylindrical Layers

**S. L. Ilmenkov**^{1,}, **A. A. Kleshchev**^{1}, **E. I. Kuznetsova**^{1}, **F. F. Legusha**^{1}, **Y. V. Chizhov**^{1}, **G. V. Chizhov**^{1}

^{1}Saint-Petersburg State Navy Technical University, Saint-Petersburg, Lotsmanskaya, Russia

### Abstract

The scattering task of stationary and impulse sound signals by the viscous-elastic and elastic cylindrical layers is studied with the help of Fourier series and fundamental solutions of the Helmholtz equation in circular cylindrical coordinates system. The reflection’s characteristics of sound (harmonic and impulse) are calculated in the large range of angles (including inverse direction).

### At a glance: Figures

**Keywords:** ** **viscous-elastic layer, impulse, Fourier series

*International Journal of Physics*, 2013 1 (5),
pp 101-105.

DOI: 10.12691/ijp-1-5-1

Received January 29, 2013; Revised August 23, 2013; Accepted August 26, 2013

**Copyright:**© 2013 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Ilmenkov, S. L., et al. "Investigation of Characteristics of the Reflection of Stationary and Pulse Sound from Visco-Elastic Cylindrical Layers."
*International Journal of Physics*1.5 (2013): 101-105.

- Ilmenkov, S. L. , Kleshchev, A. A. , Kuznetsova, E. I. , Legusha, F. F. , Chizhov, Y. V. , & Chizhov, G. V. (2013). Investigation of Characteristics of the Reflection of Stationary and Pulse Sound from Visco-Elastic Cylindrical Layers.
*International Journal of Physics*,*1*(5), 101-105.

- Ilmenkov, S. L., A. A. Kleshchev, E. I. Kuznetsova, F. F. Legusha, Y. V. Chizhov, and G. V. Chizhov. "Investigation of Characteristics of the Reflection of Stationary and Pulse Sound from Visco-Elastic Cylindrical Layers."
*International Journal of Physics*1, no. 5 (2013): 101-105.

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### 1. Introduction

The problems of scattering of stationary and non-stationary sound signals discussed in various publications ^{[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]}. In these papers the ideal or elastic bodies of infinite circular cylinders** **form are substantially examined. In the present work the scattering of stationary and impulse sound signals by the viscous-elastic cylindrical layers are studied.

**Figure 1.**The system of the viscous-elastic and elastic cylindrical layers irradiated by plane sound wave in liquid

The first part of the article substantiates the scattering of the stationary and non-stationary (impulse) sound signals by the viscous-elastic cylindrical layers. Firstly we will consider a harmonic uninterrupted sound signal of frequency ω, which irradiates a system of the elastic and viscous-elastic cylindrical layers (Figure 1). The wave vector of this wave is orthogonal as regards the axis z of system of the cylindrical layers (the plane task, Figure 1).

The system consists of four layers: external thin viscous-elastic layer (5), the viscous-elastic more thick layer (4), elastic layer (3), the inside viscous-elastic layer (2). There is vacuum on the inside region of the system (region 1), and water - on the outside region (region 6). All layers are of various characteristics of material: the density, the elastic modules and the loss coefficients. The wave numbers of flexural waves and the corresponding arguments of the Bessel’s and the Naman’s cylindrical functions by the loss in the viscous-elastic layers are complex.

The following designations of the sound pressures, scalar and vector displacement potentials one can find ^{[2, 3, 4, 5, 6, 7, 11, 12, 13]}:

1. on the inside region of the layers (region 1) - is the vacuum, the wave process is absent;

2. the viscous-elastic layer 2 is are described by scalar potential and vector potential ;

3. the elastic layer 3 - scalar potential and vector potential;

4. the viscous-elastic layer 4 - and ;

5. the viscous-elastic layer 5 - and ;

6. outside region 6 is are described by irradiate sound pressures and scattering sound pressures .

For plane task all vector potentials have only one nonzero component ,named as .

All potentials and sound pressures are also expanded by independent solutions of the Helmholtz equation in circular cylindrical coordinates system [1,2,3,4,5,6,7, 10,11,12,13]^{[1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13]}:

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

where and is a longitudinal and flexural wave numbers in elastic, viscous-elastic layers and outside liquid region correspondingly; for , for ; are the unknown coefficients of expanding, which are calculated basing at the boundary conditions ^{[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]}:

1. a strains on the boundary (vacuum - viscous-elastic layer 2 ,is equal to zero:

(11) |

(12) |

2. сontinuity of displacements and strains at external boundary of the viscous-elastic layer 2 and at boundary of the elastic layer 3 (

(13) |

(14) |

(15) |

(16) |

3. сontinuity of displacements and strains at external boundary of the elastic layer 3 and at boundary of the viscous-elastic layer 4 (

(17) |

(18) |

(19) |

(20) |

4. continuity of displacements and strains at external boundary of the viscous-elastic layer 4 and at boundary of the viscous-elastic layer 5 (

(21) |

(22) |

(23) |

(24) |

5. tangent strains are equal to zero, a normal component of a displacement vector is continuous, a normal strains is equal to total scattering pressure in liquid at an external boundary of the viscous-elastic layer 5 and liquid (:

(25) |

(26) |

(27) |

where Lame constants of the viscous-elastic layers; Lame constants of the elastic layer; **_{}_{} the solidity of the liquid environment; the circular frequency.

If we substitute (1)(10) in the boundary conditions (27), and the trigonometrical functions cos*mφ* and sin*mφ* are opthogonal, we’ll get the algebraical system of 17 order to define the unknown coefficients of expanding of the fixed index. We are interested in the coefficients *A*_{m}* *(10). Based in (10) and by using the asymptotic of Hankel function of the first kind , we get the far sound field characteristic of the Fraunhofer region (the angular scattering characteristic) ^{[9, 10, 11]}:

(28) |

At Figure 2 are shown the modulus of the angular scattering characteristics for the different wave sizes . Figure 2 (a): Figure 2 (b): Figure 2 (c):

**Figure 2.**The modulus of the angular scattering characteristics of cylindrical layers for the different wave sizes

The angular scattering characteristics in frequency range 2000 Hz (by step Hz) and in angle range (by step are represented for the calculation of impulse signals scattering charact= 200 Hz, = 2000 Hz). Period number of the filling frequency is equal to 30.

**Figure 3.**The irradiate impulse (a); the modulus of his spectrum (b)

The irradiate impulseand the modulus of his spectrum it connect ^{[14]}:

(29) |

The scattering impulseare represented ^{[14]}:

(30) |

where- the spectral characteristic of the scattering impulse.

The time characteristic of the scattering impulse it normalize: the dependence from distance to observation point is absent (for plane task the dependence are represented: .

At Figures 3 – 9 the time and spectral characteristics of irradiate and scattering impulses for the cylindrical viscous-elastic and elastic layers are shown. The scattering impulses are calculated by step in angle range: (inverse direction) - . The scattering maximum is for.

**Figure 4.**The normalized scattering impulse (inverse direction) , (a), the normalized modulus spectrum (b)

**Figure 5.**The normalized scattering impulse , (а); the normalized modulus spectrum (b)

**Figure 6.**The normalized scattering impulse , (а); the normalized modulus spectrum (b)

**Figure 7.**The normalized scattering impulse , (а); the normalized modulus spectrum (b)

**Figure 8.**The normalized scattering impulse , (а); the normalized modulus spectrum (b)

**Figure 9.**The normalized scattering impulse , (а); the normalized modulus spectrum (b)

The interaction of two systems of cylindrical layers (second parallel by first) is defined by the method for elliptic cylinders ^{[15]}.

The results presented in the article are received in the conducting of scientific research in the fra me work of State contract P 242 from 21 April 2010 FPP (“Scientific and scientific - pedagogical personnel of innovative Russia for the years 2009-2013”).

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