**Appendix**

The coefficients in Eq. (11) are listed as follows:

$$\begin{aligned} g\_0 &= n^2 \beta^2 \nu\_1 \left( 1 + \beta^2 \right) \left[ -n^2 + 2n^3 - n^4 \right] \\ &+ \Omega^2 n^2 \left[ \left[ \beta^4 \nu\_1 + 2\beta^2 \nu\_2 + \nu\_1 \right] + n^2 \left[ -2\beta^4 \nu 1 - \beta^2 \left( 4\nu\_2 - 3\nu\_1 \right) + \nu\_1 \right] \right] \\ &+ n^3 \beta^2 \left[ \beta^2 \nu\_1 + \left( 2\nu\_2 - \nu\_1 \right) \right] \\ &+ \Omega^4 \left\{ \left[ 1 + \beta^2 \right] + n^2 \left[ \beta^2 \left( \nu\_3 - 2\nu\_4 \right) + \left( \nu\_1 - 2\nu\_2 \right) \right] + n^4 \beta^2 \right\} - \Omega^6 \end{aligned} \tag{A.1}$$

Dispersion Relations and Modal Patterns of Wave in a Cylindrical Shell 101

2

  (A.7)

(A.8)

(A.9)

 

 

The coefficients in Eq. (12) are given in the following.

5 4 4 26 08 06 82 2 3 2

2 ( 4 36 ) 27( ),

*Feng Chia University, Electroacoustic Graduate Program, Taiwan, R.O.C.* 

Exposition, IMECE2002-32683, pp. 91-96, , ISBN 0791836436.

Society of America, Vol.119, No.6, pp. 3553-3557, ISSN 0001-4966.

 

*g g gg gg gg gg*

6 46 8 7 5 4 5

/ , 4

Yu-Cheng Liu, Yun-Fan Hwang and Jin-Huang Huang

 *g g*

 

*gg g*

1899, ISSN 1729-8806.

311-318, ISSN 0098-2202.

where

and

**Author details** 

**5. References** 

0022-460X.

1563962934.

*A A*

*A A*

2 / (3 ) / (3 2 ), ( / 4) 2 / 3 , ( / 2) 4 / 3 , 4 8,

1 6 8 2 4 8 3 2 8 4 4 26 08 2 2

[1] Forsberg K. (1963). Influence of boundary conditions on the modal characteristics of thin cylindrical shells. *Journal of the Acoustical Society of America*, Vol.119, No.6 , pp. 1898-

[2] Fuller C. R. (1981). The effects of wall discontinuities on the propagation of flexural waves in cylindrical shells. Journal of Sound and Vibration, Vol.75, pp. 207-228, ISSN

[3] Hwang Y. F. (2002). Structural-acoustic wave transmission and reflection in a hose- pipe system, Proceedings of 2002 ASME International Mechanical Engineering Congress and

[4] Karczub D. G. (2006). Expressions for direct evaluation of wave number in cylindrical shell vibration studies using the Flügge equations of motion. Journal of the Acoustical

[5] Lesmez M. W., Wiggert D. C., and Hatfield F. J. (1990). Modal analysis of vibrations in liquid-filled piping systems. Journal of Fluids Engineering ASME, Vol. 112, No. 3, pp.

[6] Leissa A. W. (1997). Vibration of Shells, Acoustical Society of America, ISBN

*g g gg gg g gg gg*

/ , /, /, 3 12 ,

, / (4 )

/ 2, / 4

/ 2, / 2

3 56 4 56

1 12 2 1

1 3 1/3 13 13 2 1 4 87 7 8 2 1 2 2 3 31 2 4 1 63 5 13 6 4 12

 

$$\begin{aligned} g\_2 &= n^2 \beta^2 \nu\_1 \left[ \left[ 3\beta^4 \nu\_1 + 7\beta^2 \nu\_1 + 2\left(\nu - 2\right) \right] - 2n^2 \left[ 3\beta^4 \nu\_1 + \beta^2 \left( 3\nu\_1 + 2\nu\_2 \right) + \left(\nu - 4\right) \right] \right] \\ &+ n^3 \left[ 3\beta^4 \nu\_1 - \beta^2 \left( \nu\_1 - 4\nu\_2 \right) - 4 \right] \Bigg) \\ &+ \Omega^2 \left[ \left[ 3\beta^4 \nu\_1 + \beta^2 \left( 2\nu - 3 \right) + \left( \nu^2 - \nu\_1 + 2\nu\_2 \right) \right] \right] \\ &+ n^2 \beta^2 \left[ 3\beta^2 \nu\_1 \left( \nu\_3 - 2\nu\_4 \right) + 2\left( 3\nu\_1 - 4\nu\_2 \right) + 2\nu\_1 \right] \\ &+ n^4 \left[ \beta^2 \nu\_1 \left( -7\nu\_3 + 8\nu\_4 \right) + 3\left( 2\nu\_2 - \nu\_1 \right) \right] \Bigg) \\ &+ \Omega^4 \left\{ \left[ -3\beta^2 \nu\_1 + \left( \nu\_1 - 2\nu\_2 \right) \right] + 2n^2 \beta^2 \right\} \end{aligned} \tag{A.2}$$

$$\begin{split} g\_{4} &= \left| \nu\_{1} \right| \left[ -3\beta^{4} + \beta^{2} \left( -4 + 3\nu^{2} \right) + 2\nu\_{1}\nu\_{3} \right] + 3\nu^{2}\beta^{2} \left[ -\beta^{2} \left( 2\nu\_{1} \left( -2 + \nu^{2} \right) + \nu\_{2} \right) + 2\nu\_{1} \right] \\ &+ n^{4}\beta^{2}\nu\_{1} \left[ \beta^{4}\nu^{2} + 6\beta^{2}\nu\_{2} - 6 \right] \\ &+ \Omega^{2}\beta^{2} \left[ \left( 11\nu\_{1} - 4\nu\_{2} \right) + \nu\_{1} \right] + n^{2} \left[ \beta^{2} \left( 1 + 7\nu\_{1} + \nu\_{1}^{2} \right) + 3\left( 2\nu\_{2} - \nu\_{1} \right) \right] \\ &+ \Omega^{4}\beta^{2} \end{split} \tag{A.3}$$

$$\begin{aligned} g\_6 &= \beta^2 \nu\_1 \left[ \left[ 2\nu \left( 1 + 3\beta^2 \right) \right] + n^2 \left[ 9\beta^4 \nu\_1 + \beta^2 \left( 8\nu\_2 - 5\nu\_1 \right) - 4 \right] \right] \\ &+ \Omega \beta^{22} \left\langle \nu\_1 \left( -1 + 5\beta^2 \right) + 2\nu\_2 \left( -1 + \beta^2 \right) \right\rangle \end{aligned} \tag{A.4}$$

$$\mathbf{g}\_8 = \left\{ \beta^2 \nu\_1 \left[ \mathbf{\hat{3}} \beta^4 - \mathbf{2} \beta^2 - \mathbf{1} \right] \right\} \tag{A.5}$$

where

$$\begin{aligned} \nu\_1 &= \frac{-1+\nu}{2}, & \nu\_2 &= \frac{-2+\nu}{2} \\ \nu\_3 &= \frac{1+\nu}{2}, & \nu\_4 &= \frac{2+\nu}{2} \end{aligned} \tag{A.6}$$

The coefficients in Eq. (12) are given in the following.

$$\begin{aligned} A\_1 &= \left(\sqrt{\Lambda\_1 + \Lambda\_2}\right) / 2, & A\_2 &= -\Lambda\_1 / 4\\ A\_3 &= \left(\sqrt{\Lambda\_5 - \Lambda\_6}\right) / 2, & A\_4 &= \left(\sqrt{\Lambda\_5 + \Lambda\_6}\right) / 2 \end{aligned} \tag{A.7}$$

where

100 Wave Processes in Classical and New Solids

0 1

*n*

34 2

 

> 

 

 

4 2 42 2 1 2

 

> 

 

 

 

22 2

4 2

4 2

 

 

*n*

*n*

*n*

*n*

where

 32 2

 

22 4 2 24 2

 

1 22

22 2 2 3 4

1 2

*g n n nn*

 

 

 

1 12 24 2 2

3 44

*g n n*

 

1 21

3 23 2

 

1 3 4 21 4 2 2 2 11 2

3 22

*g n*

 

 

 

 

7 8 32

 

3 2 23 4 2

13 4 1 2 1

 

*n*

 

2 2 2 2 2

6 6

*g n*

     

 

*n*

 

42 2 22 2 2 4 1 1 3 1 2 1

3 43 2 3 2 2 2

 

 

 

 

 

 

 2 42 8 1 *g* 

1 2 , 2 2

 

 

1 2 , 2 2

213 9 8 5 4

11 4 1 7 32

 

> 

15 2 1

1 2

3 4

2 2 24 2 6 1 1 21 22 2 2 1 2

 

> 

 

 

1 21 11 21

1 1 2

*n n*

 

2

 

 

22 4 2 24 2 2 111 1 12

 

1 21 2 11

 

 

 

 

> 

321 (A.5)

 

> 

(A.1)

(A.2)

(A.3)

(A.4)

(A.6)

 

 

2 21 4 3

 

 

3 7 2 2 23 3 2 4

 

 

4 2 22 42 6 34 12

*n n*

$$\begin{aligned} \Lambda\_1 &= 2^{13}\lambda\_4 / (\Im g\_8 \lambda\_7^{1/3}) + \lambda\_7^{13} / (\Im g\_8 2^{13}), & \Lambda\_2 &= (\lambda\_1^2 / 4) - \{2\lambda\_2 / 3\}, \\ \Lambda\_3 &= (\lambda\_1^2 / 2) - \{4\lambda\_2 / 3\}, & \Lambda\_4 &= -\lambda\_1^3 + 4\lambda\_6 - 8\lambda\_3, \\ \Lambda\_5 &= -\Lambda\_1 + \Lambda\_3, & \Lambda\_6 &= \Lambda\_4 / \{4\sqrt{\Lambda\_1 + \Lambda\_2}\} \end{aligned} \tag{A.8}$$

and

$$\begin{aligned} \lambda\_1 &= \mathsf{g}\_6 \wedge \mathsf{g}\_8, & \lambda\_2 &= \mathsf{g}\_4 \wedge \mathsf{g}\_{8\prime}, & \lambda\_3 &= \mathsf{g}\_2 \wedge \mathsf{g}\_{8\prime}, & \lambda\_4 &= \mathsf{g}\_4^2 - 3\mathsf{g}\_2\mathsf{g}\_6 + 12\mathsf{g}\_0\mathsf{g}\_{8\prime}, \\ \lambda\_5 &= 2\mathsf{g}\_4(\mathsf{g}\_4 - 4\mathsf{g}\_2\mathsf{g}\_6 - 36\mathsf{g}\_0\mathsf{g}\_8) + 27(\mathsf{g}\_0\mathsf{g}\_6^2 + \mathsf{g}\_8\mathsf{g}\_2^2), & \text{(A.9)} \\ \lambda\_6 &= \mathsf{g}\_4\mathsf{g}\_6 \wedge \mathsf{g}\_8^2, & \lambda\_7 &= \mathsf{\lambda}\_5 + \sqrt{-4\mathsf{\lambda}\_4^3 + \mathsf{\lambda}\_5^2} \end{aligned} \tag{A.9}$$

## **Author details**

Yu-Cheng Liu, Yun-Fan Hwang and Jin-Huang Huang *Feng Chia University, Electroacoustic Graduate Program, Taiwan, R.O.C.* 

#### **5. References**

	- [7] Skelton E. A. & James J. H. (1997). Theoretical Acoustics of Underwater Structures, Imperial College Press, ISBN 1860940854.

**Section 2** 

**Wave Features in Classical Media and Structures** 

[8] Liu Y. C., Hwang Y. F., and Huang J. H. (2009). Modes of wave propagation and dispersion relations in a cylindrical shell. ASME Journal of Vibration and Acoustic, Vol. 131, No. 4, 041011-1~9, ISSN 1048-9002.

**Wave Features in Classical Media and Structures** 

102 Wave Processes in Classical and New Solids

Imperial College Press, ISBN 1860940854.

131, No. 4, 041011-1~9, ISSN 1048-9002.

[7] Skelton E. A. & James J. H. (1997). Theoretical Acoustics of Underwater Structures,

[8] Liu Y. C., Hwang Y. F., and Huang J. H. (2009). Modes of wave propagation and dispersion relations in a cylindrical shell. ASME Journal of Vibration and Acoustic, Vol.

**Chapter 0**

**Chapter 5**

**Spatio-Temporal Feature in Two-Wave**

It is well known that the wave propagation depends mainly on its velocity and frequency in one direction for a single wave[1-3]. There are many literatures devoted researches of single wave propagation such as solitary wave, periodic wave, chirped wave, rational wave etc [4-6]. However, what can be happen when two and even more waves with different features propagate together along different directions? In the past decades, many methods have been proposed for seeking two waves and multi-wave solutions to nonlinear models in modern physics. Recently, some effective and straight methods have been proposed such as homoclinic test approach(HTA)[7-8], extended homoclinic test approach(EHTA)[9-10] and three wave method [11-12]. These methods were applied to many nonlinear models. Several exact waves with different properties have been found out, such as periodic solitary wave, breather solitary wave, breather homoclinic wave, breather heteroclinic wave, cross kink wave, kinky kink wave, periodic kink wave, two-solitary wave, doubly periodic wave, doubly breather solitary wave, and so on. Because of interaction between waves with different features in propagation process of two-wave or multi-wave, some new phenomena have been discovered and numerically simulated, for example, resonance and non resonance, fission and fusion, bifurcation and deflexion etc. Furthermore, similar to the bifurcation theory of differential dynamical system, constant equilibrium solution of nonlinear evolution equation and propagation velocity of a wave as parameters are introduced to original equation, and then by using the small perturbation of parameter at a special value, two-wave or multi-wave propagation occurs new spatiotemporal change such as bifurcation of breather multi-soliton,

This chapter mainly focus on explanation of different test methods and comprehensive applications to two-wave or multi-wave propagation. New methods will be described such as HTA, EHTA, Three-wave method and parameter small perturbation method. The spatiotemporal variety in exact two-wave and multi-wave propagation will be investigated and numerically simulated. In this chapter, some important models such as shallow water wave propagation models under the transverse long-wave disturbance Potential

and reproduction in any medium, provided the original work is properly cited.

©2012 Dai et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**and Multi-Wave Propagations**

Zhengde Dai, Jun Liu, Gui Mu, Murong Jiang

http://dx.doi.org/10.5772/46090

**1. Introduction**

Additional information is available at the end of the chapter

periodic bifurcation and soliton degeneracy and so on.

cited.
