摘要
二十世纪六十年代,自然科学的许多科学分支几乎不约而同地出现了非线性问题的研究热潮,诸方面的研究汇成了非线性的洪流,孤子、湍流、混沌、分形及复杂系统等新的物理现象被揭示,大大扩展了人们的视野,并导致了自然科学认识论和发展观的一场大革命。非线性科学已成为近代科学发展的一个重要标志,它是自然科学各科学分支共同关心的真正的基础性研究。非线性科学涉及到自然界诸多复杂现象,具有广阔应用前景。特别是非线性动力学和非线性波动的研究对于解决物理学、化学、生物学和地球物理学中遇到的复杂现象和问题有着极其重要的意义。
非线性科学发展中一个重要成就就是孤立子理论的建立。在许多非线性物理领域,已经发现一大批非线性演化方程具有孤立子解。这些方程的共同特征是具有无穷个守恒律、可用散射反演法解析求解、存在B(?)cklund变换、完全可积分等。孤立子典型的特征是在其传播过程中伴随有能量集聚,且孤立子间相互作用时表现出犹如粒子弹性碰撞一样的行为。这些特性已在流体力学、等离子体、光纤通讯等技术领域获得广泛应用。
固体力学在线性波的研究方面曾取得过辉煌的成就,为推动物理学中波动理论的发展做出过巨大贡献。近年来固体结构中非线性波的研究已开始受到关注。本文在综述了其它非线性物理领域孤立子理论的研究基础上,以弹性细杆波导为对象,考虑了固体结构中常出现的非线性源及粘性耗散效应、几何弥散性质等,研究了固体中几类非线性波的传播问题,取得了以下一些主要结果:
1.利用Hamilton变分原理,导出了计及有限变形和横向剪切及横向惯
In 1960s the research upsurge of the nonlinear problems appeared simultaneously in many science branches in natural science. The research in many aspects formed nonlinear onrush. Many new physical phenomena such as soliton, turbulence, chaos, fractal and complex system etc. were discovered, which greatly expanded the people's view, and leaded to a great revolution in epistemology and development viewpoint in natural science. The nonlinear science has been an important symbol in the development of modern science. It is the actual fundamental study jointly concerned by various science branches in natural science. The nonlinear science involves lots of complex phenomena in nature, and has broad application prospect. Especially the researches in nonlinear dynamics and nonlinear wave-motion have extraordinary significance in solving problems and complicated phenomena encountered in physics, chemistry, biology and geophysics.
The establishment of soliton theory is an important achievement in the development of nonlinear science. In many nonlinear physics fields, it has been found that large quantities of nonlinear evolution equations had soliton solutions. The common characteristic of these equations are that they have infinite conservation rules, and can be solved by inverse scattering method, and have Backlund transform, and can be completely integrated et al. The typical properties of soliton are that its propagation accompanies energy centralization and the interacting among solitons shows behaviors similar to
引文
[1] Zabusky N. J., Kruskal M. D., Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 1965, 15, 240-243
[2] Russell J. S., "Report on waves", Report ot the 14th meeting of British Association for the Advancement of Science, John Murray, London, 1844, 311-390
[3] Korteweg D. J., Vries G. de, On the change of form long waves advancing in a rectangular canal and on a new type of long stationary waves, Phil. Mag., 1895, 39, 422-443
[4] Fermi A., Pasta J., Ulam S., Lectures in Apllied Mathematics, 1974, 15, 143-196
[5] Perdng J. K., Skyrme T. H. R., A Model Unified Theory, Nuel. Phys, 1962, 31, 550-555
[6] Gardner C. S., Greene J. M., Kruskal M. D. etc., Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 1967, 19, 1095-1097
[7] Scott A. C., Chu F. Y. F., Mclaughlin D. W., The soliton -A new concept in applied science, proc. IEEE, 1973, 61, 1443-1483
[8] 刘式适,刘式达,谭本馗,非线性大气动力学,北京,国防工业出版社,1996
[9] Gardner C. S., Greene P. A., Kruskal R. M. etc., Korteweg de Vires and generalizations Ⅵ. Methods for exact solution, Commun. Pure Appl. Math., 1974, 27, 97-133
[10] Zakharov V. E., Shabat A. B., Exact theory of two-dimensional self-focusing and one-dimensional of waves in nonlinear media, Sov. Phys. JETP, 1972, 34, 62-69
[11] 谷超豪,胡和生,周子翔,孤立子理论中的达布变换,上海,上海科学出版社,1999
[12] Gu C H, Zhou Z X., On Darboux transformations for soliton equations in high dimensional space-time, Lett. Math. Phys, 1994, 32, 1-6
[13] Zhou Z X. On the Darboux transformation for 1+2 dimensional equations. Lett. Math. Phys., 1988, 16, 9-15
[14] Bluman C. W., Cole J. D., Similarity methods for differential equations, Springer-Verlag, 1974
[15] Olver P. J., Applications of Lie groups to differential equations, 2th Ed., 世界图书出版公司, 1999
[16] 田畴,李群及其在微分方程中的应用,北京,科学出版社,2001
[17] 潘祖梁,非线性问题的数学方法及其应用,杭州,浙江大学出版社,1997[18] Wahllquist H. D., Estabrook F. B., Backlund transformations for solutions of the Korteweg-de Veries equations, Physics Review Letter, 1973, 31, 1386-1390
[19] Tien C. Backlund transformation of nonlinear evolution equations, Acta Math. Appl. Sinica, 1986
[20] Rogers C., Shadwick W. K., Backlund transformations and their applications, New York: Academic Press, 1982
[21] Lou S. Y., Huang H. Y., Exact solitary waves in a convecting fluid, J. Phys., A: MathGen, 1991, 24, 587-590
[22] Zhang G. X., Li Z. B., Duan Y. S., Exact solitary solutions of nonlinear wave equations, Sci. in China, 2000, A 44(3), 396-401
[23] 周钰谦,孤立子理论与非性发展方程的精确解,[硕士学位论文],成都,四川师范大学,2004
[24] Zhang Jiefang, New solitary wave solution of the combined KdV and mKdV equation, International Journal of Theoretical Physics, 1997, 37(5), 1541-1546
[25] 闫振亚,张鸿庆,范恩贵,一类非线性演化方程新的显式行波解,物理学报,1999,48(1),1-5
[26] Xia Tiecheng, Zhang Hongqing, New explicit and exact traveling wave solution for class of nonlinear wave equation in mathematical physics, Journal of Jinzhou normal college(Natural Science Edition), 2001, 22(3), 8-10
[27] 石玉仁,非线性物理中若干问题的研究,[硕士学位论文],兰州,西北师范大学,2002
[28] 王霞,吕岿,郑春龙,一类非线性反应扩散方程的双曲函数级数解法,上饶师范学院学报,2001,21(3),p36-40
[29] 王明亮,非线性发展方程与孤立子,兰州,兰州大学出版社,1990
[30] Wang Mingliang, Exact solutions for a compound KdV-Burgers equation, Physics Letters A 213, 1996, 279-287
[31] 王明亮,李志斌,周宇斌,齐次平衡原则及其应用,兰州大学学报(自然科学版),1999,35(3),8-16
[32] 周宇斌 王明亮,物理学中的一个非线性耦合偏微分方程组的精确解,兰州大学学报(自然科学版),1996,32(3),31-36
[33] Wang Mingliang, Zhou Yubing, Li Zhibing, Apllication of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters A 216, 1996, 67-75
[34] Wang Mingliang, Solitary wave solutions for variant Boussinesq equations, Physics??Letters A 199, 1995, 169-172
[35] 范恩贵,张鸿庆,非线性孤子方程的齐次平衡法,物理学报,1998,47(3),353-362
[36] Fan E. G. Two new applications of the homogeneous balance method, Physics Letters A 265, 2000, 353-357
[37] 张辉群,齐次平衡方法的扩展及应用,数学物理学报,2001,A21(3),321-325
[38] Hirota R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solutions, Physics Review Letter 72, 1971, 1192-1194
[39] 刘式适,付遵涛,刘式达,赵强,一类非线性方程的新周期解,物理学报,2002,51(1),10-14
[40] 刘式适,付遵涛,刘式达,赵强,变系数非线性方程的Jacobi椭圆函数展开解,物理学报,2002,vol.51(9),p1923-1926
[41] Fu Zuntao, Liu Shikuo, Liu Shida, Zhao Qiang, The JEFE method and periodic solutions of two kinds of nonlinear wave equations, Communications in Nonlinear Science and Numerical Simulation 8, 2003, 67-75
[42] Fu Zuntao, Liu Shikuo, Liu Shida, Zhao Qiang, New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Physics Letters A 290, 2001, 72-76
[43] 刘式适,付遵涛,刘式达,赵强,Jacobi椭圆函数展开法及其在求解非线性波动方程中的应用,物理学报,2001,50(11),2068-2073
[44] Fu Zuntao, Liu Shikuo, Liu Shida, New transformations and new approach to find exact solutions to nonlinear equations, Physics Letters A, 2002, 299, 507-512(sine-cosine)
[45] 张善卿,李志斌,Jacobi椭圆函数展开法的新应用,物理学报,2003,52(5),1066-1070
[46] 石玉仁,郭鹏,吕克璞等,修正的Jacobi椭圆函数展开法及其应用,物理学报,2004,53(10),3265-3269
[47] Shen Shoufeng, Pan Zu liang, A note on the Jacobi elliptic function expansion method, Physics Letters A, 2003, 308, 143-148
[48] Chen Haitang, Zhang Hongqing, On the Jacobi Elliptic Function Expansion Method, Journal of Mathematical Research & Exposition, 2004, 24(3), 430-436
[49] Parkes E. J., Duffy B. R., Abbott P. C., The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations, Physics Letters A 295, 2002, 280-286
[50] Fan E. G., Hon B. Y. C., Double periodic solutions with Jacobi elliptic functions for??two generalized Hirota-Satsuma coupled KdV systems, Physics Letters A 292, 2002, 335-337
[51] Shen S. F., Pan Z. L., A note on the Jacobi elliptic function expansion method, Physics Letters A 308, 2003, 143-148
[52] 叶彩儿,几个非线性发展方程(组)的精确解与Painleve分析,[硕士学位论文],杭州,浙江大学,2003
[53] 沈守枫,几个非线性发展方程的精确解,[硕士学位论文],杭州,浙江大学,2002
[54] 闫振亚,张鸿庆,范恩贵,一类非线性演化方程新的显式行波解,物理学报,1999,48(1),1-5
[55] Zheng Xuedong, Xia Tiecheng, Zhang Hongqing, New exact traveling wave solutions for compound KdV-Burgers equations in Mathematical Physics, Apllied Mathematics E-Notes, 2 (2002), 45-50
[56] Xia Tiecheng, Zhang Hongqing, New explicit and exact traveling wave solution for class of nonlinear wave equation in mathematical physics, Journal of Jinzhou normal college (Natural Science Edition), 2001, 22(3), 8-10
[57] Xia Tiecheng, Zhang Hongqing, Yan Zhenya, New explicit and exact traveling waves solutions for a class of nonlinear evolution equations, Apllied Mathematics and Mechanics, 2001, 22(7), 788-793
[58] 郑赟,张鸿庆,一类非线性演化方程新的显式行波解,物理学报,2000,49(3),389-391
[59] Yan Zhenya, Zhang Hongqing, Explicit and exact solutions for the generalized reaction Duffing equation, Communications in Nonlinear Science & Numerical Simulation, 1999, 4(3), 224-227
[60] Yao R. X., Li Z. B., New exact solutions for three nonlinear evolution equations, Physics Letters A 297, 2002, 196-204
[61] 钱天虹,刘中飞,韩家骅,一类非线性波方程新的精确解,安徽建筑工业学院学报(自然科学版),2005,13(71),15-17
[62] Fu Zuntao, Liu Shida, Liu Shikuo, New kinds of solutions to Gardner equation, Chaos, Solitons and Fractals, 20 (2004), 301-309
[63] Fu Zuntao, Liu Shida, Liu Shikuo, Solving nonlinear wave equations by elliptic equation, Chaos, Commun. Theor. Phys., 39(2003), 531-536
[64] Liu Shikuo, Fu Zuntao, Liu Shida et. al, A simple method in finding particular solutions of some nonlinear PDE, 2001, 22(3), 326-331
[65] 刘式适,刘式达,物理学中的非线性方程,北京,北京大学出版社,2000[66] 郑赞,非线性偏微分方程求解的变换方法,[硕士学位论文],大连,大连理工大学,2000
[67] 夏铁成,吴方法及其在偏微分方程中的应用,[博士学位论文],大连,大连理工大学,2002
[68] 刘希强,非线性发展方程显式解的研究,,[博士学位论文],北京,中国工程物理研究院北京研究生部,2002
[69] EI-Kalaawy O. H., Exact soliton solutions for some nonlinear partial differential equations, Chaos, Solitons and Fractals 14, 2002, 547-552
[70] Karpman V. I., Soliton Evolution in the Presence of Perturbation, Physics Scripta, 1979, 20, 462-478
[71] Wclaughlin D. W., Scott A. C., Perturbation analysis of fluxion dynamics, Phys. Rev., 1978, A18, 1652-1680
[72] 周凌云,王瑞丽,吴光敏等,非线性物理理论与应用,北京,科学出版社,2000
[73] 陆同兴,非线性物理概论,合肥,中国科学技术出版社,2002
[74] 洪学仁,非线性物理中的孤波研究,[硕士学位论文],兰州,西北师范大学,2003
[75] 杨伯君,赵玉芳,高等数学物理方法,北京,北京邮电大学出版社,2003
[76] 牛实为,冲击波、孤立波与人,北京,科学出版社,1999
[77] 黄景宁,徐济仲,熊吟涛,孤子概念、原理和应用,北京,高等教育出版社,2004
[78] 王振东,孤立波与孤立子,力学与实践,2005,27(5),86-88
[79] 李翊神,汪克林,郭光灿等,非线性科学选讲,合肥,中国科技大学出版社,1994
[80] E. Infeld, Nonlinear Waves, Solitons and Chaos, Cambridge University Press, 1990
[81] Michel Remoissenet, Waves Called Solitons, 2nd Edition, New York: Springer-Verlag Berlin Heidelberg, 1999
[82] [美]阿肯巴赫著,徐植信,洪锦如译,弹性固体中波的传播,上海,同济大学出版社,1992
[83] 郭仲衡,非线性弹性理论,北京,科学出版社,1980
[84] Juri Engelbrecht, Nonlinear Wave Dynamics, Netherlands: Kluwer Academic Publishers, 1997
[85] 匡震邦,非线性连续介质力学基础,西安,西安交通大学出版社,1989
[86] Mumaghan F. D., Finite Deformation of an Elastic Solid, Weley Dover, 1951
[87] 钱祖文,非线性声学,北京,科学出版社,1992,324-387[88] 吴畏,各向同性弹性固体介质中二维非线性纵波方程的有限元解法,[硕士学位论文],吉林,吉林大学,2004
[89] Whitham G. B., Linear and Nonlinear Waves. New York, John Wiley & Sons, 1974, 96-113
[90] Bhatnager P. L., Nonlinear Waves in One-dimensional Dispersive System. Oxford: Clarendon Press, 1979, 61-88
[91] [日]谷内俊弥,西原功修著,徐福元,张家泰,丁启明译,非线性波动,北京,原子能出版社,1981
[92] [英]C.A.库尔森,A.里弗里著,杨善元,杨燕译,波,北京,地震出版社,1988
[93] [苏]M.B.维诺格拉多娃,O.B.鲁登科,A.П.苏哈鲁柯夫著,王珊译,波动 理论,上海,复旦大学出版社,1990
[94] 杨桂通,张善元著,弹性动力学,北京,中国铁道出版社,1988
[95] Hacinliyan I., Saadet E., A higher-order model for transverse waves in a generalized elastic solid, Chaos, Solitons and Fractals 14, 2002, 1127-1135
[96] [美]A.H.奈弗,D.T.穆克著,罗惟德等译,非线性振动,北京,高等教育出版社,1990
[97] Nariboli G. A., Sedov A., Bargers's-Kortewey-De Vries Equation for Viscoelastic Rods and Plates. J. Math. Anal Appl., 1970, 32(3): 661-677
[98] Nariboli G. A., Nonlinear longitudinal dispersive waves in elastic Rods, J. Math. hys. Sci. 4, 1970, 64-73
[99] Ostrovsky L. A., Sutin A. M., Nonlinear elastic waves in rods, Appl. Math. Mech. 41, no. 3, 1977, 543-49
[100] Tanaka K., Motoyama C., The propagation of longitudinal waves in a viscoelastic circular bar with consideration of the laterial inertia, Bull. JSME 17, no. 112, 1974, 1257-50
[101] Clarkson P.A., Leveqne R.I.,etc., Solitary Wave Interactions in Elastic Rods. Studies in Appl. Math., 1986, 2(75): 95-112
[102] Samsonov A.M., Soliton Evolution in a Rod with Variable Cross Section. Soy. Physics-Doklady, 1984, 29(7): 586-587
[103] Samsonov A. M., Sokurinskays E. V., On the Excitation of a Longitudinal Deformation Soliton in a Nonlinear Elastic Rod, Sov. Phys.-Techn. Phys., 1988a, 33(8), 989-991
[104] Samsonov A. M., On Existence of Longitudinal Strain Solitons in a Infinite Non-linearly Elastic Rod. Sov. Pyhs.- Doklady, 1988b, 33: 298-300[105] Samsonov A. M., Sokurinskays E. V., Longitudinal Solitary Wave in inhomogeneous Nonlinearly Elastic Rod. J. Appl. Mathe. Mechanics PMN, 1987, 51(3), 483-488
[106] Dreiden G.V., Ostrovsky A. M., Samsonov A. M., etc., Formation and Propagation of Strain Solitons in Non-linearly Elastic Solids. Sov. Phys. techn. Phys., 1988, 33(10): 1237-1241
[107] Dreiden G. V., Porubov A. V., Samsonov A. M. etc., Experiments in the propagation of longitudinal strain solitons in a nonlinearly elastic rod, Tech. Phys. Lett., 21, 1995, 415-417
[108] Samsonov A. V., Dreiden G. V., Porubov A. V. etc., Longitudinal-strain soliton focusing in a narrowing nonlinearly elastic rod, Phys. Rev. B 57,1998,5778-5787
[109] Semenova I. V., Dreiden G. V., Samsonov A. M. etc., Strain solitary waves in inhomogeneous wave guides, Optics Lasers in Engineering, 2000, 34,159-168
[110] Samsonov A.M., Strain solitons in solid and how to construct them, Chapman & Hall/CRC, 2001
[111] Dai Huihui, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech.127, 1998,193-207
[112] Dai Huihui, Huo Yi, Solitary shock waves and other traveling waves in a general compressible hyperelastic rod, Proc. R. Soc. Lond, A 456,2000, 331-363
[113] Dai Huihui, Huo Yi, Asmptotically approximate model equations for nonlinear dispersive waves in incompressible elastic rods, Acta Mechanica 157, 2002, 97-112
[114] Dai Huihui, Fan Xiaojun, Asmptotically approximate model equations for weakly nonlinear long waves in compressible elastic rods and their comparisons with other simplified model equations, Mathematics and Mechanics of Solids 9, 2004, 61-79
[115] Constanin A., Stauss W., A., Stability of a class of solitary waves in compressible elastic rods, Physics Letters A 270, 2000, 140-148
[116] Konyukhov B. A., Shalashov G. M., Generation of higher harmonics, On third-order effects at propagation of elastic waves solids, J. Appl. Mech. Tech. Phys. 1974, 15, 535-540
[117] Porubov A. V., Velarde M. G., On nonlinear waves in an elastic solid, Mechanics of solid and structure, 2000, 328, 165-170
[118] Porubov A. V., Velarde M. G., Dispersive-dissipative solitons in nonlinear solids, Wave Motion 31,2000,197-207
[119] Porubov A. V., Velarde M. G., Strain kinks in an elastic rod embedded in a viscoelastic medium, Wave Motion 35,2002, 189-204
[120] Porubov A. V., Maugin G. A., Longitudinal strain solitary waves in presence of cubic non-linearity, International Journal of Non-linear Mechanics 40, 2005, 1041-1048[121] Potapov A. I., Nonlinear Deformation Waves in Rods and Plates, Gorky University Press, Gorky, 1985(in Russian)
[122] Ostrovsky L. A., Potapov A. I., Modulated waves: theory and applications, Baltimore and London, The Johns Hopkins University Press, 1999
[123] Wright T. W., Nonlinear waves in rods, Proceedings of IUTAM Symposium on Finite Elasticity, eds D. E. Carlson and R. T. Shield, The Hague: Martinus Nijhoff 1982
[124] Wright T. W., Nonlinear waves in rods results for incompressible elastic materials, Stud. Appl. Math. 72, 1985, 149-160
[125] Wright T. W., Axial plastic waves in a rod, International Journal of Plasticity, 1998, Vol. 14(1-3), 25-42
[126] 朱位秋,弹性杆中的非线性波,固体力学学报,1980,2,247-253
[127] 庄蔚,杨桂通,孤波在非线性弹性杆中的传播,应用力学和数学,1986,7,7-13
[128] Zhang Shanyuan, Zhuang wei, The Strain Solitary Waves in a Nonlinear Elastic Rod. Acta Mechanica Sinica, 1987, 1(3), 62-72
[129] 韩强,郑向锋,非线性弹性杆中的孤波解及其数值分析,华南理工大学学报(自然科学版),2004,32(4),92-96
[130] 段文山,非线性弹性杆中的NLS型孤波,西北师范大学学报,1999,35(4),30-32
[131] Duan Wenshan, Zhao Jingbao, Solitary waves in a quartic nonlinear elastic rod, Chaos, Solitons and Fractals 11, 2000, 1265-1267
[132] 吕克璞,王红艳,非线性弹性杆中的速度孤立子波,西北师范大学学报,1999,35(4),33-36
[133] Zhang Shanyuan, Guo Jiangang, Zhang Nianmei, The Dynamics Behaviors and Wave Properties of Finite Deformation Elastic Rods with Viscous or Geometrical-dispersive Effects. ICNM-Ⅳ, Shanghai, Aug. 2002,728-732
[134] 郭建刚,周丽军,张善元,有限变形粘弹性杆的波动行为研究,太原理工大学学报,2002,33(5),558-560
[135] 郭建刚,有限变形弹性杆中应力波和波动行为及传播特性,[硕士学位论文],太原,太原理工大学,2003
[136] Guo Jiagang, Zhou Lijun, Zhang Shanyuan, The Geometrical Nonlinear Waves in Finite Deformation Elastic Rods, Appl. Math. Mech. 2005, 26(5), 667-674
[137] Liu Zhifang, Zhang Shanyuan, Nonlinear Waves and Periodic Solution in Finite Deformation Elastic rod, Acta Mechanica Solida Sinica, 2006, 19(1)
[138] 张瑞萍,孙家驹,弹性介质中几何和运动非线性问题的孤波解,应用力学学报,??1997,14(1),99-102
[139] Rayleigh J W S, The Theory of Sound, New York: Dover publications, 1945, p157
[140] Love A E H, A Treaties on the Mathematics Theory of Elasticity, New York: Dover publications, 1951, p428
[141] 刘志芳,张善元,非线性弹性杆中的应变孤波,太原理工大学学报,2005,36(6),651-653
[142] 刘志芳,张善元,圆杆波导中的一个非线性波动方程及精确行波解,物理学报,2006,54(2),628-633
[143] 陆万明,罗学富,弹性理论基础,1990,北京,清华大学出版社,165-209
[144] [日]鹫津久一朗著,老亮,郝松林译,弹性和塑性力学中的变分法,北京,科学出版社,1984
[145] Kappus, Zur Elastizitatstheorie endlicher Verschidbungen Z. angew. Math. Mech. 1939, Bd. 19, Nr. 5.344-360
[146] 川本脁万,山田善一等[日],杨广里,刘慧茹译,材料力学,北京,中国铁道出版社,1984,197-199
[147] Boyd J. P., Chen Guanyu, Weakly nonlinear wavepackets in the Korteweg-de Vries equation: the KdV/NLS connection, Mathematics and Computers in Simulation 55, 2001, 317-328
[148] 刘式适,刘式达,特殊函数,第二版,北京,气象出版社,2002,651-674
[149] Taniuti T., Wei C. C., Reductive perturbation mdthod in nonlinear wave propagation, J. Phys. Soc. Japan, 1968, 24, 941-946
[150] Gardner C. S., Morikawa G. K., Courant Inst. Math. Sci.Rept., 1960, 1, NYO-9082