Abstract：D’Alembert method usually is applied in solving the initial problem of an infinite-long string free vibration.Based on the idea of this method, D’Alembert analytical solution to initial-boundary value problems of wave equation with Neumann boundary was studied.For initial-boundary value problems of wave equation within a finite long time interval, the method of variable separation was usually adopted to solve them.Two kinds of the continuation methods were used to solve these problems.One kind was that through successively extending time t, the time piecewise solution to an initial-boundary value problem of wave equation with Neumann boundary was derived within a finite long time interval.Another was that through extending definition fields of the initial displacement and initial velocity, D’Alembert type solution to an initial-boundary value problem of wave equation with Neumann boundary was derived within a finite long time interval.
[1] 汪德新. 数学物理方法[M], 北京：科学出版社, 2006.
WANG De-xin. Method of mathematical physics[M]. Beijing：Science Press, 2006.
[2] 王明新. 数学物理方程[M], 北京：清华大学出版社2009.
WANG Ming-xin. Mathematical physics equation[M]. Beijing：Tsinghua University Press, 2009.
[3] Lea Sirota, Yoram Halevi. Extend D’Alembert solution of finite length second order flexible structures with damped. boundaries[J], Mechanical Systems and Signal Processing, 2013, 39(1-2):47-58.
[4] A.P.S.Selvadurai. Partial Differential Equations in Mechanics1[M], Springer,Germany, 2000.
[5] Tugce Akkaya, WimT.van Horssen. On the Transverse Vibrations of Strings and Beams on Semi-Infinite Domains[J], Procedia IUTAM, 2016, 19:266-273.
[6] Gisèle Ruiz Goldstein, Jerome A.Goldstein. Structure of solutions to Linear Evolution Equation:Extensions of d’Alembert’s Formula[J], Mathematical Analysis and Application, 1996, 201(2):461-477.
[7] Marcelo H.Kobayashi, Romain Genest. On an extension of d’Alembert solution to initial-boundary value prolems in multi-layered, multi-material domains[J], Wave Motion, 2014,51(5):768-784.
[8] Nick V. Gaiko, WimT.van Horssen. On wave reflections and energetics for a semi-infinite traveling string with a nonclassical boundary support[J], Journal of Sound and Vibration, 2016, 370:336-350.
[9] Cresus F.L. Godinho, J.Weberszpil, J.A.Helayël-Neto. Extending the D’alembert solution to space-time Modified Riemann-Liouville fractional wave equations[J], Chaos, Solitons&Fractals 2012,45(6):765-771.
[10] CARRELLA A, BRENNAN M J, WATERS T P, et al. On for Coupled Wave Initial Value Problems[J], Computers Math Applic, 1998, 35(9):1-15.
[11] 张继业，曾京，舒忠周. 杆的纵向冲击振动[J]. 振动与冲击, 1999, 18(3):56-61.
ZHANG Ji-ye, ZENG Jing, SHU Zhong-zhou. Longitudinal vibration of prismatic bar during impact[J]. Journal of Vibration and Shock ,1999, 18(3):56-61.
[12] 靖静，喻九阳，郑小涛等. 机械振打除灰装置的纵向冲击特性[J].振动与冲击, 2014, 33(5):125-131.
JING Jing, YU Jiu-yang, ZHENG Xiao-tao, et al. Longitudinal impact behaviours of mechanical rapping ash removal devices[J]. Journal of Vibration and Shock ,2014, 33(5):125-131.
[13] 王华杰. 达朗贝尔法的推广及其在简支梁自由振动中的应用[J].力学与实践, 1998,20(4), 70.
WANG Hua-jie, The extension of the D’alembert method and its application in the free vibration of the beam[J]. Mechanics in Engineering ,1998, 20(4):70.
[14] 姚端正，李中辅等. 用达朗贝尔公式求解二、三维波动方程的初值问题[J].大学物理, 1997 16(4):18-21.
YAO Duan-zheng, LI Zhong-fu, et al. Initial-value problems for two- and three-Dimentional wave equations solved by D’alembert formula[J]. College Physics ,1997, 16(4):18-21.
