Application of Y. Takahasi's Method to the Integration of the Wave Fquations

[摘要] Recently, Y. Takahasi has solved various complicated problems of heat conduction and diffusion by an excellent method of graphical integration.In the present paper, I extended his method to the integration of the wave equations and obtained the following formulae corresponding to Y. Takahasi's.If τ be sufficiently small and τ3 be negligible, the value of a function u (x, t) at the time t+τ, which satisfies ∂2u/∂t2=V2•∂2u/∂x2, may be expressed in the formThe first term of the right-band side expresses the mean value of u(x, t) at the time t over the interval (x-√3Vτ, x+√3Vτ) of x, and the second term is the effect of inertia, which does not appear in the case of heat conduction. The inertia term may be transformed into the following form;The corresponding formulae for two or three dimensioual problems may be obtained, in the same way, as follows;I introduced, also, modified formulæ which are more convenient for numerical calculations without the help If graphs. If a function u (x1, x2, … xn, t) in a n-dimensional space satisfies a wave equation where δji=1 when j=i, and 0 when j≠i.If we assume k to be eqnal to √3 and replace the integrals which appear in the right-hand side approximately by Simpson's 1/3 rule of intogration usiug the values of u at xi-Viτ, xi, xi+ViΤ, the process of calculation will be simplified.A simple example is shown: A string is stretched with a constant tension. One of its ends is fixed and the other is forced to move in a straight line perpendicular to the string with a small amplitude signified by f(t). When f(t)=0 for t<0, and sin2πt/T for t_??_0, the vibration of string is calculated.The errors of the results obtained by the above method are found to be always less than 0.001.The cases for heterogeneous medium and telegraphist's sequations may be also treated in the same manner, but with a little additional trouble.

[学科分类] 大气科学