第十一次作业

homework

6.6 An important feature Of a linear equation is that the sum of two solutions is also a solution. 0ne consequence Of this is that two wavepackets will travel independently Of cach other. An especially clear way to demonstrate this is to set up a string with an initial profile such that there are two Gaussian wavepackets located at different places at the string. These wavepackets (or components of them) may then propagate toward each other and collide. Show that the wavepackets are unaffected by these collisions. That is， show that two such wavepackets pass through each other without changing shape or speed.

摘要

本次实验主要是验证波的叠加态原理。介质中同时存在几列波时，每列波能保持各自的传播规律而不互相干扰。在波的重叠区域里各点的振动的物理量等于各列波在该点引起的物理量的矢量和。在两列波重叠的区域里，任何一个质点同时参与两个振动，其振动位移等于这两列波分别引起的位移的矢量和，当两列波振动方向在同一直线上时，这两个位移的矢量和在选定正方向后可简化为代数和。只有当波的强度较小，波动方程变现为线性方程时，波的叠加原理才普遍成立。

原理

波动方程
$\frac{∂^2y}{∂t^2}=c^2\frac{∂^2y}{∂x^2}$
$x=i*Δx$ $t=n*Δt$
∴ $y=y(i*Δx,n*Δt)$
∴$\frac{y(i,n+1)+y(i,n-1)-2y(i,n)}{Δt^2}=c^2\frac{y(i+1,n)+y(i-1,n)-2y(i,n)}{Δx^2}$
$y(i,n+1)=2(1−i^2)y(i,n)−y(i,n−1)+r^2y(i+1,n)+r^2y(i−1,n)$
程序代码
可以得到如图所示的结果

由图像可知波包之间相遇时只是对点的位移及相位产生叠加，分离时又会按照原来的情况运动。