第十四次作业

homework14
2013302290059

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Abstract

An important feature of an linear equation is that the sum of two solution is also a solution.One consequence is that the two wavepackets will travel independently of each other.An especially clear way to demonstrate this is to set up a string with initial profile such that there two Gaussian wavepackets,located at different places on the string.These wavepackets may then progate toward each other and collide.We showed that two such wavepacket pass through each other without changing shape or speed.

Background

The central equation of wave motion is:

$$\frac{\partial^2y}{\partial t^2}=c^2\frac{\partial^2y}{\partial x^2}$$
As the following picture,if we assume that string displacement is small,so that angles $\theta_i$ are also small,we can write the equation of motion as:
$$\frac{d^2y_i}{dt^2}\approx(\frac{T}{\rho})\frac{\partial^2y}{\partial x^2}$$
Set the parameter combination $r=c\Delta t/\Delta x$,loop through the interior points i=1through i=M-1.Update according to:
$$y(i,n+1)=2(1-r^2)y(i,n)-y(i,n-1)+r^2[y(i+1,n)+y(i-1,n)]$$
the ends at i=0 and i=M are fixed,so$y(0,n)=y(M,n)=0$. for all time steps n.Here we have assumed a simple ''Gaussian pluck''of the string.That is,we have taken the initial string profile to be:
$$y_0(x)=exp[-k(x-x_0)^2]$$
where the displacement is centered at $x_0$ and k is a factor that determins the width of the Gaussian envelope.

Text

code
We use $x_0=0.3 m$ and k=1000$m^{-2}$ ,$c2=c1/2,r=1$for the left part of the string and $r=1/2$ on the right side of the system.we get the fruit as follows:



wo see that two such wavepacket pass through each other without changing shape or speed.

Reference

Thanks for program of the Wentao Liu（2013301020085)