exrcise 11

problem 6.12

背景

The central equation of wave motion is

$$\frac{\partial^2y}{\partial{t}^2}=c^2\frac{\partial^2y}{\partial{x}^2}$$ We treat $x$ and $t$ as discrete variables with $i$ and $n$,That is $x=i\Delta{x}$ and $t=n\Delta{t}$.We can obtain: $$y(i,n+1)=2[1-r^2]y(i,n)-y(i,n-1)$$ $$+r^2[y(i+1,n)+y(i-1,n)]$$ where $r=c\Delta{t}/\Delta{x}$.
Tiypically,we might know the string configuration at $t=0$,which we refer to as $y_0(x)$.
The another problem is to how to treat the ends of the string.We consider fixed ends($y(0,n)=y(m,n)=0$),and then restricting the updating of $y(i,n)$ to the interior of the string(from $i=1$ to $i=m-1$).
We should choose $r=1$,if we were to cut $\Delta{x}$ in half but keep $\Delta{t}$ fixed so that $r=2$,the algorithm becomes unstable.
This signal can be decomposed into Fourier components.That is ,we can write the signal as a sum of sines and cosines with different frequencies and amplitudes.

正文

1.for $y_0(x)=exp[-k(x-x_0)^2]$

The string was excited with a Gaussian initial pluck 5 percent from its center.
2.compare different displacement

observe different point's displacement.
3.the initial string displacement consists of two straight lines.


the excitation point is at 0.3.We observe 0.3 and 0.7 point's displacement.

代码

code