第十三次作业

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一、背景
The central equation of wave motion is

$$\frac{\partial^2 y}{\partial t^2}=c^{2}\frac{\partial^2 y}{\partial x^2}$$
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: $$y\left ( i,n+1 \right )=2\left [ 1-r^{2} \right ]y\left ( i,n \right )-y\left ( i,n-1 \right )+r^{2}\left [ y\left ( i+1,n \right )+y\left ( i-1,n \right ) \right ]$$
where$r=c\Delta t/\Delta x$and then restricting the updating of y(i,n)
Tiypically,we might know the string configuration at t=0,which we refer to as $y_{0}\left ( x \right )$.
The another problem is to how to treat the ends of the string.We consider fixed ends,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.
二、代码

import numpy as np
import matplotlib.pyplot
from pylab import *
from math import *
import mpl_toolkits.mplot3d
k=1000
dx=0.01
c=300.0
dt=dx/c
y=[[0 for i in range(100)]for n in range(3000)]
x=np.linspace(0,1,100)
t=np.linspace(0,0.1,3000)
r=c*dt/dx
for i in range(31):
    y[0][i]=0.8*x[i]
    y[1][i]=0.8*x[i]
for i in range(30,100):
    y[0][i]=-2.4/70*x[i]+24/7
    y[1][i]=-2.4/70*x[i]+24/7
for n in range(3000):
    for i in range(1,99):
        y[n][i]=2*(1-r**2)*y[n-1][i]-y[n-2][i]+r**2*(y[n-1][i+1]+y[n-1][i-1])
y4=[]
for n in range(len(t)):
    y4.append(y[n][30])
figure(figsize=[16,8])
subplot(121)
plot(t,y4)
title('string signal versus time')
xlabel('time(s)')
ylabel('signal')
xlim(0,0.11)
subplot(122)
p=abs(np.fft.rfft(y4))**2
f = np.linspace(0, int(1/dt/2), len(p))
plot(f, p)
xlim(0,3000)
xlabel('Frequency(Hz)')
ylabel('Power')
title('Power spectrum')
show()

三、致谢
感谢室友进行的指导。