Exercise_13: Chapter 6 problem 6.13

一摘要：

Conder the power spectra for waves on a string as a function of where the string vibration is observed ,$x_0$.

二背景介绍:

1.the central equation of wave motion is: $$\ddot{y_i}=\frac{T}{\rho}*\frac{\partial^2 y}{\partial x^2},(6.1)$$

2.when $r=c*\frac{\Delta{t}}{\Delta{x}}$, $$y(i,n+1)=2[1-r^2]y(i,n)-y(i,n-1)+r^2[y(i+1,n)+y(i-1,n)],(6.2)$$

3.the initial string profile is $$y_0(x)=exp[-k(x-x_0)^2],(6.3)$$

三正文：

When $x_0$ is 0.1,0.2,0.3,0.4,0.5,0.7,here are the results:

from matplotlib import animation
from pylab import*
import numpy as np
c=300
dx=0.01
l=np.linspace(0,1,1000)
def w():
    y=[]
    dt=dx/c
    t=0.0
    y0=np.exp(-1000*(l-l*0.2)**2)
    y.append(y0)
    while t<0.05:
        y_next=np.zeros(1000)

        for i in range(1,998):
            y_next[i]=-y[-2][i]+y[-1][i+1]+y[-1][i-1]

        y.append(y_next)
        t=t+dt
    return y, t
a=w()[0]
'''
for iter in a01:
   print type(a01)
   '''
b=[a_iter[800] for a_iter in a]
print len(b)
e=len(b)
d=linspace(0,dx/c*e,e)
plot(d,b,'orange')
xlim(0,0.05)
plt.xticks([0.01,0.02,0.03,0.04,0.05])
xlabel('Time(s)')
ylabel('Signal(arbitary units)')
title('String signal versus time at x_0=0.2')
show()
y2=np.fft.fft(b)
f=np.fft.fftfreq(len(b),dx/c)
y2=abs(y2)**2
plot(f,y2,'orange')
xlim(0,3000)
plt.xticks([1000,2000,3000])
xlabel('Frequency(Hz)')
ylabel('Power(arbitrary units)')
title('Power spectrum:Pluck at 0.2')
show()

四结论：

From the schematic of power spectrum, we can see that the frequency is distributed mainly from 0~1000HZ,and as the $x_0$ is incresing ,the power is stronger.

五致谢:

Thanks to FangyingLi Senior.