Exercise 13

Problem 6.1:
Write a program to simulate wave motion on a string with free ends.Do this by either using boundary conditions that always give the ends of the string the same displacement as the points that are one spatial unit in from the end,or by employing(6.7).Study how the waves are reflected from the ends of the string and compare the results with the behavior with fixed ends.You should find that the reflected wave packets are not inverted.

Background:
The central equation of wave motion:

13.1.gif

Gaussian pluck:

13.2.gif

Algorithm for ideal wave motion equation:

13.3.gif

Main body

import numpy as np

import matplotlib.pyplot as plt

from matplotlib import animation

class waves:

def __init__(self,r=1,length=1,amplitude=1,n=100,update_times=5000,pluck_point=0.1,):

self.r=r

self.l=length

self.A=amplitude

self.n=n

self.N=update_times

self.p=pluck_point

self.x=np.linspace(0,length,n)

self.y0=[]

self.y1=[]

self.y=[]

self.ss=[]#某一点的震动时域

self.fss=[]#某一点震动的频域

'''fixed ends'''

def wave1(self):#初始状态和第一次update

#self.y0=np.sin(4*np.pi*np.linspace(0,self.l,self.n))#初始正弦波

self.y0=np.exp(-1000*(self.x-self.p)**2)#Gaussian plucking

self.y1=np.linspace(0,0,self.n)

for i in range(len(self.y0)-2):

self.y1[i+1]=2*(1-self.r**2)*self.y0[i+1]-self.y0[i+1]+self.r**2*(self.y0[i+2]+self.y0[i])

self.y.append(self.y0)

self.y.append(self.y1)

def propagate1(self):#propagate

counter=1

while(1):

counter+=1

temp_y=np.linspace(0,0,self.n)

for i in range(len(self.y0)-2):

temp_y[i+1]=2*(1-self.r**2)*self.y[-1][i+1]-self.y[-2][i+1]+self.r**2*(self.y[-1][i+2]+self.y[-1][i])

self.y.append(temp_y)

if counter>self.N:

break

'''free ends'''

def wave2(self):

self.y0=np.sin(4*np.pi*np.linspace(0,self.l,self.n))#初始正弦波

#self.y0=np.exp(-1000*(self.x-self.p)**2)#Gaussian plucking

self.y1=np.linspace(0,0,self.n)

for i in range(len(self.y0)-2):

self.y1[i+1]=2*(1-self.r**2)*self.y0[i+1]-self.y0[i+1]+self.r**2*(self.y0[i+2]+self.y0[i])

self.y1[0]=2*self.y1[1]-self.y1[2]

self.y1[-1]=2*self.y1[-2]-self.y1[-3]

self.y.append(self.y0)

self.y.append(self.y1)

def propagate2(self):

counter=1

while(1):

counter+=1

temp_y=np.linspace(0,0,self.n)

for i in range(len(self.y0)-2):

temp_y[i+1]=2*(1-self.r**2)*self.y[-1][i+1]-self.y[-2][i+1]+self.r**2*(self.y[-1][i+2]+self.y[-1][i])

temp_y[0]=2*temp_y[1]-temp_y[2]

temp_y[-1]=2*temp_y[-2]-temp_y[-3]

self.y.append(temp_y)

if counter>self.N:

break

a=waves()

a.wave2()

a.propagate2()

#show animated result

fig = plt.figure()

ax = plt.axes(xlim=(0,1), ylim=(-1, 1))

line, = ax.plot([], [], lw=2)

def init():

line.set_data([], [])

return line,

def animate(i):

x = a.x

y = a.y[i]

line.set_data(list(x), list(y))

return line,

anim=animation.FuncAnimation(fig, animate, init_func=init, frames=90, interval=25)

plt.grid(True)

plt.xlabel('x')

plt.ylabel('y')

plt.show()

''' spectra'''

class fourier_tr(waves):

def vibrate(self):

for i in range(len(self.y)):

self.ss.append(self.y[i][int(self.n/2)])

def show_vib(self):

temp_t=np.linspace(0,len(self.ss),len(self.ss))

plt.plot(temp_t,self.ss,label='pluck point=%.2f'%self.p)

plt.xlabel('t')

plt.ylabel('signal')

plt.title('string signal versus time')

plt.grid(True)

plt.legend(frameon=True)

def ft(self):

temp=np.fft.fft(self.ss)

for i in temp:

self.fss.append(i.real**2+i.imag**2)

temp_f=np.linspace(0,int(len(self.fss)),int(len(self.fss)))

plt.figure(figsize=(15,3))

plt.plot(temp_f,self.fss,label='pluck point=%.2f'%self.p)

plt.xlim(0,len(temp_f)/2)

plt.title('power spectrum')

plt.xlabel('f')

plt.ylabel('power')

plt.xlim(0,1000)

plt.legend(frameon=True)

b=fourier_tr()

b.wave1()

b.propagate1()

b.vibrate()

b.show_vib()

Conclusion:

13.4.gif

13.5.gif

13.6.gif

We can see the ends condition have a huge affect on the string motion.