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=[]#某一點的震動時域
self.fss=[]#某一點震動的頻域
'''fixed ends'''
def wave1(self):#初始狀態(tài)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.
