Contents

(1) Beat phenomenon when there is no dispersibility. At this time, the phase velocity of the wave and the group velocity match.

(2) Beat phenomenon when there is dispersibility. The group velocity does not match the phase velocity.

(3) Combine the animations created in (1) and (2) above.

Code (1): When there is no dispersibility

"""
Beat phenomenon:No dispersibility

"""

%matplotlib nbagg 
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation


fig = plt.figure(figsize = (8, 6))

L, t_max = 6,12
Nx= 240
Nt = 100

delta_x=L/Nx
delta_t=t_max/Nt
xx=list(range(Nx))

a=1
k1= 2*np.pi/(L/10)
omega1=10
phi1=0

k2= (1+0.15)*k1
omega2=(1+0.15)*omega1
phi2=0

u1=np.zeros([Nx,Nt])
u2=np.zeros([Nx,Nt])
utot=np.zeros([Nx,Nt])


xlis=np.zeros([Nx])

for i in range(Nx):
    x = i*delta_x
    xlis[i] = x
    for j in range(Nt):
        t = j*delta_t
        u1[i,j]=a*np.cos(k1*x-omega1*t-phi1)
        u2[i,j]=a*np.cos(k2*x-omega2*t-phi2)
        utot[i,j]= u1[i,j]+u2[i,j]


def update(j,utot,fig_title):
    if j != 0:
        plt.cla()
    plt.ylim(-3, 3)
    plt.xlim(0, L)
    plt.plot( xlis,u1[:,j],  '--', color='red', linewidth = 1)
    plt.plot( xlis,u2[:,j],  '--', color='blue',linewidth = 1)
    plt.plot( xlis,utot[:,j],  '-', color='purple', linewidth =4)

    plt.title(fig_title  + str("{0:.1f}".format(j*delta_t)))

    plt.xlabel('X')
    plt.ylabel('U')
    plt.legend(['u1','u2', 'u1+u2'], loc='upper right')


ani = animation.FuncAnimation(fig,update,fargs = (utot,'Beat: t ='), interval =1, frames = Nt,blit=False)
fig.show()

ani.save("output.gif", writer="imagemagick")

Result (1) No dispersibility

A beat phenomenon (a phenomenon in which the amplitude of the composite wave fluctuates periodically) is observed. At this time, the phase velocity and the group velocity match.

Code (2): Distributed

Investigate what happens when heterogeneous waves, that is, dispersed waves, are superimposed.

"""
Dispersion relation ω=f(k)If there is
"""

%matplotlib nbagg 
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation

fig = plt.figure(figsize = (8, 6))


L, t_max = 6,12
Nx= 240
Nt = 100

delta_x=L/Nx
delta_t=t_max/Nt
xx=list(range(Nx))
#X,T = np.meshgrid(x,t)

omega=10
omega0=30

a=1
k1= 2*np.pi/(L/10)

V0= omega/k1

omega1=np.sqrt((V0*k1)**2+omega0**2)
phi1=0

k2= (1+0.15)*k1
omega2=np.sqrt((V0*k2)**2+omega0**2)
phi2=0

u1=np.zeros([Nx,Nt])
u2=np.zeros([Nx,Nt])
utot=np.zeros([Nx,Nt])


xlis=np.zeros([Nx])
for i in range(Nx):
    x = i*delta_x
    xlis[i] = x
    for j in range(Nt):
        t = j*delta_t
        u1[i,j]=a*np.cos(k1*x-omega1*t-phi1)
        u2[i,j]=a*np.cos(k2*x-omega2*t-phi2)
        utot[i,j]= u1[i,j]+u2[i,j]



def update(j,utot,fig_title):
    if j != 0:
        plt.cla()
    plt.ylim(-3, 3)
    plt.xlim(0, L)
    plt.plot( xlis,u1[:,j],  '--', color='red', linewidth = 1)
    plt.plot( xlis,u2[:,j],  '--', color='blue',linewidth = 1)
    plt.plot( xlis,utot[:,j],  '-', color='green', linewidth =4)

    plt.title(fig_title  + str("{0:.1f}".format(j*delta_t)))

    plt.xlabel('X')
    plt.ylabel('U')
    plt.legend(['u1','u2', 'u1+u2'], loc='upper right')


ani = animation.FuncAnimation(fig,update,fargs = (utot,'Beat: t ='), interval =1, frames = Nt,blit=False)
fig.show()

ani.save("output2.gif", writer="imagemagick")

Result (2) Dispersive

** In this case, there is a difference between the group velocity (the velocity at which the envelope ("beat waveform") moves) and the phase velocity (the velocity at which the wave itself moves). The "beat waveform" moves at group velocity, and the wave itself moves at phase velocity. ** **

Result (3): Dispersive and non-dispersive animation: Comparison

The difference in motion between dispersive (green line) and non-dispersive (purple line) is clear.

Addendum

(1) Dispersion relation used in this paper

One-dimensional string wave equation, $\frac{\partial^2 u(x,t)}{\partial t^2} = v^2 \frac{\partial^2 u(x,t)}{\partial x^2} \tag{1} $ When the restoring force $-\ omega_0 ^ 2 u (x, t) \ tag {2} $ is added to hold the constituent particles of the string at each position, the wave equation becomes

\frac{\partial^2 u(x,t)}{\partial t^2} = v^2 \frac{\partial^2 u(x,t)}{\partial x^2} -\omega_0^2 u(x,t) \tag{3} Will be. Some vibrations that occur in actual matter follow a similar equation (such as the vibration of intraatomic electrons).

The solution of this wave equation is

As $ \ omega (k) = (v ^ 2k ^ 2 + \ omega_0 ^ 2) ^ \ frac {1} {2} \ tag {4} $

u(x,t) = a cos(k x-\omega(k)t-\phi \tag{5}

It turns out that In the content (2) of this paper, the above dispersion relation (Equation .4) was used.

(2) Various dispersion relations are known in various media. Let me give you an example.

• ** Dispersion formula of waves propagating on the surface of deep water **: $ \ omega (k) = \ sqrt (gk + T k ^ 3 / \ rho) \ tag {6} $ (g is gravitational acceleration, T is Surface tension, $ \ rho $ is the density.

• ** Diatomic system (each ion mass is $ m_1 $ and $ m_2 $) ** ** lattice vibration ** considering only closest interaction Dispersion relation of wiki /% E6% A0% BC% E5% AD% 90% E6% 8C% AF% E5% 8B% 95): \omega^2(k)= \frac{f}{m_\mu} \left ( 1 \pm \sqrt{1-\frac{4m_\mu^{\, 2}}{m_1m_2} \sin^2{ka} } \right ), \quad \frac{1}{m_\mu}=\frac{1}{m_1} + \frac{1}{m_2}\tag{7}

References

[1] For animation creation, Qiita article by AnchorBlues: Flexible animation creation using animation.FuncAnimation of matplotlib was helpful and easy to understand. ..

[2] Regarding group velocity, for example, by Yosuke Nagaoka ["Vibration and Waves"](https://www.amazon.co.jp/%E6%8C%AF%E5%8B%95%E3%81% A8% E6% B3% A2-% E9% 95% B7% E5% B2% A1-% E6% B4% 8B% E4% BB% 8B / dp / 4785320451) has an elementary and easy-to-understand explanation.