Dispersion des ondes
Fondamental : Diaporama du micro-contenu (rafraichir la page si redimensionnée)
Texte légal : Polycopié du micro-contenu
Simulation : Programme Python
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# -*- coding: utf-8 -*-2
"""3
Dispersion4
O. Thual, 21/08/20215
"""6
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# clear all8
for iglob in list(globals().keys()):
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if(iglob[0] != '_'):
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exec('del {}'.format(iglob))
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# import libraries12
import numpy as np
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import matplotlib.pyplot as plt
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import os
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# Sous programmes 17
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def inifig(xaxe=0,yaxe=0,xlab='x',ylab='y'):
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plt.figure(2)
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plt.axvline(xaxe)
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plt.axhline(yaxe)
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plt.xticks(fontsize=12)
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plt.yticks(fontsize=12)
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plt.xlabel(xlab,fontsize=16 )
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plt.ylabel(ylab,fontsize=16)
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def zfi(x,le=2):
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miss=le-len(str(x))
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a='0'*miss+str(x)
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return a
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def reladi(k):
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#omega=al*k-be*k**34
om=2*np.pi
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omega=np.sqrt(om**2+k**2)
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return omega
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def vgroup(k):
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#v=al-3*be*k**240
om=2*np.pi
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omega=np.sqrt(om**2+k**2)
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v=k/omega
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return v
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def gauss(k):
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g=np.exp(-(k-k0)**2/(2*var))
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return g
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def signalN(x,t):
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psi=np.zeros((N,Nx))
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for n in range(0,N):
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kn=k[n]; omn=om[n];
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psin=np.cos(kn*x-omn*t)*gauss(kn);
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psi[n,:]=psin;
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return psi
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def plotrelaN(name,kmin,kmax,k):
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kplt=np.linspace(0.01,kmax,501);
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om=reladi(k); cg=vgroup(k); cp=om/k;
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omplt=reladi(kplt); cgplt=vgroup(kplt); cpplt=omplt/kplt;
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fig=plt.figure(2,figsize=(7,4))
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plt.xlabel(r'$k$',fontsize=16 )
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plt.ylabel(r'$c_g, c_p, \omega$',fontsize=16)
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plt.xlim(kmin,kmax)
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plt.ylim(0,10)
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titlefig=name
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plt.title(titlefig,fontsize=16)
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plt.grid(color='black', axis='y', linestyle='-', linewidth=1)
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plt.grid(color='black', axis='x', linestyle='-', linewidth=1)
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plt.plot(kplt,omplt,color='black',linewidth=3)
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plt.plot(kplt,cgplt,color='magenta',linewidth=3)
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plt.plot(kplt,cpplt,color='blue',linewidth=3)
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plt.plot(kplt,fac*gauss(kplt),color='black',linewidth=3)
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# modes 75
plt.scatter(k,fac*amp,marker='o',color='blue',s=50)
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plt.scatter(k,om,marker='o',color='blue',s=50)
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plt.scatter(k,cg,marker='o',color='blue',s=50)
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plt.scatter(k,cp,marker='o',color='blue',s=50)
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# extremes80
plt.scatter(k1,fac*gauss(k1),marker='o',color='red',s=100)
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plt.scatter(k1,reladi(k1),marker='o',color='red',s=100)
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plt.scatter(k1,vgroup(k1),marker='o',color='red',s=100)
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plt.scatter(k1,reladi(k1)/k1,marker='o',color='red',s=100)
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plt.scatter(k2,fac*gauss(k2),marker='o',color='green',s=100)
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plt.scatter(k2,reladi(k2),marker='o',color='green',s=100)
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plt.scatter(k2,vgroup(k2),marker='o',color='green',s=100)
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plt.scatter(k2,reladi(k2)/k2,marker='o',color='green',s=100)
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plt.savefig(name)
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plt.show()
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def reladisp(name,kmin,kmax,k):
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kplt=np.linspace(0.01,kmax,501);
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om=reladi(k); cg=vgroup(k); cp=om/k;
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omplt=reladi(kplt); cgplt=vgroup(kplt); cpplt=omplt/kplt;
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fig=plt.figure(2,figsize=(7,4))
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plt.xlabel(r'$k$',fontsize=16 )
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plt.ylabel(r'$c_g, c_p, \omega$',fontsize=16)
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plt.xlim(kmin,kmax)
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plt.ylim(0,10)
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titlefig=name
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plt.title(titlefig,fontsize=16)
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plt.grid(color='black', axis='y', linestyle='-', linewidth=1)
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plt.grid(color='black', axis='x', linestyle='-', linewidth=1)
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plt.plot(kplt,omplt,color='black',linewidth=3)
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plt.plot(kplt,cgplt,color='magenta',linewidth=3)
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plt.plot(kplt,cpplt,color='blue',linewidth=3)
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plt.plot([0,kmax],[1,1],color='black',linestyle='-.',linewidth=3)
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plt.savefig(name)
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plt.show()
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def animationN(name):
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t=0;
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dt=Time/Nt
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for i in range(0,Nt):
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t=dt*i;
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title=name+" i="+zfi(i)
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titlefig=name
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print(title)
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fig=plt.figure(1,figsize=(7,4))
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plt.xlabel('x',fontsize=16 )
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plt.ylabel(r'$\theta$',fontsize=16)
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plt.title(titlefig,fontsize=16)
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# signaux127
psi=signalN(x,t)
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xp=np.zeros(M); psip=np.zeros(M)
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for m in range(M):
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xp[m]=-m*L0+v*t; psip[m]=0;
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psi1=np.cos(k1*x-om1*t)*gauss(k1)
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psi2=np.cos(k2*x-om2*t)*gauss(k2)
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# 134
psitot=x*0;
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for n in range(N):
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psin=psi[n,:]
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psitot=psitot+psin/N
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for m in range(M):
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psip[m]=psip[m]+np.cos(k[n]*xp[m]-om[n]*t)*gauss(k[n])/N
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plt.plot(x,psin,color='blue',linewidth=1)
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plt.plot(x,psi1,color='red',linewidth=2)
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plt.plot(x,psi2,color='green',linewidth=2)
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plt.plot(x,2*psitot,color='black',linewidth=3)
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plt.scatter(xp,2*psip,marker='o',color='magenta',s=160)
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plt.xlim(-L,L)
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plt.ylim(ymin,ymax)
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namei=name+zfi(i)+'.png';
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plt.grid(color='black', axis='y', linestyle='-', linewidth=1)
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plt.grid(color='black', axis='x', linestyle='-', linewidth=1)
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plt.savefig(namei)
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plt.show()
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plt.close()
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gifanim="Anim"+name+".gif"
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#os.system('/opt/local/bin/convert -set delay '+delay+' '+name+'* '+gifanim);155
os.system('/usr/local/bin/convert -set delay '+delay+' '+name+'* '+gifanim);
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if Flagrm: os.system('rm '+name+'*');
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# Main 161
F=False; T=True
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Flagrm=T;
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# N ondes167
# relation de dispersion168
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withanim=T;
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k0=4; de=.05; dk=de*k0; k1=k0-dk; k2=k0+dk;
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var=dk**2; fac=6;
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for N in range(2,6):
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k=np.linspace(k1,k2,N); amp=gauss(k);
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om=reladi(k); cg=vgroup(k); cp=om/k;
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om1=reladi(k1); om2=reladi(k2)
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M=10*N # nombre de temoins de phase
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v=(om1+om2)/(k1+k2)
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name="Klein-Gordon-"+zfi(N)+"-ondes"
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#180
L0=10*4*np.pi/(k1+k2);
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L=N*5*4*np.pi/(k1+k2);
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Nx=1001; x=np.linspace(-L,L,Nx);
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Nt=(N+1)*10; Time=.5*L/vgroup(k0);
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ymin=-2; ymax=2;
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delay="40"
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nom="Paquet-"+zfi(N)+"-ondes"
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print(nom)
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if withanim: animationN(nom)
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name="Klein-Gordon-"+zfi(N)+"-ondes-Zoom"
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kmin=3.6; kmax=4.4
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plotrelaN(name,kmin,kmax,k)
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name="Klein-Gordon-"+zfi(N)+"-ondes"
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kmin=.01; kmax=5
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plotrelaN(name,kmin,kmax,k)
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kmin=.01; kmax=10
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reladisp("Klein-Gordon",kmin,kmax,k)
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