m17.mws  020   4.6.4 1)   wave-1-DR.mws     WAVE EQUATION    X(0)=0,  X'(L)+HX(L)=0    (Dirichlet-Robin)

>

restart;

>	L:=10;H[2]:=3;a:=2;

characteristic equation:

>	w(x):=x*cos(x*L)+H[2]*sin(x*L);

>	plot(w(x),x=0..2);

Eigenvalues:

>	lambda:=array(1..100);

>	n:=1: for m from 1 to 100 do z:=fsolve(w(x)=0,x=m/5..(m+1)/5): if type(z,float) then lambda[n]:=z: n:=n+1 fi od:

>	for i to 5 do lambda[i] od;

>

N:=n-1;

>	n:='n':i:='i':

Eigenfunctions:

>	X[n]:=sin(lambda[n]*x);

Squared norm:

>	NX2[n]:=int(X[n]^2,x=0..L);

Initial conditions:

sinusoidal:

>	u0(x):=sin(x/L);u1(x):=sin(2*x/L);

impulse:

>	u0(x):=(Heaviside(x-6)-Heaviside(x-7));u1(x):=0;

parabolic:

>	u0(x):=x*(L-x);u1(x):=1;

Fourier coefficients:

>	b[n]:=int(u0(x)*X[n],x=0..L)/NX2[n];

>	d[n]:=int(u1(x)*X[n],x=0..L)/NX2[n]/lambda[n]/a;

Solution:

>	u(x,t):=sum(X[n]*(b[n]*cos(lambda[n]*a*t)+d[n]*sin(lambda[n]*a*t)),n=1..N):

>	plot3d({u(x,t)},x=0..L,t=0..21,axes=boxed,style=wireframe,projection=0.9,color=black);

>	with(plots):

Warning, the name changecoords has been redefined

>	animate({u(x,t),u0(x)},x=0..L,t=0..21,axes=boxed,frames=500);

>

SINGLE  WAVE:

>	u0(x):=(Heaviside(x-6)-Heaviside(x-7));u1(x):=0;

Fourier coefficients:

>	b[n]:=int(u0(x)*X[n],x=0..L)/NX2[n];

>	d[n]:=int(u1(x)*X[n],x=0..L)/NX2[n]/lambda[n]/a;

Solution:

>	u(x,t):=sum(X[n]*(b[n]*cos(lambda[n]*a*t)+d[n]*sin(lambda[n]*a*t)),n=1..N):

>	plot3d({u(x,t)},x=0..L,t=0..12,axes=boxed,projection=0.9,style=wireframe,grid=[150,150]);

>	with(plots):

>	animate({u(x,t),u0(x)},x=0..L,t=0..21,axes=boxed,frames=300,numpoints=500);

NORMAL MODES:

fundamental mode:

>	m1:=subs(n=1,X[n]*(b[n]*cos(lambda[n]*a*t)+d[n]*sin(lambda[n]*a*t))):

>	animate({m1},x=0..L,t=0..9);

1st overtone:

>	m2:=subs(n=2,X[n]*(b[n]*cos(lambda[n]*a*t)+d[n]*sin(lambda[n]*a*t))):

>	animate({m2},x=0..L,t=0..9);

2nd overtone:

>	m3:=subs(n=3,X[n]*(b[n]*cos(lambda[n]*a*t)+d[n]*sin(lambda[n]*a*t))):

>	animate({m3},x=0..L,t=0..9);

3rd overtone:

>	m4:=subs(n=4,X[n]*(b[n]*cos(lambda[n]*a*t)+d[n]*sin(lambda[n]*a*t))):

>	animate({m4},x=0..L,t=0..9);

>

Standing Waves:       initial condition is in the form of overtone:

>	u0(x):=subs(n=8,X[n]*(b[n]*cos(lambda[n]*a*t)+d[n]*sin(lambda[n]*a*t))):u1(x):=0;

Fourier coefficients:

>	b[n]:=int(u0(x)*X[n],x=0..L)/NX2[n];

>	d[n]:=int(u1(x)*X[n],x=0..L)/NX2[n]/lambda[n]/a;

Solution:

>	u(x,t):=sum(X[n]*(b[n]*cos(lambda[n]*a*t)+d[n]*sin(lambda[n]*a*t)),n=1..N):

>	with(plots):

>	animate({u(x,t),u0(x)},x=0..L,t=0..21,axes=boxed,frames=300,numpoints=500);