Schwingung einer gespannten Saite mit fixierten Endpunkten
(Waves on a stretched string with fixed ends)restart:with(plots):Solution of wave equation:l := 1; c:=1;Spatial solutionX[n](x):= sin(n*Pi/l*x);X[m](x):=subs(n=m,X[n](x)):Time-dependent solution T[n](t):= (A(n)*cos(n*Pi*c/l*t)+B(n)*sin(n*Pi*c/l*t));u[n](x,t):=T[n](t)*X[n](x):Eigenfunction expansion u(x,t):=Sum(u[n](x,t),n=1..infinity);Initial conditionsu0(x):=x*(1-x);v(x):=x*(1-x);Plot of u0:plot(u0(x) , x=0..l ,thickness=5);Evaluation of coefficients for specific ICsA(n):=(2/l)*Int(u0(x)*X[n](x) ,x=0..l);A(n):=expand(value(%)):A(n):=simplify(subs({sin(n*Pi)=0,cos(n*Pi)=(-1)^n},A(n)));B(n):=(2/(n*Pi*c))*Int( v(x)*X[n](x),x=0...l);B(n):=expand(value(%)):B(n):=radsimp(subs({sin(n*Pi)=0,cos(n*Pi)=(-1)^n},%));(Observe convergence of series.)u[n](x,t):=eval(T[n](t)*X[n](x)):Series solution u(x,t):=Sum(u[n](x,t),n=1..infinity);First few terms of sum u(x,t):=sum(u[n](x,t),n=1..5);Animationanimate(u(x,t),x=0...l,t=0..20,color=red,thickness=5 , frames=600 );Different initial conditions:u0(x):=x*(1-x);v(x):=0;Evaluation of coefficients for specific ICsA(n):=(2/l)*Int(u0(x)*X[n](x) ,x=0..l);A(n):=expand(value(%)):A(n):=simplify(subs({sin(n*Pi)=0,cos(n*Pi)=(-1)^n},A(n)));B(n):=(2/(n*Pi*c))*Int( v(x)*X[n](x),x=0...l);B(n):=expand(value(%)):B(n):=radsimp(subs({sin(n*Pi)=0,cos(n*Pi)=(-1)^n},%));u[n](x,t):=eval(T[n](t)*X[n](x)):Series solution u(x,t):=Sum(u[n](x,t),n=1..infinity);First few terms of sum u(x,t):=sum(u[n](x,t),n=1..5);Animationanimate(u(x,t),x=0...l,t=0..20,color=red,thickness=5 , frames=600 );Different initial conditions (again):l:=1:u0(x):=x^3*(1-x^3);v(x):=0;Plot of u0plot(u0(x), x=0..l ,thickness=5);Evaluation of coefficients for specific ICsA(n):=(2/l)*Int(u0(x)*X[n](x) ,x=0..l);A(n):=expand(value(%)):A(n):=simplify(subs({sin(n*Pi)=0,cos(n*Pi)=(-1)^n},A(n)));B(n):=(2/(n*Pi*c))*Int( v(x)*X[n](x),x=0...l);B(n):=expand(value(%)):B(n):=radsimp(subs({sin(n*Pi)=0,cos(n*Pi)=(-1)^n},%));u[n](x,t):=eval(T[n](t)*X[n](x)):Series solution u(x,t):=Sum(u[n](x,t),n=1..infinity);First few terms of sum u(x,t):=sum(u[n](x,t),n=1..15);Animationanimate(u(x,t),x=0...l,t=0..20,color=red,thickness=5 , frames=600 );Different initial conditions (again, again):Initially flat string with nonzero initial velocity.u0(x):=0;v(x):=x^3*(1-x^3);Plot of u0plot(u0(x) , x=0..l ,thickness=5);Evaluation of coefficients for specific ICsA(n):=(2/l)*Int(u0(x)*X[n](x) ,x=0..l);A(n):=expand(value(%)):A(n):=simplify(subs({sin(n*Pi)=0,cos(n*Pi)=(-1)^n},A(n)));B(n):=(2/(n*Pi*c))*Int( v(x)*X[n](x),x=0...l);B(n):=expand(value(%)):B(n):=radsimp(subs({sin(n*Pi)=0,cos(n*Pi)=(-1)^n},%));u[n](x,t):=eval(T[n](t)*X[n](x)):Series solution u(x,t):=Sum(u[n](x,t),n=1..infinity);First few terms of sum u(x,t):=sum(u[n](x,t),n=1..5);Animationanimate(u(x,t),x=0...l,t=0..20,color=red,thickness=5 , frames=600 );