MATH 373 -- Applied Mathematics

January 25, 2001

D'Alembert's general solution of the 1D wave equation

is , which expresses the

solution as the superposition of two traveling waves -- one

moving to the right with constant speed and the other moving

to the left with the same constant speed . For example,

with , we have that

> u:=(x,t)->sin(t)*cos(x);

is a solution:

> diff(u(x,t),t$2)-diff(u(x,t),x$2);

Here is an animation showing how looks as a function of

> with(plots):

> standing1:=animate(u(x,t),x=-3*Pi..3*Pi,t=-Pi..Pi,frames=50,color=green):

> display({standing1},insequence=true);

This is called a "standing wave solution". The key to writing this in D'Alembert's

form is the trig identity:

> travleft:=animate(sin(x+t),x=-3*Pi..3*Pi,t=-Pi..Pi,frames=50,color=red):

> travright:=animate(sin(x-t),x=-3*Pi..3*Pi,t=-Pi..Pi,frames=50,color=blue):

> display({travleft,travright});

> standing2:=animate((sin(x+t)+sin(x-t))/2,x=-3*Pi..3*Pi,t=-Pi..Pi,frames=50):

> display({standing2},insequence=true);

>