Realize that you want to calculate
.
There will two answers depending of the value of .
(this is where your identity comes in)
Hey guys having trouble getting the same answer in the text book.
Question: Consider solving wave equation (c=1) for string length Pi with fixed endpoints u(0,t)=0 and u(pi,t)=0 and initial velocity = 0 and displacement = 0.02sinx.
So after doing the usual steps i arrived at
Using the initial conditions and fourier coefficient formula's i have Bn=0
and
Can i treat these both as sin(x)? and use the trig indentity 1/2 -1/2cos(2x) to solve?
because the answer is u(x,t)=0.02costsinx and if i do that it will work.
but i just dont understand where the 'n' went in my solution :
seeing that it should be an infinite series and the answer given is not!
A standard technique for integrating such a thing is to use the trig identity
sin(a)sin(b)= (1/2)(cos(a-b)- cos(a+b) so that sin(x)sin(nx)= (1/2)(cos((1-n)x)- cos((1+n)x)
But, in fact, the functions "sin(nx)" and "sin(mx)" are "orthogonal" if then as well as sin(nx) being orthogonal to cos(mx) for any m.
[/quote]Can i treat these both as sin(x)? and use the trig indentity 1/2 -1/2cos(2x) to solve?
because the answer is u(x,t)=0.02costsinx and if i do that it will work.
but i just dont understand where the 'n' went in my solution :
seeing that it should be an infinite series and the answer given is not![/QUOTE]
as long as while for n= 1
You have while for n> 1 and for all n.
Since