Orthogonal Properties for Sine Don't Hold if Pi is involded?

  • Thread starter mmmboh
  • Start date
  • #1
407
0
Orthogonal Properties for Sine Don't Hold if Pi is involded??

Normally I know [tex]
\int_{-L}^L \sin \frac{n x}{L} \sin \frac{\m x}{L} ~ dx = 0\mbox{ if }n\not =m , \ =L \mbox{ if }n=m
[/tex] but apparently this doesn't work for [tex]
\int_{-L}^L \sin \frac{\pi n x}{L} \sin \frac{\pi m x}{L} ~ dx
[/tex]

I am trying to find fourier coefficients, and my integral is [tex]\int_{0}^{L/2} \sin \frac{2\pi x}{L} \sin \frac{\pi n x}{L} ~ dx
[/tex]

For even n that aren't 2, the integral is 0, if n=2, then the integral is L/2, but if n is odd, then the integral doesn't equal 0 (it's actually a fairly complex answer) even thought it looks like it should by orthogonal properties ...why doesn't this work?

The integrals I am doing take very long to do, is there a property that would allow me to shorten them?
 
Last edited:

Answers and Replies

  • #2
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,556
767


It does work. If [itex]n \ne m[/itex] are positive integers then

[tex]\int_0^L\ \sin(\frac{n\pi x}{L})\sin(\frac{m\pi x}{L})\, dx = 0[/tex]

Sturm-Liouville theory would guarantee this without working it out.
 
  • #3
407
0


Sorry the limits are actually 0 to L/2, in this case it doesn't work, is there another property that would help evaluate the integral quicker or do I have to go through the whole thing every time?
 
  • #4
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,556
767


The set of functions

[tex]\left \{ \sin\frac{2n\pi x}{L}\right\}[/tex]

are orthogonal on [0, L/2].
 
  • #5
407
0


Is there anything like that for [tex]
\left \{ \sin\frac{(2n)\pi x}{L}\right\ \left \sin\frac{(2n+1)\pi x}{L}\right\}
[/tex]?
 
  • #6
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,556
767


Is there anything like that for [tex]
\left \{ \sin\frac{(2n)\pi x}{L}\right\ \left \sin\frac{(2n+1)\pi x}{L}\right\}
[/tex]?
I doubt you will get zero, but why don't you just integrate it and see?
 

Related Threads on Orthogonal Properties for Sine Don't Hold if Pi is involded?

Replies
15
Views
7K
Replies
7
Views
1K
Replies
1
Views
5K
Replies
1
Views
422
Replies
1
Views
2K
Replies
6
Views
1K
Replies
14
Views
5K
Replies
2
Views
572
Top