Is cosx+cos(sqrt(2)x) periodic?

  • Thread starter peripatein
  • Start date
  • #1
880
0

Homework Statement



Is the function cosx + cos(sqrt(2)x) is periodic?

Homework Equations



cos(x)=cos(x+2pi)

The Attempt at a Solution



For the above function to be periodic:
cosx + cos(sqrt(2)x) = cos(x+T) + cos(sqrt(2)(x + T))
Does that imply that 2pi = T AND 2pi = sqrt(2)T, ergo there is no such T?
 

Answers and Replies

  • #2
296
0
It doesn't exactly imply that. However, by the definition of periodicity, you need integer T to satisfy the equation. For example, the function cos(2x)+cos(3x) is periodic, one has period [itex]\pi[/itex] and the other has period [itex]2\pi/3[/itex]. It is not hard to see that both of these functions are also periodic by [itex]2\pi[/itex], and hence their sum is also periodic. Returning to your question, your answer is correct, the function is not periodic. However, your equations seem a bit fallacious to me. All you need is that the period of the sum is the least common multiple of the periods of the summands.

Can you see such a least common multiple?
 

Related Threads on Is cosx+cos(sqrt(2)x) periodic?

Replies
3
Views
2K
  • Last Post
Replies
4
Views
22K
  • Last Post
Replies
3
Views
29K
Replies
4
Views
4K
Replies
18
Views
864
  • Last Post
Replies
2
Views
12K
Replies
5
Views
1K
  • Last Post
Replies
1
Views
7K
Replies
4
Views
17K
  • Last Post
Replies
5
Views
15K
Top