just checking

[futurebird]

f(x) = sin(x) + sin(x*sqrt(2)) has no fundamental period.

Right?

Comments

[detroitfather]

You want the mathematician answer or the engineer answer?



I think in precise mathematical terms you are correct, because sqrt(2) is irrational.

But, we engineers live with approximations (sometimes pretty crude ones) every day of our lives. So, you and I as friends, can agree to call sqrt(2) = 1.4, why then the combined signal does have a fundamental period.

What does it say about my social life that I have time to answer this on a Friday night?!

Typo ...

[detroitfather]

That should have read: So, if you and I as friends can agree ...

[futurebird]

So, you and I as friends, can agree to call sqrt(2) = 1.4

Only if you make that a WAVY equal sign!

This all makes me so nervous.

[detroitfather]

sqrt(2) ≈ 1.4

Works for me.

So, in summary, your function has various values of x for which it is "approximately periodic" ... If you are willing to make the even rougher assumption that sqrt(2) ≈ 1.5, then you see that it has an even shorter (smaller) "approximate fundamental period".

Or, if you wish to go in the other direction (i.e., less deviation from mathematical exactness), you can say sqrt(2) ≈ 1.4142 (or any other rational approximation), and then the "approximate fundamental period" will be much longer (larger).

[futurebird]

Isn't it going to get way out of synch for large numbers?

[detroitfather]

No. The fast frequency keeps on catching up and passing the slow one (each one is periodic by itself, remember). It's just that, as the fast one passes the slow one, the never come back to zero at exactly the same time. They will get arbitrarily close to zero at the same time, if you wait long enough, just as a rational approximation can be written which is as close to sqrt(2) as you want.