Oscillation Frequency of superposition of two oscillations of different frequencies

[Sidnv]

Homework Statement

Find the frequency of combined motion of the following

(a) x = sin (12pi.t) + cos(13pi.t + pi/4)
(b) x = sin(3t) - cos(pi.t)

Homework Equations

The book I'm using states that if the periods are commensurable ie if there exist 2 integers n1 and n2 such that n1T1 = n2T2 then the period is given by T= n1T1 = n2T2 where n1 and n2 are the smallest possible integers satisfying these conditions.

The Attempt at a Solution

I tried using the formula but the answer I got for a was off and part b should not have a period at all since it is not commensurable. Could anyone help me out please? It would also be really helpful if you could explain how to look at such problems using complex exponentials or else provide me with a link to some material I can read.

Thanks

 

[ehild]

Show your work, please. What are the periods of the sine and cosine functions in question A?

ehild

 

[Sidnv]

The period is given by the sine and cosine functions.

The angular frequency of the sine oscillations in part a is 12 pi and that of the cosine oscillations is 13 pi.

Thus T1 is 1/6 sec and T2 is 2/13 sec.

Thus the period of combined oscillations T = 12*1/6=13*2/13=2sec

Thus frequency should be 0.5 hz. But the answer given is 6.25 hz.

Similarly in part b the ratio of the Time periods is irrational. Hence they are not commensurable and do not give rise to periodic motion.