# Finding the periods of combinations of a fundamental frequency and some overtones

## Homework Statement:

$$\sin{\pi t}+\sin{2\pi t}+\frac{1}{3} \sin{3\pi t}$$

## Relevant Equations:

The sum has the period of the fundamental.
This problem came from Problems, Section 3 Chapter 7 in ML Boas, Mathematical Methods in Physical Sciences. Boas suggested to make a computer plot. From my computer plot (I use online graphing calculator) and find that the period of the sum is 2.

Instead of using computer, I want to find the period of combinations of fundamental and overtone analytically. Based on Boas, I have to find the fundamental and its period. So, my biggest problem is I can't find which one is the fundamental terms from 3 terms in the problem.

Could you please explain how to find the fundamental? Thanks.

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pasmith
Homework Helper
If $f(x) = f(x + L)$ for every $x$, what is $f(k(x + \frac Lk))$ for $k >0$?

If $k > 1$, is $L/k$ shorter or longer than $L$?

etotheipi
Gold Member
2019 Award
Here's a second way of looking at it. Generalise your function to $$f(t) = \sum_i a_i \sin{(\omega_i t)}$$ Now like @pasmith alludes to, ##T## is the period if it is the smallest number for which $$f(t + T) = f(t) \quad, \forall t \in D$$Also notice that $$f(t + T) = \sum_i a_i \sin{(\omega_i t + \omega_i T)}$$which means that we have ##2\pi | \omega_i T## for all ##\omega_i##. Hence, for all ##i##, $$\frac{T}{\left(\frac{2\pi}{\omega_i}\right)} = \frac{T}{T_i} \in \mathbb{Z}$$where ##T_i## are the periods of the individual terms. Now, with that in mind, how does ##T## relate to all of the other individual time periods?

If $f(x) = f(x + L)$ for every $x$, what is $f(k(x + \frac Lk))$ for $k >0$?

If $k > 1$, is $L/k$ shorter or longer than $L$?
If ##k>1## then the period ##L/k## shorter than ##L##. So, the frequency is higher. I still don't get it?

Oh, wait. So, the longer period means that this period is the fundamental's period? Because overtone means that the frequency is higher so the period is shorter than the fundamental.

In my problems, I have 3 terms of sine function. The first one have ##T_1=2##; ##T_1=1##; ##T_1=\frac{2}{3}##. So, the period of the fundamental is 2. Is that right?

Here's a second way of looking at it. Generalise your function to $$f(t) = \sum_i a_i \sin{(\omega_i t)}$$ Now like @pasmith alludes to, ##T## is the period if it is the smallest number for which $$f(t + T) = f(t) \quad, \forall t \in D$$Also notice that $$f(t + T) = \sum_i a_i \sin{(\omega_i t + \omega_i T)}$$which means that we have ##2\pi | \omega_i T## for all ##\omega_i##. Hence, for all ##i##, $$\frac{T}{\left(\frac{2\pi}{\omega_i}\right)} = \frac{T}{T_i} \in \mathbb{Z}$$where ##T_i## are the periods of the individual terms. Now, with that in mind, how does ##T## relate to all of the other individual time periods?
I don't understand what fraction is that? If I have ##T_1=2##; ##T_1=1##; ##T_3=\frac{2}{3}## so I have 3 fraction, which are ##\frac{1}{2} T##;##\frac{1}{1} T##; ##\frac{3}{2} T##.

From other thread, I read that one can find the fundamental period by finding the least common multiple of given sine function. Is it related to our discussion here? Thanks.

But, with this method, I find the period first, then the fundamental sine function. But, with Boas suggestion, one should find the fundamental sine function first right?

etotheipi
Gold Member
2019 Award
My point was, that if ##\frac{T}{T_i}## is always an integer, then ##T## is the lowest common multiple of the time periods of the individual waves.

My point was, that if ##\frac{T}{T_i}## is always an integer, then ##T## is the lowest common multiple of the time periods of the individual waves.
What is D in #3? And, if ##\frac{T}{T_i}## is an integer so T is multiplication of T_i. Is that right? So, because we want the shorter period, we must seek the lowest common multiple. Is my logic true? Please CMIIW. Thankss

etotheipi
Gold Member
2019 Award
What is D in #3? And, if ##\frac{T}{T_i}## is an integer so T is multiplication of T_i. Is that right? So, because we want the shorter period, we must seek the lowest common multiple. Is my logic true? Please CMIIW. Thankss
Oh ##D## is just the domain of times we are interested in, it's not very important. But I think your reasoning is correct, we want ##T## to be the smallest possible number that is divisible by all of the ##T_i##; and that's the definition of the lowest common multiple.

Oh alright. Is there any exception, that ##\frac{T}{T_i}## is not an integer? If not, why the fraction is always be an integer number? Are there any detail from the definition of periodic function that misunderstood by me?

##2\pi | \omega_i T## for all ##\omega_i##.
Wait, what does it mean?

etotheipi
Gold Member
2019 Award
##2\pi | \omega_i T## for all ##\omega_i##.
Wait, what does it mean?
It's just notation for the fact that ##2\pi## divides into ##\omega_i T##, for all ##\omega_i##.