Finding the Fundamental Frequency of a Combination: Boas's Method

[agnimusayoti]

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.

 

[pasmith]

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

If [itex] k > 1[/itex], is [itex]L/k[/itex] shorter or longer than [itex]L[/itex]?

 

[etotheipi]

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?

 

[agnimusayoti]

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

If [itex] k > 1[/itex], is [itex]L/k[/itex] shorter or longer than [itex]L[/itex]?

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

 

[agnimusayoti]

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?

 

[agnimusayoti]

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##.

 

[agnimusayoti]

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.

 

[agnimusayoti]

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]

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.

 

[agnimusayoti]

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]

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.

 

[agnimusayoti]

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?

 

[agnimusayoti]

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

 

[etotheipi]

##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##.