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 $f(x) = f(x + L)$ 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]

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 $f(x) = f(x + L)$ 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?

 

[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 $\omega_i$.