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

In summary: The period of a periodic function is the smallest number for which its derivative is zero. In this case, the period is 2 pi.
  • #1
agnimusayoti
240
23
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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  • #2
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]?
 
  • #3
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?
 
  • #4
pasmith said:
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?
 
  • #5
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?
 
  • #6
etotheipi said:
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##.
 
  • #7
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.
 
  • #8
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?
 
  • #9
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.
 
  • #10
etotheipi said:
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
 
  • #11
agnimusayoti said:
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.
 
  • #12
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?
 
  • #13
##2\pi | \omega_i T## for all ##\omega_i##.
Wait, what does it mean?
 
  • #14
agnimusayoti said:
##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##.
 

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

1. What is the fundamental frequency?

The fundamental frequency is the lowest frequency at which a system can vibrate, and it is the basis for all other frequencies in a complex waveform.

2. How is the fundamental frequency of a combination determined using Boas's method?

Boas's method involves finding the greatest common divisor (GCD) of the frequencies in the combination. The GCD represents the fundamental frequency of the combination.

3. Can Boas's method be used for any combination of frequencies?

Yes, Boas's method can be used for any combination of frequencies, as long as the frequencies are whole numbers and not irrational numbers.

4. What is the significance of finding the fundamental frequency of a combination?

Finding the fundamental frequency of a combination is important in understanding the underlying frequencies that make up a complex waveform. It can also help in identifying patterns and predicting future frequencies in the combination.

5. Are there any limitations to using Boas's method for finding the fundamental frequency?

One limitation of Boas's method is that it only works for combinations of whole number frequencies. It also assumes that the combination is a linear superposition of pure tones, which may not always be the case in real-world scenarios.

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