# Beat frequency of superimposition of three sine waves

Basically we have the function

$$f(t)=\sin(2π\nu_1 t)+\sin(2π\nu_2 t)+\sin(2π\nu_3 t)$$

Let $T$ be the fundamental period of $f(t)$.

Beat frequency is defined as the number of peaks in intensity per unit time for the resultant wave ....

Which means $$f'(t)=0 = 2π\nu_1 \cos(2π\nu_1 t) +2π\nu_2 \cos(2π\nu_2 t) +2π\nu_3 \cos(2π\nu_3 t)$$

This equation is also periodic with period $T$.

So beat frequency can be defined as the number of roots of $f'(t)=0$ where $t \in [0,T)$ divided by T.

Is there any simple way of calculating this? (I don't need a formula ... A simple algorithm will also do)

Does it help if all three frequencies are natural numbers (because that's usually the case when dealing with tuning fork related problems)?

I tried considering three vectors of lengths $\nu_1, \nu_2,\nu_3$. The vectors are rotating with angular speeds $2π\nu_1,2π\nu_2,2π\nu_3$.

If the frequencies are natural numbers I think $T=1/\gcd (\nu_1,\nu_2,\nu_3)$

Now we need to find the number of times the cos component of the vector sum of the rotating vectors gives zero.

My teacher had a weird but simple way of doing this ... He said that all 3 vectors must either be parallel, or 2 vectors parallel and 1 antiparallel, or the 3 vectors should form a triangle.

I didn't completely understand the method or the reasoning behind it.

Does such a method indeed exist?

• Your teacher was using the idea of a phasor with the phasors rotating at different angular speeds. en.wikipedia.org/wiki/Phasor – Farcher Apr 20 '18 at 9:41
• @Farcher I do have some idea about phasors and I've been using them to try solving it myself. What I didn't understand is the exact solution to this question. – ghosts_in_the_code Apr 20 '18 at 9:43
• Wave beats - physicskey.com/40/wave-beats-and-doppler-effect – Saral Dec 29 '18 at 6:42
• @Saral that's just for two beats :( not relevant – ghosts_in_the_code Dec 29 '18 at 6:51

Using phasors can be useful to illustrate what happens but for three frequencies which are not simple multiples of one another an algebraic approach will probably be easier.

To start with let me use the phasor method for 2 frequencies of $1.0\, \rm Hz$ and $1.1\, \rm Hz$ where after one second the phase between the two phasors changes by $36^{\circ}$ as illustrated below.

Which as a graph of displacement against time looks like with a beat frequency of $0.1 \,\rm Hz$.

Three frequencies is more difficult and I have chosen a relatively simple example with the three frequencies, $1.0\, \rm Hz, \, 1.1\, \rm Hz$ and $1.2\, \rm Hz$

What your teacher told you to look for are additions of the type shown below.

If you want to analyse more complex groups of frequencies it might help you if you got WolframAlpha or some similar package to plot the graphs out for you as I did for the three frequencies that I chose.

• Thanks, those diagrams at the end are exactly what he was talking about. However you have not actually provided a method of finding the beat frequency .... Does that mean there is no simple method to do it? – ghosts_in_the_code Apr 20 '18 at 11:48
• @ghosts_in_the_code It all depends on the values of the three frequencies and how they are related to one another. That is why I suggest that you investigate further and see how it might be done. – Farcher Apr 20 '18 at 11:52
• I tried googling as well as solving it myself, both failed hence I posted it here. – ghosts_in_the_code Apr 20 '18 at 12:03
• @ghosts_in_the_code So what have you tried so far? – Farcher Apr 20 '18 at 12:06
• After seeing tfb's answer I realise we can multiply and divide the expression for f(t) by 2, then apply the result for superposition for two waves pairwise .... This gives the expression $f_b (t)$ in tfb's answer. But even that doesn't help me figure out how to calculate the beat frequency. Is the beat frequency the LCM of the pairwise differences of the original frequencies? – ghosts_in_the_code Apr 23 '18 at 9:05

I'm pretty convinced the result is that there may not be a beat frequency: the thing can be aperiodic.

In particular consider three frequencies, $f_1$, $f_2$, $f_3$. Then it's easy to know what the beat frequency between any pair of these is, and in particular

\begin{align} f_{1,2} &= |f_1 - f_2|\\ f_{1,3} &= |f_1 - f_3|\\ f_{2,3} &= |f_2 - f_3| \end{align}

($f_{i,j} = f_{j,i}$ of course). But now we need to know when, and if, the combined waveform repeats. So the beat waveform $f_b(t)$ is something like

$$f_b(t) = a\sin(2\pi f_{1,2} t) + b\sin(2\pi f_{1,3}t) + c\sin(2\pi f_{2,3}t)$$

and this is periodic only if there is a number $x$ such that $x = lf_{1,2} = mf_{1,3} + nf_{2,3}$ for some $l,m,n \in \mathbb{N}$. And that's not generally true. So, in general, the waveform is aperiodic.

(Note that this is still true considering just any pair of the beat frequencies, and an earlier version of this answer did that.)