Beat frequency for 3 waves 
Consider 3 waves of frequency 101, 103, 106 hz, and of same intensity. What should be the beat frequencies?  

Now I can calc it for 2 waves, and i know how to write the combined equation of the two. But the addition of a third causes lots of problems...the equation is getting cumbersome. Can you please help me here? I dug around online, and some responses were neglecting the third wave as it was too close to one of the waves. If I want to avoid that, is there any other solution? 
 A: A simple Python script gives some insight - here is what I calculate for the sum of the three frequencies:

There is obviously a strong beat at 1 Hz - this is when all three frequencies are (back) in phase. There are minor peaks in between - of which I would consider 5 Hz the most visible component. 
This matches intuition - you would expect the difference frequencies of 2 Hz, 3 Hz and perhaps 5 Hz to show up, but they will be muddled. And sure enough, 5 Hz is there; the two bigger peaks around 0.5 s are indicative of the 2 Hz signal; while the 3 Hz signal cannot be separately seen.
Here is the code I used to plot this graph:
# beat frequencies
import numpy as np
import matplotlib.pyplot as plt
from math import pi

f = [101,103,106];
Ns = 8*1024
t = np.linspace(0,2,Ns);
a = np.zeros((Ns),'double')
for fi in f:
    a = a + np.cos(2*pi*fi*t)

plt.figure(figsize=(10,5))
plt.plot(t, a)
plt.title('three frequencies beating')
plt.xlabel('time (s)')
plt.ylabel('amplitude')
plt.show()

A: The waves with separately produce beat with frequencies of $2, 3$ and $5$ $Hz$.
These beats have time periods $1/2, 1/3,$and $ 1/5 s$, and beats can be heard after these time intervals. Their LCM is $1 s$. So, after $1 s$, the whole wave will have the same phase and amplitude, and there will be a strong beat. The frequency of this strong beat is $1 Hz$.
See the other answer for a numerical verification.
