Two waves on the string of a musical instrument superposition to give you a
third wave. One wave is given by the equation $y =\sin(30t)$ and the other is given by
$y = \sin(32t)$. The superposition of the two waves makes a sound (a hum) of a certain
frequency, the amplitude of which varies periodically, making beats. What is the frequency
of the hum and what is the frequency of the beats that you hear? Assume that time is given
in seconds.
I don't know how to approach this computationally, but this is the graph of the superposition function $\sin(30t)+\sin(32t)$.
My understanding, if correct, is that each individual peak and trough implies finding the frequency of the beats, whereas the periodicity of the "larger" sinusoidal wave represents the frequency of the hum.
How can I approach this computationally? To find the beat frequency, do I set the superposition function equal to $0$ and then solve? For the hum frequency, equal to $1$? Or what should I do?
Best Answer
Use a trig formula to combine the terms:
$$sinx+siny = 2sin((x+y)/2)cos((x-y)/2)$$
In our case we get:
$2sin(31t)cos(1t) %and the amplitude is between %[-2,2]$
If I'm not mistaken, the beat frequency is given as $f_{beats}=|f_1-f_2|=2$. The cos is responsible for the "beat effect", and is the envelope for the other wave, which is the hum. The frequency is 2*pi*31.