Sharedmaster / Sine_waves.sagewsOpen in CoCalc

# Sine waves

### Declare variables

var('t')

t

## Sine functions

$y = \sin{(2\pi t)}$

plot(sin(2*pi*t), (t, -1, 2), ymin = -2, ymax =  2)


$y = \sin{(2\pi t)} + \sin{(2\pi t)}$

show(plot(sin(2*pi*t) + sin(2*pi*t), (t, -1, 2), ymin = -2, ymax =  2) +
plot(sin(2*pi*t), (t, -1, 2), ymin = -2, ymax =  2,  color = 'red', thickness = 0.5))


$y = \sin{(2\pi t)} + \sin{(2\pi t - 2\pi(1/2))}$

show(plot(sin(2*pi*t) + sin(2*pi*t - 2*pi*(1/2)), (t, -1, 2), ymin = -2, ymax = 2) +
plot(sin(2*pi*t), (t, -1, 2), ymin = -2, ymax = 2, color = 'red', thickness = 0.5) +
plot(sin(2*pi*t - 2*pi*(1/2)), (t, -1, 2), ymin = -2, ymax = 2, color = 'green', thickness = 0.5))


$y = \sin{(2\pi t)} + \sin{(2\pi (2t))}$

show(plot(sin(2*pi*t) + sin(2*pi*(2*t)), (t, -1, 2), ymin = -2, ymax = 2) +
plot(sin(2*pi*t), (t, -1, 2), ymin = -2, ymax = 2,  color = 'red', thickness = 0.5) +
plot(sin(2*pi*(2*t)), (t, -1, 2), ymin = -2, ymax = 2,  color = 'green', thickness = 0.5))



$y = \sin{(2\pi t)} + \frac{1}{3}\sin{(2\pi (3t))}+ \frac{1}{5}\sin{(2\pi (5t))} + \frac{1}{7}\sin{(2\pi (7t))}$

show(plot(sin(2*pi*t) + (1/3)*sin(2*pi*(3*t)) + (1/5)*sin(2*pi*(5*t)) + (1/7)*sin(2*pi*(7*t)), (t, -1, 2), ymin = -2, ymax = 2) +
plot(sin(2*pi*t), (t, -1, 2), ymin = -2, ymax = 2, color = 'red', thickness = 0.5) +
plot((1/3)*sin(2*pi*(3*t)), (t, -1, 2), ymin = -2, ymax = 2, color = 'red', thickness = 0.5) +
plot((1/5)*sin(2*pi*(5*t)), (t, -1, 2), ymin = -2, ymax = 2, color = 'red', thickness = 0.5) +
plot((1/7)*sin(2*pi*(7*t)), (t, -1, 2), ymin = -2, ymax = 2, color = 'red', thickness = 0.5))


## Beats

$y = \sin{(2\pi (22t))} + \sin{(2\pi (20t))}$

plot(sin(2*pi*(22*t)) + sin(2*pi*(20*t)), (t, -1, 2), ymin = -2, ymax = 2,  thickness = 0.5)


### Beat derivation

Here is a useful trigonometric identity:

$\sin(a) + \sin(b) = 2 \sin{\left( \frac{a + b}{2} \right)} \cos{\left( \frac{a - b}{2} \right)}.$

A more convenient formulation is

For example, suppose we have two frequencies that are close together: one at 20 Hz and the other 2 Hz higher (so 22 Hz). According to the above formula, with $f = 20$ and $\epsilon = 2$:

Think of this as $\Big[ \sin{\left( 2\pi (21t) \right)} \Big] \Big[ 2 \cos{\left( 2\pi t \right)} \Big].$

The blue graph below is $\sin(2\pi (21t))$. The red graph is $2 \cos(2\pi t)$. (The dotted curve is just $-2 \cos(2\pi t)$.)

show(plot(sin(2*pi*(21*t)), (t, -1, 2), ymin = -2, ymax = 2, thickness = 0.5) +
plot(2 * cos(2*pi*t), (t, -1, 2), ymin = -2, ymax = 2,  color = 'red',  thickness = 2) +
plot(-2 * cos(2*pi*t), (t, -1, 2), ymin = -2, ymax = 2,  color = 'red', thickness = 2, linestyle="--"))

show(plot(sin(2*pi*(22*t)) + sin(2*pi*(20*t)), (t, -1, 2), ymin = -2, ymax = 2, thickness = 0.5) +
plot(2 * cos(2*pi*t), (t, -1, 2), ymin = -2, ymax = 2,  color = 'red',  thickness = 2) +
plot(-2 * cos(2*pi*t), (t, -1, 2), ymin = -2, ymax = 2,  color = 'red', thickness = 2, linestyle="--"))


Therefore, in addition to the 21 Hz tone (the average of the two frequencies, 22 Hz and 20 Hz), there will be a perceptible beat every half-second—in other words, a 2 Hz beat.

That last fact needs a little explanation. The function $2 \cos(2\pi t)$ has a frequency of 1 Hz, so why are the beats heard at 2 Hz? If you look at the graph above, you can see that the 21 Hz wave "fills up" both halves of the cosine graph, so it repeats every half-cycle of the cosine graph.

The net effect is that the beats are heard at the freuency $\epsilon$, and here, $\epsilon = 2$.