Beats

When two oscillations with similar frequencies are superimposed, a phenomenon called beats occurs. For simplicity, assume that both oscillations have equal amplitude. The corresponding wave equations can be written as:

$$
y_1(t) = A \sin(2\pi f_1 t)
$$

$$
y_2(t) = A \sin(2\pi f_2 t)
$$

Here, the frequency $f$ is used instead of angular frequency $\omega$, as it is more intuitive.

The resulting signal is the sum of the two individual displacements:

$$
y(t) = y_1(t) + y_2(t) = A \left[ \sin(2\pi f_1 t) + \sin(2\pi f_2 t) \right]
$$

Using the trigonometric identity:

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

the expression becomes:

$$
y(t) = 2A \sin\left(2\pi \frac{f_1 + f_2}{2} t\right) \cos\left(2\pi \frac{f_1 - f_2}{2} t\right)
$$

This is a product of two trigonometric functions. The sine term contains the average frequency:

$$
\boxed{\bar{f} = \frac{f_1 + f_2}{2}}
$$

and the cosine term contains the difference frequency:

$$
\boxed{\Delta f = \frac{f_1 - f_2}{2}}
$$

So the full expression becomes:

$$
\boxed{y(t) = 2A \sin(2\pi \bar{f} t) \cos(2\pi \Delta f t)}
$$

The beat frequency $\Delta f$ increases with the frequency difference between $f_1$ and $f_2$.

The time-dependent behavior of such a beat is shown in the figure above. The frequency $\bar{f}$ defines the pitch of the perceived sound, while the envelope, shown as a dashed line, oscillates with frequency $\Delta f$, resulting in periodic changes in loudness.