Question

When two pure notes that are close in frequency are played together, their sounds interfere to produce beats; that is, the loudness (or amplitude) of the sound alternately increases and decreases. If the two notes are given by f1(t) = cos(11t) and f2(t) = cos(13t) the resulting sound is f(t) = f1(t) + f2(t). Graph y = 2 cos(t) and y = −2 cos(t), together with the graph in part (a), in the same viewing rectangle. How do these graphs describe the variation in the loudness of the sound?

Answer 1

When two pure notes with close frequencies are played together, their sounds interfere and produce beats, which result in a variation in the loudness of the sound. Graphing the individual amplitudes of the notes and their combination can depict the fluctuation in loudness.

Step-by-step explanation:

When two pure notes that are close in frequency are played together, their sounds interfere to produce beats, which is a variation in loudness. Beats occur due to the superposition of waves with slightly different frequencies but the same amplitude. In this case, the two notes f1(t) = cos(11t) and f2(t) = cos(13t) are given, and the resulting sound is f(t) = f1(t) + f2(t).

The loudness of the resulting sound from the combination of these two notes can be described by graphing y = 2 cos(t) and y = -2 cos(t), which represent the amplitudes of the individual notes. The graph of f(t) will show alternating constructive and destructive interference, resulting in a fluctuation in loudness. The peaks and troughs on the graph represent the increase and decrease in loudness respectively.