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). Verify that f(t) = 2 cos(t) cos(12t).

Answer 1

Beats occur when two waves of similar frequencies are superimposed and result in a fluctuating amplitude called the beat frequency. To verify the resulting sound of two waves f1(t) = cos(11t) and f2(t) = cos(13t), we can manipulate the equation using trigonometric identities.

Step-by-step explanation:

Beats occur when two waves of similar frequencies are superimposed. The resulting wave fluctuates in amplitude, with a frequency called the beat frequency. To verify that f(t) = 2 cos(t) cos(12t) is the resultant sound of f1(t) = cos(11t) and f2(t) = cos(13t), we can use trigonometric identities and manipulate the equation.

1. Start with f(t) = cos(11t) + cos(13t).
2. Apply the cosine sum formula: f(t) = 2cos((11t+13t)/2)cos((13t-11t)/2).
3. Simplify: f(t) = 2cos(12t)cos(t).