Question

I don't understand how to use beats using sum-to-product identity, or this assignment at all.

Answer 1

When two waves are in phase they reinforce each other and when they are out of phase they tend to cancel each other out. When two waves that have nearly but not exactly the same frequency combine they might start off reinforcing each other, but as time passes the difference in frequency means that they start to cancel, and then later still they reinforce each other again.

Consider the combination of a 50 Hz tone and a 55 Hz tone In a second the overall amplitude reaches a maximum five times. Listening to the combination one hears a single tone half-way between 50 and 55 Hz that appears to vary in loudness five times a second, i.e. at 5 Hz. You don't hear two separate tones one at 50 Hz and one at 55 Hz.

The 5 Hz tone is called a beat frequency.

The sum to product identities are useful for modeling what happens with sound frequencies. Think of two different tones represented by sine curves. Add them together, and they beat against each other with a warble — how much depends on their individual frequencies. The identities give a function modeling what’s happening.

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