Today in my recitation we discussed several problems in acoustics. One of them involved beats. This happens when two tones which are very close in pitch are played at the same time. There’s a demonstration on the Wikipedia article. I’ll solve the problem here since if it confused people in class there’s probably people googling it. It’s an easy problem, the difficulty comes from a lack of clarity in this section of the book.

This problem is Young and Geller 12.54:

A violinist is tuning her instrument to concert A (440 Hz). She plays a note while listening to an electronically generated tone of exactly that frequency and hears a beat frequency of 3 Hz, which increases to 4 Hz when she tightens the strong slightly. What was the frequency of her violin when she heard the 3 Hz beat?

Naively, you look up the equation and solve. It’s a simple equation, what could go wrong?

The problem is that this equation is a little confusing in that it has the possibility of a negative beat frequency. While the given equation is single-valued on its face, in reality there are two possibilities because clearly the beat frequency should be invariant under the interchange of source 1 and source 2. I think it makes much more sense to write the equation like this, and it’s what I have my students do now:

This way it’s clear that only the difference matters, and there’s no weirdness with negative numbers. For the particular problem under consideration here, it means that there’s two possible starting frequencies for the violin that produce a 3 Hz beat frequency. It could be 443 Hz or 437 Hz. Only given the beat frequency and the reference tone, that’s all we can say.

Fortunately the problem gives us a little more to work with. She tightens the string slightly and the beat frequency increases. This breaks the symmetry of the problem and let’s up find an answer. Tightening the string increases the pitch of the string. So if making the string frequency higher makes the beat frequency higher, then the distance between the two pitches is increasing. That can only happen if we were too high to begin with. Therefore the 443 Hz answer is the correct one.