Piano Tuner's Guide: Solving Beat Frequency in Tuning Fork and Piano Notes

[miyuki9]

Special Help Plz!

Somone please show me how to answer the question below and please explain to me why. thanks
Suppose a piano tuner hears 2 beats per second when listening to the combined sound from her tuning fork and the piano note being tuned. After slightly tightening the string, she hears 1 beat per second. Should she loosen or shoud she further tighten the string?
I think the answer for this one is that she should loosen the string... but I'm not exactly sure and don't know why..." Plz explain to me. Thanks

 

[J-Man]

miyuki9 said:
Suppose a piano tuner hears 2 beats per second when listening to the combined sound from her tuning fork and the piano note being tuned. After slightly tightening the string, she hears 1 beat per second. Should she loosen or shoud she further tighten the string?

Lets examine the first sentence. The key information is that there are 2 beats per second and two sources of sound. Let's define the frequency of the tuning fork as fa, the frequency of the piano string as fb, and the interference frequency (the beats) as fbeat. We are given the original frequency of fbeat; fbeat = 2 Hz.

Now let's examine the second sentence. The key information here is the string is tightened and the new fbeat frequency is 1 Hz. So to decrease fbeat we tighten the string.

At this point we need some information to relate fbeat, fa and fb. So look in your physics book for an equation with these three variables. (Big hint: one of the variables is equal to the difference of the other two.)

Now we need to think, what does it mean to "tune" a piano? It means that we want the string and tuning-fork to sound the same, in other words, they should have the same frequency. Therefore fa = fb when the piano is in tune, the desired outcome.

So we need to think some more (and look at our equation)... What should fbeat be equal to when fa = fb?
When the string was tightened, fbeat decreased, so if we tighten the string fbeat will decrease again, if we loosen it fbeat will increase. So the problem comes down to, do we need to decrease fbeat (tighten) or increase fbeat (loosen) to get the desired fbeat frequency when fa = fb?

(edit: added another bold code.)