EK Physics #690 - Beat Frequency (Question and Answer included)

[ilovemcat]

Question 691: A piano tuner plays an out of tune A note on his piano and then strikes his 440 Hz tuning fork. He notices a beat of 2 Hz. When he tightens the piano string and plays the note again, the beat remains at 2 Hz. What was the frequency of the note before he tightened the string? (Note: The A note is 440 Hz).

A. 438 Hz
B. 440 Hz
C. 442 Hz
D. 444 Hz




The issue I'm having with this problem is the relationship between velocity with wavelength and frequency. The book solution explains, "When the piano string was tightened, the velocity of the wave, and thus the frequency, must have increased. The wavelength remains constant because the string length is not changed."

I have no problem solving this question once I realized that frequency responds to to the increase in velocity. However, what I'm having trouble with is reaching this conclusion. How exactly can this be true. Normally we are told to assume that for a given wave moving from one medium to another, frequency stays the same but wavelength changes (increases/decreases). But here we have a situation similar to what we'd see in a standing wave problem, where waves can only occupy fixed wavelengths on a given wave. How than could we reasonably explain that wavelength doesn't respond to a change in velocity (ie. a different harmonic wavelength) but that frequency respondes to the change instead? Why cant wavelength change ...or both even?

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[ilovemcat]

The answer was A if anyone cares.

 

[Rabolisk]

If you know anything about music and stringed instruments, tightening a string raises its pitch, and the only answer that makes sense is that his note changed from 438 Hz to 442 Hz. Tightening a string is analogous to increasing the stiffness of the spring; both increases the restoring force and thus the frequency.

Another way to raise pitch is to shorten the string, which is what a guitarist does when he puts his fingers down. He effectively shortens the string, which decreases wavelength and increases frequency. For standing waves, wavelength can only change if the length of the medium (e.g. string) changes. This is a very easy question if you are familiar with music, but it's somewhat hard to explain to a non-music person, unfortunately.

 

[ilovemcat]

If you know anything about music and stringed instruments, tightening a string raises its pitch, and the only answer that makes sense is that his note changed from 438 Hz to 442 Hz. Tightening a string is analogous to increasing the stiffness of the spring; both increases the restoring force and thus the frequency.

Another way to raise pitch is to shorten the string, which is what a guitarist does when he puts his fingers down. He effectively shortens the string, which decreases wavelength and increases frequency. For standing waves, wavelength can only change if the length of the medium (e.g. string) changes. This is a very easy question if you are familiar with music, but it's somewhat hard to explain to a non-music person, unfortunately.

Even though I barely use musical instruments myself (sad I know), I was able to reason the changes you mentioned using equations I learned in EK. But those are some great examples you gave which really help with my understanding, so thanks!

I have another question if you don't mind. In EK 1001 they explain the idea of having an instrument that has all the harmonics combined for a particular tone. How exactly does this work?

Here's a little intro to a group of questions they gave:

"When an instrument plays a note, the resulting sound is a combination of all the possible harmonics for that instrument in its momentary configuration. For instance, a musician changes notes on a violin by pressing the strings against the neck of the instrument, thus shortening the string length and changing the possible harmonics. A given shortened string will play at one time all the possible harmonics allowable by its string length. A given note is the same set of harmonics for all instruments." Then they give a diagram showing sound waves of the same note for 3 different instruments.

So if we look at the equations for a standing wave:
2L/n = wavelength

I'm guessing that as the string shortens, wavelength decreases gradually which in turn results in a decrease in wavelength (and increase in frequency). But wouldn't changing the length result in a different set of harmonic wavelengths... How exactly can shortening a string account for all possible harmonics. I'm a little confused with what they mean and can't seem to find any online source that could elaborate on this. You think you can help me figure this one out? Thanks

 

[Rabolisk]

What they mean is that when a string is shortened, the harmonic wavelengths and frequencies change, and the note that is played (i.e. the sound heard) is a combination of all the harmonics associated with that length. (not just the first harmonic, although that is the most dominant). I think that's the best I could explain. Someone smarter than me could probably do better.

 

[ilovemcat]

What they mean is that when a string is shortened, the harmonic wavelengths and frequencies change, and the note that is played (i.e. the sound heard) is a combination of all the harmonics associated with that length. (not just the first harmonic, although that is the most dominant). I think that's the best I could explain. Someone smarter than me could probably do better.

Thanks.

 

[obliquemoth]

Hey guys,

I realize that this is a very late reply, but hopefully this answer will help other people who may have come across this question in the EK physics practice book.

For a string of a given length L, the set of all harmonic wavelengths will not change. L = n*(λ/2) must be used in this problem, since both ends of a piano string's ends are fixed (i.e. they are nodes). Geometrically speaking, only full crests and troughs can comprise standing waves, due to the fixed ends. This is because a node (point of zero vertical displacement) can only occur 1) halfway through a wave's period or 2) at the end of a full period. In other words, the standing wave can only accommodate integral multiples of λ/2 that allow both ends of the string to be nodes. As long as L is constant (which it is if you're just tuning the string by tightening or loosening it), solving for the harmonic wavelengths using n = {1, 2, 3...} will yield the same set of λs every time.

However, there are other ways to change the pitch of a string's vibrations without modulating wavelength. If we remember that v = λ*f and v = sqrt(tension/&#956, we can see that tightening the string (i.e. increasing tension) will cause v to increase (since a stiffer medium allows waves to propagate more rapidly). However, in order to continue fulfilling the relationship v = λ*f without changing the wavelength λ, frequency f must also increase.

Moving on to the rest of the problem: before the string was tightened, we knew that the beat frequency was 3 Hz, but we didn't know whether this was due to a string frequency that was lower or higher than the tuning fork frequency. If we had tightened the string and heard a decrease in the beat frequency, we would have been able to infer that the absolute value difference between the tuning fork and string frequencies was becoming smaller; this can only be true if the original string frequency was smaller than the tuning fork frequency. However, since we hear an increase in the beat frequency, that means that the absolute value difference between the tuning fork and string frequencies is getting larger, which can only happen if the original string frequency was greater than the tuning fork frequency to begin with.

Hope this helps.