Guitar string - standing wave question

[thejoyofmeth]

This question is for a conceptual physics class (no trig involved).

1. The wave speed on a tightened guitar string is 880 m/s. What is the shortest length of string that will produce standing waves of 440-hertz frequency? (Be very CAREFUL!)

a. 0.5m
b. 1.0m
c. 1.5m
d. 2.0m
e. none of these


2. velocity = frequency x wavelength


3. 880/440 = 2m


Is 2m correct? I ask because of the warning the instructor placed at the end of the question leads me to believe that the answer isn't so straightforward. If it is incorrect, can you offer any pointers as to what I did wrong or did not take into consideration?

Thanks




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[stevenb]

Is 2m correct? I ask because of the warning the instructor placed at the end of the question leads me to believe that the answer isn't so straightforward. If it is incorrect, can you offer any pointers as to what I did wrong or did not take into consideration?

First, ask yourself if you've ever seen a guitar with 2 meter strings. That's a clue.

What formula are you trying to apply? That's another clue?

What is the correct formula for frequency as a function of length? What other things does the formula depend on?

If I had a 0.5 meter string that produced an A note (440 Hz), why couldn't I just tune the note down in pitch slightly and then put my finger on the first fret and get an A note with a shorter length?

 

[thejoyofmeth]

First, ask yourself if you've ever seen a guitar with 2 meter strings. That's a clue.

What formula are you trying to apply? That's another clue?

What is the correct formula for frequency as a function of length? What other things does the formula depend on?

If I had a 0.5 meter string that produced an A note (440 Hz), why couldn't I just tune the note down in pitch slightly and then put my finger on the first fret and get an A note with a shorter length?

Christ, I don't know. I rode the short bus to school and so a lot of this isn't quite obvious to me. Is it 1 meter?

 

[stevenb]

Christ, I don't know. I rode the short bus to school and so a lot of this isn't quite obvious to me. Is it 1 meter?

OK, let's start with some guitar and music facts, which aren't really needed to answer the question. Most guitars have string length just over 0.6 meters (about 25 inches). The 440 Hz is an A note which is the fifth fret (3/4 of the scale length) on the string with highest notes. This is the first string, which is tuned to an E. So right here, we see a real guitar has an A note with length less than 0.5 meters. (OK real guitars may not correspond with the question exactly, but it gives you a feel)

Now, you need to come up with a theoretical explanation based on physics.

The longest string which can be tuned to 440 Hz will be based on the material strength. For example, steel will break somewhere around 0.7 meters with 440 Hz. This is why you don't see standard length guitars with a string tuned to 440 Hz, even though some musicians would like to have that. Some musicians try to use thinner and thinner strings thinking that the high A note can be obtained with 0.625 meter length and reasonable string tension, but the string always breaks after a short amount of time. This is the material strength limit. (OK, so the practical limits may or may not apply, but again it gives you a feel)

Since you are being asked this question, you must be responsible for knowing the relevant formulas here. Put them together and see if you can prove the above practical knowledge mathematically. What does the sound speed tell you about the tension, and how can this be related to the frequency and length. Have you studied strength limits and do you think the question is aimed at that aspect?