When tightening a guitar string, what happens to velocity, wavelength, and frequency? [closed]

Question

I'm thinking about the following physics problem. (I'm a freshman in high school.)

A guitarist plucks a string, producing string wave #1 (fundamental vibration) in the string. He then tightens its tension and plucks again, producing string wave #2 (fundamental vibration). Compare the following factors for string wave #1 vs string wave #2:

• velocity
• wavelength
• frequency
• which string wave creates the sound wave that makes your ear drum vibrate the fastest?

I think the velocity should increase after tightening, because the velocity of a wave on a string is $\sqrt{T/\mu}$, and tightening increases $T$.

I think the frequency should increase, too, but that's only based on empirical evidence: I know that when someone tightens a guitar string and then plucks again, the pitch is higher. Is there any way to justify my reasoning besides appealing to this empirical evidence?

I'm not sure what to say about wavelength. I know $v=\lambda f$, but if $v$ and $f$ increase, I can't conclude anything about $\lambda$. If $v$ and $f$ increase by the same factor, then $\lambda$ stays constant; if they increase by different factors, then $\lambda$ could increase or decrease. So I don't know what to think about the wavelength.

I think the last question is asking about which sound wave has a higher frequency. Since the frequency of a string wave is the same as the frequency of the induced sound wave, the second string wave should create the sound wave that makes my ear drum vibrate faster, right?

Answer 1

As you tighten the guitar string, the length of the string does not change; therefore the wavelength of the standing wave on the string that is producing the sound is the unchanged.

A string stretched between two points, such as on a stringed instrument, will have tension. Tension refers to how tightly the string is stretched. Tightening the string gives it a higher frequency while loosening it lowers the frequency.

Tightening the strings on a guitar increases the tensile force on the string. This force limits the amount of movement the string can make when you pluck it, and as such, the distance it vibrates decreases (which translates into a higher frequency of vibration). This higher frequency of vibration means that the air particles around the string carry a higher frequency sound wave to your ear, which perceives the higher frequency as a higher pitch.

Hope this helps.