I have a question about standing waves on strings. I'll try to explain the best I can, I searched and researched the whole day yesterday but I am confused still:
- Every frequency has a single, and only one possible wavelength. This is what I come across over and over again, with no other conditions mentioned. eg. E2 is 82.407hz with a wavelength of 4.129m. The wavelength seems to be referenced as a constant, nonchanging measurement. Is this correct?
- The above is proven false with simple tuning of the guitar, where the standing wave is clearly the same length (the half-length of it from nut to bridge, from node to node) yet the frequency is changed by increasing tension. The same length of the string produces for example fundamental pitch of the note E2 and F2. Does this prove the above statement false?
- Does this mean that the wavelength of the standing wave and the wavelength of the sound that it produces can be different?
EDIT: For simplicity's sake, unless it's relevant, I am talking about the fundamental only. Also, I am assuming that the speed of sound is the same, through the air, same temperature same everything, the string are the same etc.
It is really puzzling to me how seemingly same wavelengths can create different frequency on the same guitar, with the same string. And further how a guitar with a shorter nut-to-neck distance can create the same frequency. Thanks a lot in advance for any help figuring this out.
Best Answer
You overlooked the fact the speed varies when tension varies. Note that since $v=f\lambda$ a variation in both $v$ and frequency $f$ can mean wavelength $\lambda$ stays constant.
Note that the speed of the wave on a string is given by $$v=\sqrt{\frac{T}{\mu}}$$ where $T$ is the tension in the string and $\mu$ is the mass per unit length (linear mass density).
When you increase the tension in the string, the wave speed increases. This also means that if you decrease the tension, the speed decreases. In either case, the wavelength remains constant.
More specifically, as the tension increases, so does the speed and frequency (so you hear a higher pitch), but the wavelength remains constant. Of course the converse is true, and as the tension decreases, so does the speed and frequency (so you hear a lower pitch), but again the wavelength remains constant.
The point is, even though the string wavelength always stays the same, the frequency need not.