Q.The string of a guitar is 80 cm long and has a fundamental frequency of 112 Hz. If a guitarist wishes to produce a frequency of 160 Hz, where should the person press the string?

[!TLDR] Wave speed on the string: $v=2L_0n_0=2(0.8)(112)=179.2$ m/s (fixed for the same wire/tension). New length for 160 Hz: $L'=v/2n'=179.2/320=0 …

The guitar string's full length is $L_0=80$ cm $=0.8$ m, with fundamental frequency $n_0=112$ Hz. From $n_0=\dfrac{v}{2L_0}$ (a string fixed at both ends, fundamental mode), the wave speed on the string is $v=2L_0n_0=2\times0.8\times112=179.2$ m/s -- this speed depends only on the string's own tension and linear density (via $v=\sqrt{T/m}$), NEITHER of which changes when the guitarist presses the string at a different point, so v stays exactly 179.2 m/s throughout. …

Find the constant wave speed v on the string from its original fundamental-mode length and frequency, then use $v=2L'n'$ to sol …

Assuming the wave speed v changes when the vibrating length changes (it does not -- v depends only on tension and linear density, both unaffected by where the string is presse …