Question

the wave speed of a string of length L with both ends fixed is v=600 m/s . the frequency of oscillation of the standing wave with three antinodes is f=1200 hz. find the length of the string.

Answer 1

Three anti nodes means that the string is vibrating in the 3rd (n=3) harmonic with frequency $f_3=1200\;Hz$ . The wave speed $v=600\;m/s$ does not depend on the harmonic level.

We can calculate the frequency of the fundamental harmonic (n=1) by:

$f_n=nf_1\Rightarrow f_1=\frac{f_n}{n}=\frac{f_3}{3}=\frac{1200}{3}=400\;Hz$

The fundamental frequency and velocity of wave are related by:

$f_1=\frac{v}{2L}\Rightarrow L=\frac{v}{2f_1}=\frac{600}{2\times400}=\boldsymbol{0.75\;m}$