Given data:
Frequency of fundamental harmonics, f = 196 Hz
Length of string, L = 0.34 m
Linear mass density of string,
[tex]\mu=4\times10^{-3}\text{ kg/m}[/tex]
Formula of frequency of standing wave on a string is as follows:
[tex]f_1=\frac{\sqrt[]{\frac{T}{\mu}}}{2L}[/tex]
Substitute given values in above equation,
[tex]196=\frac{\sqrt[]{\frac{T}{4\times10^{-3}}}}{2\times0.34}[/tex]
Taking square of above equation,
[tex]\begin{gathered} 52245.76=\frac{T}{4\times10^{-3}} \\ T=208.98\text{ N} \end{gathered}[/tex]
Formula of velocity is as follows:
[tex]v=\sqrt[]{\frac{T}{\mu}}[/tex]
Here, T is tension
Substitute known values in above equation,
[tex]\begin{gathered} v=\sqrt[]{\frac{208.98}{4\times10^{-3}}} \\ v=\sqrt[]{52245} \\ v=228.57\text{ m/s} \end{gathered}[/tex]
Formula of frequency of nth harmonics is as follows:
[tex]f_n=nf_1[/tex]
Now, In the given case,
we have to find frequency of first three harmonics
Frequency of fundamental harmonics is given.
Hence,
Frequency of second harmonics is as follows:
Substitute values of frequency fundamental harmonics in above equation,
[tex]\begin{gathered} f_2=2\times196\text{ Hz} \\ f_2=392\text{ Hz} \end{gathered}[/tex]
Frequency of third harmonics is as follows:
[tex]\begin{gathered} f_3=3\times196\text{ Hz} \\ f_3=588\text{ Hz} \end{gathered}[/tex]
