Transverse arrangement:
The fork is placed in the transverse position and the lengthy of string is fixed. By changing the load in the pan, well-defined loops are formed. This is due to the formation of stationary waves due to the superposition of waves from the prong and the reflected waves from the pulley. Well-defined loops are formed when the frequency of each segment coincides with the frequency of the fork. The frequency can be given by
$$\eta = \frac{1}{2l}\sqrt{\frac{T}{\mu}}$$Where,
\(\eta\) = Frequency of Tuning Fork,
\(l\) = Length of each loop,
\(T\) =Tension applied,
\(\mu\) = Mass per unit length of string.
When the fork is placed in the longitudinal position and the string vibrates, the frequency of the stretched string will be half of the frequency η of the tuning fork. That is, when the well-defined loops formed on the string, the frequency of each vibrating segment is exactly half of the frequency of the tuning fork.
During longitudinal vibrations, when the prong is in its right extreme position, the string corresponding to a loop gets slackened and it moves up to its initial horizontal position and becomes light. But when the prong is again in its right extreme position, it completes one vibration. So, the string goes up; its inertia carrying it onward and thereby completing half a vibration. The frequency of each loop is $$\eta = \frac{1}{2l}\sqrt{\frac{T}{\mu}}$$
And the frequency of tuning fork is $$\eta = \frac{1}{l}\sqrt{\frac{T}{\mu}}$$
Where,
\(\eta\) = Frequency of Tuning Fork,
\(l\) = Length of each loop,
\(T\) =Tension applied,
\(\mu\) = Mass per unit length of string.