Melde's Exp: Why Transverse & Longitudinal Arrangements Differ?

[TachyonLord]

Homework Statement

Why working formula for transverse and longitudinal arrangement in melde experiment different in Melde's experiment ?

Homework Equations

None. The corresponding equations are all derived from the same fact.

The Attempt at a Solution

So, I have understood that the tuning fork and the string complete one oscillation simultaneously in tranverse mode whereas its half the oscillation in longitudinal mode, but why is it so ? What causes the longitudinal oscillations to have a lower frequency ?

 

[TSny]

The standing wave of the string is driven by the vibration of the prong of the tuning fork. Note that the standing wave of the string is a transverse wave for both the transverse and longitudinal arrangements of the the tuning fork.

When a string vibrates transversely, the tension varies a little during the vibration due to the fact that the string must stretch a little as it displaces from equilibrium.
(1) During each full cycle of vibration of the string, how many times does the tension reach a maximum?

Suppose the prong of the fork vibrates transversely to the line along which the string is stretched.
(2) For each full cycle of vibration of the prong, how many times does the prong tend to increase the tension in the string?

Now suppose the prong of the fork vibrates parallel to the line along which the string is streched.
(3) For each full cycle of vibration of the prong, how many times does the prong tend to increase the tension in the string?