Question

A uniform rope hangs vertically from a rigid support with a slab of mass 2 kg attached to the free end of the rope. The rope has a length of 12 m and a mass of 6 kg. A transverse pulse of wavelength 0.06 m is produced at the free end of the rope. Determine the wavelength of the pulse when it reaches the top of the rope?

Answer 1

 The wave velocity will be where F is the tension in rope at a point and µ is mass per unit length of the string. As F is varying along the length of the rope so the velocity will vary along the length of the rope. As source frequency is constant λ will vary. As the rope is stretched using a slab, its tension will be different at different points along the length of the rope. The tension at the free end will be (2 kg) g while at the upper end it will be (8kg) g.

Since the frequency of the wave pulse is dependent only on the frequency of the source, it will be consistent across the length of the rope. The mass per unit length will also be consistent for the entire rope as the rope is uniform. Thus, by 

Where λ₁ is the wavelength at the top of the rope, this gives λ₁ =0.12m.