Transverse waves are being generated on a rope under constant tension. Determine the factor by which the required power is changed in each of the following situations.

(a) The angular frequency is doubled, and the amplitude is halved.

final/initial =

(b) Both the wavelength and the amplitude are halved.

final/initial =

(c) The length of the rope is doubled, and the angular frequency remains constant.

final/initial =

(d) Both the wavelength and the length of the rope are halved.

final/initial =

Explanation

Energy of a wave is proportional to square of both frequency and amplitude.

a)

if frequency is doubled and amplitude is halved, energy and therefore power will remain constant. so required ratio = 1

b)

If wave length is halved, frequency will be doubled. Hence, effect will be similar to first case, hence required ratio = 1

c) If angular frequency remains constant, power will remain constant. Hence required ratio = 1

d)

If wavelength is halved, frequency is doubled, hence power will be 4 times. Hence required ratio = 4.