In the figure shown a sinusoidal wave is generated at the end `'A'`. The wave travels along positive `x-`axis and during its motions it encounter another string `BC` at the junction `'B'` at `x = 0`. The density of strings `AB` and `BC` and `rho` and `9rho` respectively and their radii of cross sections are `2r` and `r`. The wave funcation. amplitude and wavelength of incident wavs are respectively `y_(r), A_(r), lambda_(r)` and `y_(t), A_(t), lambda_(t)` repectively

The ratio of amplitudes `A_(r)` to `A_(t)` is (i.e. A_(r) : A_(t)`) will be

Options

A. `Deltaphi = 0`, between `y_(i)` and `y_(r)`
B. `Deltaphi = 0`, between `y_(r)` and `y_(t)`
C. `Deltaphi = pi`, between `y_(i)` and `y_(t)`
D. `Deltaphi = pi`, between `y_(r)` and `y_(t)`

Answer

D

Explanation

Medium of string `BC` is denser. These will be `pi` phase difference between reflected and transmitted wave.
`(lambda_(r))/(lambda_(t)) = (v_(1))/(v_(2)) = sqrt((mu_(2))/(mu_(1)))`
Now `(mu_(1))/(mu_(2)) = (4)/(9)` so `(lambda_(r))/(lambda_(t)) = (3)/(2)`
`(A_(r))/(A_(t)) = ((v_(2)-v_(1))/(v_(1)+v_(2))Ai)/((2v_(2))/(v_(1)+v_(2)).Ai)=(v_(2)-v_(1))/(2v_(2))=(1)/(4)`