The displacement of the medium in a sound wave is given by the equation `y_(1) = A cos(ax + bt)` where `A`, `a and b` are positive constants. The wave is reflected by an obstacle situated at `x = 0`. The intensity of the reflected wave is `0.64` times that of the incident wave.
(a) What are the wavelength and ffrequency of incident wave?
(b) Write the equation for the reflected wave.
( c ) In the resultant wave formed after reflection, find the maximum and minimum values of the particle speeds in the medium.
(d) Express the resultant wave as a superposition of a standing wave and a travelling wave. what are the positions of the antinodes of the standing wave ? What is the direction of propagation of travelling wave?

Answer

A, B, C, D

Explanation

(a) KEY CONCEPT : Use the equation of a plane progressive wave which is as follows.
`y = A cos((2pi)/(lambda)x + 2pivt)`
The given equation is
`y-(1) = A cos (ax + bt)`
On comparing, we get `(2pi)/(lambda) = a` rArr `lambda = (2pi)/(a)`
Also, `2piv = b`
rArr `v = (b)/(2pi)`
(b) Since the wave is reflected by an obstacle, it will suffer a phase difference of `pi`. The intensity of the reflected wave is `0.64` times of the incident wave.
Intensity of orginal wave `I prop A^(2)`
Intensity of reflected wave `I' = 0.64 I`
rArr `I' prop A'^(2)` rArr `0.64 I prop A'^(2)`
rArr `0.64 A^(2) prop A'^(2)` rArr `A' prop 0.8A`
So the equation of resultant wave becomes
`y_(2) = 0.8A cos (ax - bt + pi) = -0.8 A cos (ax - bt)`
( c ) KEY CONCEPT : The resultant wave equation can be found by superposition principle
`y = y_(1) + y_(2)`
`= A cos (ax + bt) + [ - 0.8 A cos (ax - bt)]`
The partcle velocity can be found by differentiating the above equation
`v = (dy)/(dt) = - Ab sin(ax + bt) - 0.8 Ab sin (ax - bt)`
`= - Ab [sin (ax + bt)+ 0.8 sin (ax - bt)]`
`= -Ab [sin axcos bt+ cos ax sin bt + 0.8 sin ax cos bt - 0.8cos ax sin bt]`
`v = -Ab[1.8 sin ax cos bt + 0.2 cos ax sin bt]`
The maximum velocity will occur when `sin ax= 1` and `cos bt = 1` under these condition `cos ax = 0` and `sin bt = 0`
:. |v_(max)| = 1/8 Ab`
Also, `|v_(min)| = 0`
(d) `y = [A cos (ax + bt)] - [0.8 A cos (ax - bt)]`
`= [0.8 A cos (ax + bt) + 0.2 A cos (ax + bt)] - [0.8 A cos (ax - bt)]`
`= [0.8 A cos (ax + bt) + 0.8 A cos (ax - bt)] + 0.2 A cos (ax + bt)]`
`= 0.8 A [-2 sin {((ax + bt)+ (ax - bt))/(2)}sin{(ax + bt) - (ax - bt)/(2)}] (0.2 A cos (ax + bt)]`
rArr `y = -1.6 A sin ax sin bt + 0.2 A cos (ax + bt)`
where `(-1.6 A sin ax sin bt)` is the equation of a standing wave and `0.2 A cos (ax + bt)` is the equation of travelling wave.
The wave is travelling in -x direction.