For the travelling harmonic wave, `y(x,t)=2.0cos2pi[10t-0.0080x+0.35]`. Where x and y are in cm and t is s, what is the phase difference between oscillatory motino at two points separated by a distance of (i) 4m (ii) 0.5m (iii) ` lambda//2` (iv) `3lambda//4` ?

Explanation

Given equation of a travelling harmonic waves is
`y(x,t)=2.0cos2pi(10t-0.0080x+0.35)` . . . (i)
The standard equation of travelling harmonic wave is
`y(x,t)=Acos[(2pi)/(T)t-(2pi)/(lamda)x+phi_(0)]` . . . (ii)
Comparing equation (i) and (ii), we get
`(2pi)/(lamda)=2pixx0.0080cm^(-1)` . . . (iii)
`(2pi)/(T)=2pixx10 and phi_(0)=0.35`
We know that phase difference `=(2pi)/(lamda)xx`path difference . . . (iv)
(i) When path difference = 4m = 400 cm, then from (iv)
Phase difference`=(2pi)/(lamda)xx400`
`=2pixx0.0080xx400` [by using (iii)]
=`6.4pirad`
(ii) When path difference=0.5m=50cm, then phase difference=`2pixx0.0080xx50=0.8pi ` rad
(iii) When path difference `=(lamda)/(2)`, then phase difference`=(2pi)/(lamda)xx(lamda)/(2)=pi` rad
(iv) When path difference`=(3lamda)/(4)` phase difference`=(2pi)/(lamda)xx(3lamda)/(4)=(3pi)/(2) rad=(pi+(pi)/(2))`