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

The given equation is `y=2cos2pi(10t-0.0080x+3.5)`
Compare it with standard form of equation of progressive wave
`y=rcos[(2pit)/(T)=(2pix)/(lambda)+phi]`
`r=2cm`
`(2pi)/(T)= 20pi, T=(1)/(10)=0.1s`
`(2pi)/(lambda)=2pixx0.008, lambda=(2pi)/(2pixx0.008)=125cm=1.25m`
`phi=2pixx3.5radian`
(a) Phase diff. between oscillatory motion at two pts. separated by a distance `x=4m`
`phi_(1)=(2pi)/(lambda)x=(2pi)/(1.25)xx4=6.4rad.`
(b) When `x=0.5m,` `phi_(2)=(2pi)/(lambda)x=(2pi)/(1.25)xx0.5=0.8pirad.`
(c) When `x=lambda//2m,` `phi_(3)=(2pi)/(lambda)xx(lambda)/(2)=pirad.`
(d) When `x=(3lambda)/(4), phi_(4)=(2pi)/(lambda)xx(3lambda)/(4)=(3pi)/(2)rad.`
(e) At `t=T,phi=(2pi)/(T)=(2pi)/(0.1)=20pirad` and at `t=5s, phi^(')=(2pi)/(T)xxt=(2pi)/(0.1)xx5=100pirad`
`:.` Phase diff. `=phi^(')-phi=100pi-20pi=80pirad.`