Question

For the harmonic travelling wave y=2cos2pi(10t-0.0080x+3.5)` where x and y ar in cm and t is in second. What is the phase diffference between the oscillatory motion at two points separated by a distance of
A. 4m
B. `0.5 m`
C. `(lambda)/(2)`
D. `(3lambda)/(4)` (at a given instant of time)

Answer 1

Given , wave functions are
`y=2cos 2pi (10t-000080x+3.5)`
`" " =2cos (20pit-0.016pix+7pi)`
Now, standard eqaution of a travelling wave can be wirtten as
`", " y=acos(omegat-kx+ophi)`
On comaring with above eqaution, we get
`" " a=2cm`
`" " omega=20pi "rad"//s`
` " " k=0.016pi`
`" "` Path difference =4 cm
(a) Phase difference `Deltaphi=(2pi)/(lambda)xx` Path difference
`:. " " Deltaphi=0.016pixx4xx100" " :. ((2pi)/(lambda)=K)`
`" " =6.4pi` rad
(b) `Deltaphi=(2pi)/(lambda)xx(0.5xx100) " " [ :. ` Path difference =0.5]
`" " =0.016pixx0.5xx100`
`" " =0.8pi "rad" `
(c) `Delta phi=(2pi)/(lambda)xx((lambda)/(2))=pi "rad" " " [ :.` Path difference `=lambda//2`]
(d) `Deltaphi=(2pi)/(lambda)xx(3lambda)/(4)=(3pi)/(2)` rad
(e) `T=(2pi)/(omega)=(2pi)/(20pi)=(1)/(10S)`
`:. " " "At" x=100 cm`
`" " t=T`
`" "phi_(1)=20piT-0.016pi(100)+7pi`
`" " =20pi((1)/(10))-1.6pi+7pi=2pi-1.6pi+7pi" "....(i)`
Again at x=100 cm, t=5s
`" " phi_(2)=20pi(5)-0.016pi(100)+7pi`
`" " =100pi-(0.016xx100)pi + 7pi`
`" " =100 pi-1.6+7pi" "...(ii)`
`:.` From Eqs. (i) and (ii) we get
`Deltaphi=` phase difference `=phi_(2)-phi_(1)`
`" " =(100pi-1.6pi+7pi)-(2pi-1.6pi+7pi)`
`" " =100pi-2pi=98pi` rad