Two waves, each having a frequency of 100 Hz and a wavelength of 2.0 cm, are travelling in the same direction on a string. What is the phase difference between the waves (a) if the second wave was produced 0.015 s later than the first one at the same place, (b) if the two waves were produced at the same instant but the first one was produced a distance 4.0 cm behind the second one ? (c) If each of the waves has an amplitude of 2.0 mm, what would be the amplitudes of the resultant waves in part (a) and (b) ?

Answer

a)`phi = 3pi`

b) `phi = 4pi`

c) `0, 4mm`

Explanation

`f=100Hz`
`y=2cm=2xx10^-2m`
wave speed `v=fy=100xx2xx10^-2m/s`
`=2m/s`
a. In 0.015 sec 1st wave has travelled
`x=0.015xx2`
`=0.03m`
=path difference
:.Corresponding phase diff.
`phi=(2pix)/lamda`
`={(2pi)/((2xx10^-2))}xx0.03=3pi`
b. path difference
`x=4cm=0.04m`
`rarr pi=((2pi)/lamda)x`
`={((2pi)/(2xx10^-2))xx0.04}`

`phi = 4pi`
c. The waves have same frequency same wavelenth and same amplitude.
`Let y_1=rsinwt`
`y_2=rsin(wt+phi)`
`rarr y=y_1+y_2`
`=r[sinwt+(wt+phi)]`
`=2rsin(wt+phi/2)cos(phi/2)`
:. Resultant amplitude =`2rcosphi/2`
S, when `phi=3xd`
`r=2xx10^-3m`
`R_(res)=2xx(2xx10^-3)cos((3pi)/2)`
=0
Again when `phi=4pi`
`R_(res)=2xx(2xx10^-3)cos((4pi)/2)`
`=4xx10^1-3xx(1)m`
`=4mm`