Question

In the given progressive wave `y=5sin (100pit-0.4pix)` where y and x are in m, t in s, what is the (a) amplitude (b) wavelength (c) frequency (d) wave velocity (e) particle velocity amplitude.
A. amplitude?
B. wavelength ?
C. frequency ?
D. wave velocity?

Answer 1

Standard equation of a progressive wave is give by
`" " y=asin (omegat-kx+phi)`
This is travelling along positive x-direction
Given equation is `" " y=5sin(100pit-0.4pix)`
Comparing with the standard equation
(a) Amplitude =5m
`k=(2pi)/(lambda)=0.4pi`
`:. " " "wavelength" lambda=(2pi)/(k)=(2pi)/(0.4pi)=(20)/(4)=5m`
(c) `omega=100pi`
`" " omega=2piv=100pi`
`:. " " "Frequency" v=(100pi)/(2pi)=50Hz`
(d) Wave velocity `v=(omega)/(k)` where k is wae number and`k=(2pi)/(lambda)`
`" " =(100pi)/(0.4pi)=(100)/(4)`
`" " =250 m//s`
(e)` y=5sin (100pit-0.4pix)" ".....(i)`
`" " (dy)/(dt)=` particle velocity
Form Eq. (i)
`" " (dt=y)/(dt)=5(100pi)cos[100pit-0.4pix]`
For particle velocity amplitude `((dy)/(dt))_("max")`
Which will be for `{cos[100pit-0.4pix]}_("max")=1`
`:.` Particle velocity amplitude
`" " =((dy)/(dt))_("max")" "5(100pi)xx1`
`" "=500pim//s`