Question

The equation of a wave is y = 60 cos (1800t - 6x), where y is in microns, t in seconds, and x in metres. The ratio of maximum particle velocity to the wave velocity of wave propagation is ______.

Options

• 3.6

• 3.6 x 10^-6

• 3.6 x 10^-11

• 3.6 x 10^-4

Answer 1

The equation of a wave is y = 60 cos (1800t - 6x), where y is in microns, t in seconds, and x in metres. The ratio of maximum particle velocity to the wave velocity of wave propagation is 3.6 x 10^-4.

Explanation:

Comparing equation with y = Acos(`omega`t - kx),

y = 60 cos (1800 t - 6x)

A= 60, `omega` = 1800, k = 6

Velocity of wave propagation is

` "v"_"w" = "n"lambda ; "n" = omega/(2pi) = 18000/(2pi)`,

`lambda  (2pi)/"k" = (2pi)/6`

` "v"_"w"= 18000/(2pi) xx (2pi)/6 = 300` m/s

Velocity of particle is

` "v"_"p" = (del"y")/(del"t") = 1800 xx 60` COS (1800 t- 6x)

∴ `"v"_"pmax" = 1800 xx 60` µm/s

∴ `"v"_"pmax"= 1800 xx 60 xx 10^-6` m/s

∴ `"v"_"pmax"/ "v"_"w"= (1800 xx 60 xx 10^-6)/300` 

= `360 xx10^-6 = 3.6 xx 10^-4`