A wave travelling along a strong is described by
`y(x,t)=0.005 sin (80.0x-3.0t)`
in which the numerical constants are in SI units `(0.005m, 80.0 rad m^(-1)` and `3.0 rad s^( -1))`. Calculate (a) the amplitude. (b) the wavelength (c) the period and frequency of the wave. Also , calculate the displacement y of the wave at a distance `x=30.0` cm and time t=20 s?

Explanation

Given, `y(x, t) = 0.005 sin (80.0 x - 3.0 t)`
The standard equation of displacement for a harmonic wave is
`y (x, t) = A sin ((2pi)/(lambda) x - (2 pi t)/(T))`
On camparing the above two equations, we get
`A = 0.005 m, (2pi)/(lambda) = 80 rad//s, (2pi)/(T) = 3 rad//s`
(a) Amplitude, A = 0.005 m
(b) Wavelength, `lambda = (2pi rad)/(80 rad//s) = 7.85 xx 10^(-2) m`
(c) Time period, `T = (2pi rad)/(3 rad//s) = 2.09 s`
Frequency, `v = (1)/(T) = (1)/(2.09) s = 0.48 Hz`
Displacement of the wave at a distance x = 30.0 cm and time t = 20 s is
`{:(y = (0.005m) sin (80.0 xx 0.3 - 3.0 xx 20)),(" "= 0.005 xx sin (-36 rad)),(y = 0.005 sin (-36 + 12 pi) rArr y = (0.005) sin (1.699)),(y = (0.005) sin (97.39) rArr y = (0.005) xx 0.99 ),(rArr y = "0.005 m or 5 mm"):}`