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

On comparing this displacement equation with Eq. (15.2),
`y(x,t)=a"sin"(kx-omegat).`
we find
(a) the amplitude of the wave is 0.005 m = 5 mm.
(b) the angular wave number k and angular frequency `omega` are
`l=80.0 m^(-1)` and `omega=3.0 s^(-1)`
We, then, related the wavelength `lambda` to k through Eq. (15.6),
`lambda=2pi//k`
`=(2pi)/(80.0 m^(-1))`
=7.85 cm
(c) Now, we related T to `omega` by the relation `T=2pi//omega`
`=(2pi)/(3.0 s^(-1))`
`=2.09 s`
and frequency, v=1/T=0.48 Hz
The displacement y at x=30.0 cm and time t=20 s is given by
`y=(0.005 m) "sin"(80xx0.3-3.0 xx 20)`
`=(0.005 m) "sin"(-36 + 12pi)`
`=(0.005 m)"sin"(1.699)`
`=(0.005 m) "sin"(97^(@))~=5 mm`