The equation of a wave is
`y=(x,t)=0.05 sin [(pi)/(2)(10x-40t)-(pi)/(4)]m`
find: (a) the wavelength, the frequency and the wave velocity
(b) the participle velocity and acceleration at `x=0.5m`and `t = 0.05s`.

Explanation

(a) The equation may be rewritten as, `y(x,t) = 0.05 sin (5 pi x - 20 pi t - (pi)/(4))m`
Comparing this with equation of plane progressive harmonic wave,
`y(x,t)=A sin(kx-omegat+phi)` we have number `k = (2pi)/(lamda) = 5 pi rad//m :. Lamda = 0.4 m`
The angular frequency is, `omega=2pif=20 pirad//s :. f =1 0 Hz`
The wave velocity is, `v = f lamda = (omega)/(k) = 4 ms^(-1)` in -x direction
(b) The particle velocity and acceleration are, `V_(p)=(dely)/(delt)=-(20pi)(0.05)cos((5pi)/(2)-pi-(pi)/(4))=2.22 m//s`
`a_(p)=(del^(2)y)/(delt^(2))==(20pi)^(2)(0.05)sin((5pi)/(2)-pi-(pi)/(4))=140 m//s^(2)`