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

Explanation

The equation may be rewritten as,
`y(x,t)=0.05sin(5pix-20pit-(pi)/(4))m`
Comparing this with equation of plane progressive harmonic wave,
`y(x,t)=Asin(kx-omegat+phi)`
Wave number `k=(2pi)/(lamda)=5pi` rad/m
`:.lamda=0.4m`
The angular frequency `omega=2pif=20pi` rad/s
`:.f=10Hz`
The wave velocity,
`v=flamda=(omega)/(k)`
`=4ms^(-1)` in + x direction.
The particle velocity and acceleration are,
`v_(p)=(dy)/(dt)`
`=-(20pi)(0.05)cos((5pi)/(2)-pi-(pi)/(4))`
`=2.22m//s`
`a_(p)=(d^(2)y)/(dt^(2))`
`=-(20pi)^(2)(0.05)sin((5pi)/(2)-pi-(pi)/(4))`
`=140m//s^(2)`