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.

The equation of a wave is, y(x,t)=0.05sin[(pi)/(2)(10x-40t)-(pi)/(4)]

1.	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.
Answer» Solution :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)`