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=0.5mand t = 0.05s.
Answers
(a) The wavelength, the frequency and the wave velocity is :
• Given : equation of a wave is y=(x,t)=0.05 sin [(pi)/(2)(10x-40t)-(pi)/(4)]m
y = 0.05sin [ 5×pi×x - 20×pi×t - pi/4 ]
Amplitude, A = 0.05, wave number, k = 5pi, angular frequency, w = 20pi
• We know that, k = 2pi / λ
λ = 2pi / k = 2pi / 5pi = 2/5 = 0.4m
Also w = 2×pi×f
Where f = frequency
• So, f = w / 2×pi = 20×pi / 2×pi = 20/2 = 10 Hz
• Wave velocity, v = w / k
= 20×pi / 5×pi = 20/5 = 4 m/s
(b) The participle velocity and acceleration at x=0.5m and t = 0.05s is given as
• v = dy/dt = 0.05×(-20×pi) cos[5×pi×x - 20×pi×t - pi/4 ]
When x=0.5m and t = 0.05s ,
v = -20×pi×0.05 cos [5×pi×0.5 - 20×pi×0.05 - pi/4 ] = 2.22 m/s
• For acceleration,
a = d^2y/dt^2 = (20×pi)^2×0.05 sin[5×pi×x - 20×pi×t - pi/4 ]
Putting x=0.5m and t = 0.05s
a = (20×pi)^2×0.05 sin[5×pi×0.5 - 20×pi×0.05 - pi/4 ] = 140 m/s^2