A simple-harmonic wave train of amplitude 3 cm and frequency 200 Hz travels in a +ve direction of x-axis with a velocity of 20 m/s. Calculate the displacement, velocity, and acceleration of a particle situated at 50 cm from the origin at t=2 s.
Anonymous
Let y be the displacement at distance x and time t.
Then y = Asin{(2π/λ)(x - vt)}, where
A = amplitude
λ = wavelength
v = wave velocity
Here A = 0.03m, v = 20 m/s, f = 200 Hz
We have λf = v → λ = v/f = 20/200 = 0.1m
So y = 0.03sin{2π/0.1(x - 20t)} = 0.03sin{20π(x - 20t)}
u (particle) = ∂y/∂t = -0.03 x 400πcos{20π(x - 20t)}
a = ∂²y/∂t² = -0.03 x 400π x 400πsin{20π(x - 20t)}
We want y, u and a when x = 0.5, t = 2
Subsituting:
y = -2 x 10^(-8)m = -2 x 10^(-6)cm
u = 37.7m/s
a = 0.03m/s²
The particle is 2 x 10^(-6)cm below the x-axis moving at 37.7m/s upwards and accelerating at 0.03m/s² upwards.