Question

The following two waves are sent in opposite directions on a horizontal string so as to create a standing wave in a vertical plane:
y₁(x, t) = (6.80 mm) sin(4.80πx - 460πt)
y₂(x, t) = (6.80 mm) sin(4.80πx + 460πt),
with x in meters and t in seconds. An antinode is located at point A. In the time interval that point takes to move from maximum upward displacement to maximum downward displacement, how far does each wave move along the string?​ ***( SHOW ALL WORK & INCLUDE UNITS)***

Answer 1

y1(x, t) = (6.80 mm) sin(4.80πx - 460πt)
y2(x, t) = (6.80 mm) sin(4.80πx + 460πt)

Comparing with the wave eqn

y = Asin(kx - ωt)

Wave number, k = 4.8π

Angular frequency, ω = 460π

Speed of the wave is given by, v_w = ω/k

v_w = 460π/4.8π = 95.83 m/s

For the standing wave the time period of oscillation of each medium particle is given by

T = 2π/ω

T = 2π/460π

T = 0.004347 s

The time taken for medium particle to move from move from maximum upward displacement to maximum downward displacement = T/2 = 0.002174 s

During this time the distance travelled by wave, x = v_w x T/2

x = 95.83 x 0.002174

x = 0.2083 m