Question

Two waves are described by y1 = 0.21 sin[?(9x - 190t)] and y2 = 0.21 sin[?(9x - 190t) + ?/4], where y1, y2, and x are in meters and t is in seconds. When these two waves are combined, a traveling wave is produced. What are the (a) amplitude, (b) wave speed, and (c) wavelength of that traveling wave?

Answer 1

using the principle of super position

if y1 (x,t) = A sin (k1 x + wt)

and y2(x,t) = A sin(k2x + wt)

the resultant of these waves

is y (x,t) = Y1 + Y2 = 2 A cos (phi/2) sin (Kx+wt + phi/2)

where 2A cos(phi/2) is the amplitude of the wave

so here given

y1 = 0.21 sin[?(9x - 190t)]

and y2 = 0.21 sin[?(9x - 190t) + ?/4],

so Y (x,t) = (2* 0.21) cos (?/8) sin (9?x -190?t) + ?/8)

Y(x,t) = 0.42 cos ( 22.5) sin ( 1620 x - 34200 t) + ?/8)

so Amplitude A = 0.42 Cos (22.5) = 0.388 m

Wave number K = 2pi/L

Wavelength L = 2pi/K = 2pi/9pi = 2/9  

angular speed W = 2pif  

f = W/2pi

Wave speed v = Lf

v = 2/9 * 190 pi/2pi

V = 21.11 m/s