Answer to Question #279286 in Mechanics | Relativity for Ahmad

Question #279286

Two sinusoidal waves travel in the same direction with the same amplitude, wavelength, and speed. Their resultant wave function is given by: y(x,t) = Ares sin(kx-ωt+π/4). If Ares = 8 cm, then the amplitude, A, of each of the original sinusoidal waves producing this resultant wave is:

1
Expert's answer
2021-12-13T16:55:15-0500

If two identical waves are traveling in the same direction, with the same frequency, wavelength and amplitude; BUT differ in phase the waves add together.   

"y = y_1 + y_2" where

"y = A sin (kx - \u03c9t) \\space \\space and \\space \\space y_2 = A sin (kx - \u03c9t + \u03c6)"

"y = A sin(kx - \u03c9t) + A sin(kx - \u03c9t + \u03c6)"

Apply trig identity: "sin a + sin b = 2 cos((a-b)\/2) sin((a+b)\/2)"

"A sin ( a ) + A sin ( b ) = 2A cos((a-b)\/2) sin((a+b)\/2)"

"y = 2A cos (\u03c6 \/2) sin (kx - \u03c9t + \u03c6\/2)"

The resultant sinusoidal wave has the same frequency and wavelength as the original waves, but the amplitude has changed: 

"Amplitude \\space equals \\space 2A cos (\u03c6 \/2)" with a phase angle of φ/2

here "\u03c6 = \\pi \/4"

Ares = "2A cos (\u03c6 \/2) = 2Acos(\\pi\/8)=8 cm \\to A = 4.32cm"

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