Answer to Question #279286 in Mechanics | Relativity for Ahmad

Question

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:

Answer 1

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

$y = A sin (kx - ωt) \space \space and \space \space y_2 = A sin (kx - ωt + φ)$

$y = A sin(kx - ωt) + A sin(kx - ωt + φ)$

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

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

$y = 2A cos (φ /2) sin (kx - ωt + φ/2)$

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

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

here $φ = \pi /4$

Ares = $2A cos (φ /2) = 2Acos(\pi/8)=8 cm \to A = 4.32cm$