Question

These two waves travel along the same string: y1 = (3.52 mm) sin(1.75?x - 470?t) y2 = (5.93 mm) sin(1.75?x - 470?t + 0.736?rad).

What are (a) the amplitude and (b) the phase angle (relative to wave 1) of the resultant wave? (c) If a third wave of amplitude 5.48 mm is also to be sent along the string in the same direction as the first two waves, what should be its phase angle in order to maximize the amplitude of the new resultant wave?

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       =               y₁         +            y₂              where y₁ = A sin (kx - ?t)    and y₂ = A sin (kx - ?t + ?)

y       = A sin(kx - ?t) + A sin(kx - ?t + ?)

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 (? /2) sin (kx - ?t + ?/2)

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

Amplitude equals 2A cos (? /2) with a phase angle of ?/2

3) If a third wave of amplitude 5.48 mm is also to be sent along the string in the same direction as the first two waves, in order to maximize the amplitude of the new resultant wave the phase angle should be more than the resultant of the two waves phase and should be in phase .