If two waves, each of amplitude #z#, produce a resultant wave of amplitude #z#, then what is the phase difference between them? Is there a specific equation or formula I can use?

Seems like the phase difference is between 0 and 180 because 0 (inphase) would imply a resultant of 2z and 180 would give a resultant of 0.....

1 Answer
Nov 21, 2017

For the amplitude of the superposition of two waves to have the same amplitude as the original waves, their phase difference must be #120^o = (2\pi)/3# #radians#

Explanation:

Consider the superposition of two sinusoidal waves of identical amplitude #\psi_0#, separated just by a phase shift #\phi#:

The mathematical expressions for the waves #\psi_1(x)# and #\psi_2(x)# are:
#\psi_1(x)=\psi_0\sin(kx); \qquad \qquad \psi_2(x)=\psi_0\sin(kx+\phi)#

The mathematical expression for the superposition of these two waves is :
#\psi_{12}(x) = [2\psi_0\cos(\phi/2)]\sin(kx + \phi/2)# ...... (derived below)

This shows that the superposition is another sinusoidal wave of amplitude #2\psi_0\cos(\phi/2)#.
This amplitude is the same as #\psi_0# for #\phi = 120^o = (2\pi)/3# #radians#.

Derivation of the expression for the superposition of two waves:

#\psi_{12}(x) = \psi_1(x) + \psi_2(x);#
#\psi_{12}(x) = \psi_0\sin(kx) + \psi_0\sin(kx + \phi)#

Use the trigonometric identity (TI3) to expand #\sin(kx+\phi)# term,
#\psi_{12}(x) = \psi_0\sin(kx) + \psi_0[\sin(kx)\cos\phi + \cos(kx)\sin\phi]#
#\psi_{12}(x) = \psi_0[\sin(kx)(1+\cos\phi) + \cos(kx)\sin\phi]#

Use trigonometric identity (TI1) to rewrite the #(1 + \cos\phi)# term and the identity (TI2) to rewrite the #\sin\phi# term, in forms which would allow further simplification

#\psi_{12}(x)= \psi_0[\sin(kx)(2\cos^2(\phi/2)) + \cos(kx)(2\sin(\phi/2)\cos(\phi/2))]#

#\psi_{12}(x) = 2\psi_0\cos(\phi/2)[\sin(kx)\cos(\phi/2) + \cos(kx)\sin(\phi/2)]#
Use the trigonometric identity (TI3) again to simplify the terms inside the square bracket
#\psi_{12}(x) = 2\psi_0\cos(\phi/2)\sin(kx + \phi/2)#

Useful trigonometric identities:
#[ 1 + \cos(a)] = 2\cos^2(a/2)# ...... (TI1)
#\sin(a) = 2\sin(a/2)\cos(a/2)# ...... (TI2)
#\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b)# ...... (TI3)