Question

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.....

Answer 1

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)