When waves of equal amplitude and different phase are superimposed, the amplitude of the resultant wave will be maximum when the phase difference between the two waves is either 0 degrees or a multiple of 360 degrees.
To understand why this is the case, let's consider two waves:
Wave 1: y1 = A sin(ωt + φ1) Wave 2: y2 = A sin(ωt + φ2)
Here, A represents the amplitude, ω is the angular frequency, t is time, and φ1 and φ2 are the phase angles of the respective waves.
The superposition of these waves is given by the sum of the individual wave equations:
Resultant wave: y = y1 + y2 = A sin(ωt + φ1) + A sin(ωt + φ2)
To find the maximum amplitude of the resultant wave, we can express the above equation in terms of trigonometric identities:
y = A [sin(ωt + φ1) + sin(ωt + φ2)] y = A [2sin((ωt + φ1 + ωt + φ2)/2)cos((ωt + φ1 - ωt - φ2)/2)]
Using the trigonometric identity sin(A) + sin(B) = 2sin((A + B)/2)cos((A - B)/2), we can simplify the equation further:
y = 2A [sin((2ωt + φ1 + φ2)/2)cos((φ1 - φ2)/2)]
Now, for the amplitude of the resultant wave to be maximum, the term sin((2ωt + φ1 + φ2)/2) must equal 1 (since the amplitude of the sine function is always between -1 and 1).
This happens when the argument of the sine function is equal to an odd multiple of 90 degrees or π/2 radians:
(2ωt + φ1 + φ2)/2 = (2n + 1)(π/2) where n is an integer
Simplifying the equation, we find:
2ωt + φ1 + φ2 = (2n + 1)(π/2)
Now, if we subtract φ1 + φ2 from both sides, we get:
2ωt = (2n + 1)(π/2) - (φ1 + φ2)
Since ω is the angular frequency and is equal to 2πf (where f is the frequency), we can rewrite the equation as:
4πft = (2n + 1)(π/2) - (φ1 + φ2)
Simplifying further:
2ft = (2n + 1)(1/2) - (φ1 + φ2)/(4π)
Therefore, for the amplitude of the resultant wave to be maximum, the phase difference (φ1 + φ2) between the two waves must be such that (φ1 + φ2)/(4π) is an integer.
In other words, the phase difference must be an integer multiple of 360 degrees or 2π radians.
So, the amplitude of the resultant wave will be maximum when the phase difference between the two waves is either 0 degrees or a multiple of 360 degrees.