When two plane waves of equal amplitude and wavelength interfere constructively, it means that the crests of one wave align with the crests of the other wave, and the troughs align with the troughs. The maximum angle at which constructive interference occurs depends on the phase difference between the two waves.
For two plane waves with the same amplitude and wavelength, the maximum angle for constructive interference happens when the phase difference between the waves is zero (0 degrees or a multiple of 360 degrees). In other words, the two waves are in phase with each other.
To calculate the maximum angle, you can use the concept of path difference. The path difference is the extra distance traveled by one wave compared to the other wave. For constructive interference to occur, the path difference should be an integer multiple of the wavelength.
The formula to calculate the path difference (Δx) for waves arriving at an angle (θ) is:
Δx = d * sin(θ)
Where:
- Δx is the path difference
- d is the separation between the sources of the waves (such as the distance between two slits in a double-slit experiment)
- θ is the angle at which the waves are measured
For constructive interference, the path difference (Δx) should be an integer multiple of the wavelength (λ). Therefore:
Δx = m * λ
Where:
- m is an integer (0, 1, 2, 3, ...)
By equating the formulas for path difference, we get:
m * λ = d * sin(θ)
To find the maximum angle, we can consider the case where m = 1, which gives:
λ = d * sin(θ_max)
Rearranging the equation, we get:
sin(θ_max) = λ / d
To find the maximum angle (θ_max), you can take the inverse sine (arcsin) of both sides of the equation:
θ_max = arcsin(λ / d)
Therefore, the maximum angle at which two plane waves of equal amplitude and wavelength can interfere constructively is given by the arcsin of the ratio of the wavelength to the separation between the sources (d).