When you add two identical but out-of-phase sinusoidal waves, the result depends on the phase difference between them.
If the two waves are completely out of phase, meaning they have a phase difference of 180 degrees (or π radians), their sum will be zero. This is because at every point in time, the positive amplitude of one wave cancels out the negative amplitude of the other wave, resulting in a net amplitude of zero.
Mathematically, if two sinusoidal waves are given by the equations:
y₁(t) = A sin(ωt + φ₁) y₂(t) = A sin(ωt + φ₂)
where A represents the amplitude, ω is the angular frequency, t is the time variable, and φ₁ and φ₂ are the phase angles of the two waves, then the sum of the two waves would be:
y(t) = y₁(t) + y₂(t) = A sin(ωt + φ₁) + A sin(ωt + φ₂) = 2A sin((φ₁ + φ₂)/2) cos((φ₁ - φ₂)/2) sin(ωt + (φ₁ + φ₂)/2)
If φ₁ - φ₂ = π (180 degrees), then cos((φ₁ - φ₂)/2) = cos(π/2) = 0, and the resulting wave becomes:
y(t) = 2A sin((φ₁ + φ₂)/2) sin(ωt + (φ₁ + φ₂)/2)
In this case, the sum of the two waves will be zero at all times, resulting in destructive interference.