Formalization of the Point of Oblivion in Waves

Given:

A wave described by: y₁(x, t) = A sin(kx - ωt + φ)
Its reflected counterpart with a negative amplitude: y₂(x, t) = -A sin(kx - ωt + φ)

Point of Oblivion:

A point in space (x) and time (t) where the superposition of the original and reflected waves results in zero displacement (y(x, t) = 0).

Mathematical Formulation:

1. Superposition:

The total displacement is the sum of the individual waves:

y(x, t) = y₁(x, t) + y₂(x, t) = A sin(kx - ωt + φ) - A sin(kx - ωt + φ)

2. Cancellation using Trigonometric Identity:

Using the identity sin(a) - sin(b) = 2 sin((a - b)/2) cos((a + b)/2):

y(x, t) = 2 A sin((kx - ωt + φ - (kx - ωt + φ)) / 2) cos((kx - ωt + φ + (kx - ωt + φ)) / 2)

Simplifying:

y(x, t) = 2 A cos(φ) cos(kx - ωt)

3. Conditions for Point of Oblivion:

For y(x, t) = 0, both cosine terms must equal zero:cos(φ) = 0: 

• This occurs when the phase constant (φ) is specifically 90° (π/2) or 270° (3π/2). These values correspond to zero points of the original sine wave.
• cos(kx - ωt) = 0: This happens when the argument (kx - ωt) is a multiple of 90°. This translates to specific combinations of x and t where the original wave is zero.

Conclusion:

The point of oblivion exists when the reflected wave (with a 180° phase shift) perfectly cancels the original wave. This requires a specific phase constant (φ) and a specific relationship between wave number (k) and angular frequency (ω) that leads to zero displacement at a particular location (x, t).