Why is it that in the expression of a wave, there's a negative for a wave moving to the positive x and has a positive for a wave moving to the negative x

Let's consider a general wave function, such as y(x, t) = A sin(kx ± ωt). For a specific point on the wave (like a crest or a trough) to maintain its shape as it propagates, its phase (the argument of the sine function) must remain constant.

1. If the phase is (kx - ωt) = constant: As time (t) increases, for the phase to remain constant, x must also increase. This means the wave is moving in the positive x-direction.

2. If the phase is (kx + ωt) = constant: As time (t) increases, for the phase to remain constant, x must decrease. This means the wave is moving in the negative x-direction.

Therefore, a negative sign (kx - ωt) indicates propagation in the positive x-direction, and a positive sign (kx + ωt) indicates propagation in the negative x-direction.