Consider two wave functions that differ only by a phase shift, y₁(x,t) = Acos(kx−ωt) and y₂(x,t) = Acos(kx−ωt+ϕ).
Use the trigonometric identities cosu + cosv = 2cos(u−v²)cos(u+v²) and cos(-θ) = cos(θ) to find a wave equation for the wave resulting from the superposition of the two waves.
Does the resulting wave function come as a surprise to you?