The phase (ΔΦ) of the wave changes with a given displacement (Δx):

Derivation:

Considering two points on a wave separated by a displacement Δx. 

Let's denote the phase difference between these points as ΔΦ.

Now, we know that the phase difference ΔΦ corresponds to the fraction of the wavelength (λ) represented by the displacement Δx. 

Therefore, we can write: 

ΔΦ = (Fraction of λ/λ) × 2π

Since Fraction of λ = Δx/λ, 

we substitute this expression into the equation: ΔΦ =(Δx/λ) × 2π

Simplifying, we get: ΔΦ = 2π/λ × Δx

Where:

• ΔΦ: The phase difference between two points on a wave, measured in radians. It indicates how much the phase of the wave changes between the two points.

• λ: The wavelength of the wave, representing the distance between two consecutive points in phase (e.g., crests, troughs). It's typically measured in meters.

• Δx: The displacement between the two points on the wave along the direction of propagation, measured in the same units as the wavelength. It represents the distance travelled by the wave.

This is the derived formula for the phase difference ΔΦ in terms of the wavelength (λ) and the displacement Δx along the direction of propagation. It shows how much the phase of the wave changes with a given displacement.