Why does a wave undergo a 180° phase change at a fixed boundary?

The phase change of can be theoretically understood as follows:

For transmission or propagation of waves between media the wave motion should maintain a principle of continuity meaning that the wave function at the interface should be continuous and diffrentiable at the interface.

At the point of incidence there are 2 types of waves reflected wave and the incident wave. Now the principle of continuity dictates that the sum of the phases of the above 2 waves should be same as that of transmitted wave. If we use these relations we notice that the reflected wave shall either change it's phase by or will not change it's phase depending on the relationship between the refractive indices of the incident and the reflecting medium. For a solid boundary a phase change of occurs.

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d₁ = displacement of tern from arctic to antarctic = 11000 miles

d₂ = displacement of tern from antarctic to arctic =   - 11000 miles

d = total displacement for the journey

total displacement for the journey is given as

d = d₁ + d₂

d = 11000 + (-11000)

d = 0 miles

t₁ = time of travel from arctic to antarctic = 3 months = 3 x 30 days = 90 days

t₂ = time of travel from antarctic to arctic = 3 months = 3 x 30 days = 90 days

t = total time of travel

total time of travel is given as

t = t₁ + t₂

t = 90 + 90

t = 180 days

average velocity is given as

average velocity = total displacement/ total time

v = d/t

v = 0/180

v = 0 miles/day