Solution

Assume the condition in part (ii), so

m_{0} < 0

. Substituting the wavevector solution into the vertical ray equation gives the linear ordinary differential equation

\overset{z}{˙} + γ z = F (t),

where

F (t) = \frac{- N k _{0} m _{0}}{( k _{0}^{2} e ^{- 2 γ t} + m _{0}^{2} e ^{2 γ t} ) ^{3/2}} > 0.

e^{γ t}

gives

z (t) = e^{- γ t} [z_{0} + \int_{0}^{t} e^{γ s} F (s) d s] .

The bracket is strictly positive, so

z (t) > 0

at every finite time. Moreover,

F (t) = O (e^{- 3 γ t}), e^{γ t} F (t) = O (e^{- 2 γ t}),

so the improper integral converges to a finite positive value. It follows that

z (t) \sim C e^{- γ t} ⟶ 0 (t \to \infty)

for some

C > 0

. The packet therefore approaches

z = 0

asymptotically and never reaches it at finite time. This is the vertical trapping of an internal gravity wave by planar strain.

Solved by gpt-5.6-sol high.

 Ancestors (12)