Surface-Forced Waves - Edubirdie

University:
Massachusetts Institute of Technology
Course:
12.802 | Wave Motions in the Ocean and Atmosphere
Academic year:

2022
Views:

69

Pages:

2
Author:

intricas72p

Surface-forced waves Suppose we specify the surface pressure P = P_{0} * cos(kx - omega*t) = \mathfrak{R}[P_{0} * e ^ (i(kx - omega*t))] what is the structure and the nature of the disturbances below? We can derive an equation for the pressure in the water column using nabla^ 2 P= partial partial z b w=- 1/(N ^ 2) partial partial t b partial partial t w=- partial partial z P + b Eliminating w from the last two and taking a z derivative gives [ 1/(N ^ 2) partial^ 2 partial t^ 2 +1] partial b partial z = partial^ 2 P partial z^ 2 [ 1/(N ^ 2) partial^ 2 partial t^ 2 +1] nabla^ 2 P= partial^ 2 P partial z^ 2 partial^ 2 partial t ^ 2 nabla^ 2 P=-N^ 2 nabla h ^ 2 P as before. The pressure will have the same horizontal and temporal structure as the forcing. P = P(z) * e ^ (i(kr - omega*t)) partial^ 2 partial z^ 2 P = - ((N ^ 2)/(omega ^ 2) - 1) * k ^ 2 * P and the vertical structure satisfies or which can also be written as The pressure has a vertical structure like exp(ams) with m ^ 2 = ((N ^ 2)/(omega ^ 2) - 1) * k ^ 2 (which is equivalent to our standard dispersion relation omega ^ 2 = N ^ 2 * (k ^ 2)/(k ^ 2 + m ^ 2) but solved for m rather than w). Trapped disturbances When w > N, we have imaginary vertical wavenumbers Mim, so that the solutions are exponentially growing or decaying with depth P(z) = Po exp(+Mz). Since the forcing is at the surface, we have to choose the solutions which decay PR[PoeMeks-wt)] Pycos(kz-wt)eMs The solutions are in phase with depth and decay at a rate N21/2 M=k(1) Propagating disturbances When N, the forcing is in the range of frequencies for which free waves exist, and m is real. To resolve which sign we use, we argue that the energy should be propagating downwards, away from the source, so that C-N mk (k2+m²)3/2 <0 Thus, we must choose the positive sign for m, and the solution becomes PR[Poet(kx+ms-wt) Po cos(ka+mz-wt) The solutions now change phase with depth and flux energy deep into the water.

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