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PS of a single wave

A proof of the PS resulting from a single wave propagating upon any background wind abiding WKB can be found in Pulido and Caranti (2000), here we just show the linear background case in order to compare results and procedures with the broad spectrum case.

The solution (2) of the wave equation for a wave propagating in an environment where the wind changes linearly with height at a particular time and a fixed horizontal position can be expressed as

$\begin{displaymath} u(z)=u_a (1- m_0 \mu^{-1} z)^{-1/2- i \mu} \end{displaymath}$ (9)

Taking into account the Parseval theorem,

$\begin{displaymath} \frac{L}{2} \int PS dm = \int u(z) u^*(z) dz \end{displaymath}$ (10)

Replacing the expression (9) into (10),

$\begin{displaymath} \int PS dm =u_a^2 \int (1- m_0 \mu^{-1} z)^{-1} dz \end{displaymath}$ (11)

Changing variable from to $m=\left(\frac{1}{m_0}-\frac{z}{\mu}\right)^{-1}$

$\begin{displaymath} \int PS dm= \frac{2 u_a^2}{L} \int \frac{\mu}{m_0 m} dm \end{displaymath}$ (12)

Then, as the extremes of the integration interval are free, the integrands must be equal,

$\begin{displaymath} PS=\frac{2 \mu u_a^2}{L m_0 m}. \end{displaymath}$ (13)

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