6.2.3: Ecuación de las trayectorias
- Page ID
- 51147
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Definimos una nueva cantidad, \(\mathcal{L}(\mathbf{r})\), a través de la relación
\[
g(\mathbf{r})=\frac{\omega}{c} \mathcal{L}(\mathbf{r}) \notag
\]
con esta expresión para la fase el vector de ondas queda
\[
\mathbf{k}(\mathbf{r})=\nabla g(\mathbf{r})=\frac{\omega}{c} \nabla \mathcal{L}(\mathbf{r}) \notag
\]
las ecMm aplicadas a una onda aproximadamente plana en un medio aproximadamente homogéneo dictan que
\[
|\mathbf{k}(\mathbf{r})|=n(\mathbf{r}) \frac{\omega}{c} \notag
\]
de suerte que obtenemos la ecuación eikonal
\[
|\nabla \mathcal{L}(\mathbf{r})|=n(\mathbf{r}) \notag
\]
en muchos textos la aproximación que hemos explicado se encuentra bajo el nombre de aproximación eikonal. Estamos listos para demostrar la ecuación de las trayectorias, que es la ecuación diferencial siguiente sobre el parámetro de arco \(s\)
\[
\frac{\mathrm{d}}{\mathrm{d} s}\left(n \frac{\mathrm{d} \mathbf{r}}{\mathrm{d} s}\right)=\nabla n \notag
\]
Para hacerlo hay que observar que \(\frac{\mathrm{d} \mathbf{r}}{\mathrm{d} s}\) es tangente y unitario y por lo tanto debe ser proporcional a k. Es decir
\[
\frac{\mathrm{d} \mathbf{r}}{\mathrm{d} s}=\frac{\mathbf{k}}{|\mathbf{k}|}=\frac{\frac{\omega}{c} \nabla \mathcal{L}}{\frac{\omega}{c} n}=\frac{1}{n} \nabla \mathcal{L} \notag
\]
o bien
\[
n \frac{\mathrm{d} \mathbf{r}}{\mathrm{d} s}=\nabla \mathcal{L} \notag
\]
Derivamos respecto al parámetro de arco ambos miembros y
\[
\frac{\mathrm{d}}{\mathrm{d} s}\left(n \frac{\mathrm{d} \mathbf{r}}{\mathrm{d} s}\right)=\frac{\mathrm{d}}{\mathrm{d} s} \nabla \mathcal{L} \notag
\]
intecambiando el orden de las derivadas
\[
\frac{\mathrm{d}}{\mathrm{d} s}\left(n \frac{\mathrm{d} \mathbf{r}}{\mathrm{d} s}\right)=\nabla\left(\frac{\mathrm{d} \mathcal{L}}{\mathrm{d} s}\right) \notag
\]
donde se puede aplicar la regla de la cadena para obtener
\[
\begin{align}
\frac{\mathrm{d} \mathcal{L}}{\mathrm{d} s} &=\nabla \mathcal{L} \frac{\mathrm{d} \mathbf{r}}{\mathrm{d} s} \notag \\
&=\frac{1}{n}(\nabla \mathcal{L})^{2} \notag \\
\frac{\mathrm{d} \mathcal{L}}{\mathrm{d} s} &=n
\end{align}
\]
queda demostrada la ecuación de las trayectorias según las hemos definido más arriba.