Conservation Priciples and More Orthogonality

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We showed earlier how problems in stratified media reduce to a first-order matrix differential equation of the form

$\begin{displaymath} \frac{\partial}{\partial z} \mbox{\bf y} \eq \mbox{\bf Ay} + \mbox{\bf s}\end{displaymath}$

(1)

It turns out that many problems in the form of (9-5-1) can be reformulated into what we will call the Atkinson form. It is

$\begin{displaymath} \mbox{\bf J}\frac{\partial}{\partial z} \mbox{\bf y} \eq [\mbox{\bf G}(z) + \lambda\mbox{\bf H}(z)]\mbox{\bf y}\end{displaymath}$

(2)

where $\mbox{\bf J}$ is a skew-Hermitian matrix $(\mbox{\bf J}^{\ast} = -\mbox{\bf J})$ independent of z, $\mbox{\bf G}(z)$ and $\mbox{\bf H}(z)$ are Hermitian matrices $(\mbox{\bf H}^{\ast} = \mbox{\bf H})$ ,and $\lambda$ is a scalar which will come to play the role of an eigenvalue. For example, in acoustics we have

$\begin{displaymath} \frac{\partial}{\partial z}\left[ \begin{array} {l} P \\ W ... ... + \left[ \begin{array} {l} s_{p} \\ s_{w} \end{array} \right]\end{displaymath}$

(3)

which can be premultiplied by a skew-Hermitian matrix to give

$\begin{displaymath} \left[ \begin{array} {rr} 0 & -i \\ -i & 0 \end{array} \rig... ... + \left[ \begin{array} {l} s_{p} \\ s_{w} \end{array} \right]\end{displaymath}$

(4)

The significant thing about (9-5-4) is that the operators are self-adjoint, meaning that the right-hand matrix is Hermitian and so is the left-hand operator. To understand why $\mbox{\bf J}(\partial/\partial z)$ is Hermitian, write it out as a difference approximation

	$\begin{eqnarray} \mbox{\bf J}\frac{\partial}{\partial z} &=& \frac{i}{\triangle ... ...& & \\ & & $i$\space & $-i$\space & & & & \\ \end{array} \right]\end{eqnarray}$
		(5)

Inspecting (9-5-5) we see that it is two rows short of being square. Choosing two boundary conditions will be like obtaining two more rows. Clearly (9-5-5) is so close to being Hermitian that two more rows can be chosen to make it Hermitian. For example, the two rows

$\begin{displaymath} \left[ \begin{array} {rrrrrrrr} & & & & & & & i \\ -i & & & & & & & \end{array} \right] \end{displaymath}$

could be squeezed between the top and bottom halves of (9-5-5). Since the operator (9-5-5) can be made Hermitian by choice of suitable boundary conditions and since the other operators in (9-5-4) are already Hermitian, it seems that the Atkinson form applies to physical problems in which the reciprocity principle is applicable. Reciprocity does apply to most geophysical prospecting problems. A simple physical situation in which reciprocity does not apply is sound waves in a windy atmosphere. Physically it is because waves go more slowly upwind than downwind, and mathematically it is because no $\mbox{\bf J}$ matrix can be found to convert (9-5-1) into the form (9-5-2). Only in a source-free region can we convert (9-5-1) to (9-5-2). If we choose to let $\omega$ play the role of the eigenvalue, then taking source terms to be zero we split (9-5-4) into

$\begin{displaymath} \left[ \begin{array} {rr} 0 & -i \\ -i & 0\end{array} \righ... ...\right] \, \left[ \begin{array} {c} P \\ W \end{array} \right]\end{displaymath}$

(6)

Here $\mbox{\bf G}(z)$ has turned out to vanish and $k^{2}_{x}/\omega^{2}$ ,which is proportional to the sine of the incident angle, is to be regarded as a constant for variable values of the eigenvalue $\omega$ .Alternatively, we could choose -k²_x to be the eigenvalue, and then (9-5-4) would become

$\begin{displaymath} \left[ \begin{array} {rr} 0 & -i \\ -i & 0\end{array} \righ... ... \right] \left[ \begin{array} {c} P \\ W \end{array} \right]\end{displaymath}$

(7)

Obviously, still another possibility is to let the angle variable $-k^{2}_{x}/\omega^{2}$ be the eigenvalue for fixed $\omega$ .

The Atkinson form (9-5-2) leads directly to various conservation principles. Let us computer the vertical derivative of the quadratic form $\mbox{\bf y}^{\ast}\mbox{\bf Jy}$ .

$\begin{eqnarray} \frac{\partial}{\partial z} \mbox{\bf y}^{\ast}\mbox{\bf Jy} &=... ...\\ &=& (\lambda - \lambda^{\ast})\mbox{\bf y}^{\ast}\mbox{\bf Hy}\end{eqnarray}$

(8)

Very often we take the eigenvalues $\omega$ ,-k²_x, or $-k^{2}_{x}/\omega^{2}$ to be real, and in such a case we have $\lambda - \lambda^{\ast} = 0$ and (9-5-8) shows the $\mbox{\bf y}^{\ast}\mbox{\bf Jy}$ is a quadratic function of the wave variables which is invariant with z. In the acoustic example, this quadratic invariant is proportional to the energy flux. Specifically

$\begin{eqnarray} \mbox{\bf y}^{\ast}\mbox{\bf Jy} &=& - i [P^{\ast} \hspace{.2in... ...\ast}W + W^{\ast}P) \eq -2i \; \mbox{\rm Re} (P^{\ast}W) \nonumber\end{eqnarray}$ (9)

If we wish to consider a complex frequency $\omega = \omega_{r} + i\omega_{i}$ , then in the first acoustic example (9-5-6) equation (9-5-8) becomes

$\begin{displaymath} -\frac{\partial}{\partial z} \mbox{\rm Re} (P^{\ast}W) \eq +... ...^{2}_{x}}{\omega^{2}\rho}\bigg)P^{\ast}P + \rho W^{\ast}W\bigg]\end{displaymath}$ (10)

Noting that if P and W have time dependence $\mbox{\rm exp}[-i(\omega_{r} + i\omega_{i})t] = \mbox{\rm exp} (-i\omega_{r}t + \omega_{x}t)$ , then quadratics like $P^{\ast}P$ and $W^{\ast}W$ have time dependence $e^{2\omega_{i}t}$ and we see that the multiplier $2\omega_{i}$ can be regarded as a time derivative. Hence (9-5-10) becomes

$\begin{displaymath} -\frac{\partial}{\partial z} \mbox{\rm Re} (P^{\ast}W) \eq +... ...ast}P + \rho W^{\ast}W \bigg\} \eq \frac{\partial}{\partial t}E\end{displaymath}$ (11)

Equation (9-5-11) is interpreted as saying that the time derivative of the energy density E at a point is proportional to the negative of the divergence of energy flux at that point. In other problems the quadratic forms need not always turn out to involve energy. Sometimes momentum in involved.

A well-known theorem in matrix theory is that Hermitian matrices have real eigenvalues. Why then did we consider the possibility of a complex eigenvalue in (9-5-8)? The answer is that the finite difference operator matrix need not be chosen to have boundary conditions which make the operators Hermitian. In particular, for $\partial E/\partial t$ to be nonzero, energy must leak in or out at a boundary.

Now, let us suppose boundary conditions have been chosen to make $\mbox{\bf J}\partial/\partial z$ symmetric so the eigenvalues become real. Let y_n(z) be a solution to (9-5-2) with eigenvalue $\lambda_{n}$ ,and let y_m(z) be another solution with a different eigenvalue $\lambda_{m}$ .The reasoning which led up to (9-5-8) can be used to obtain

$\begin{displaymath} \frac{\partial}{\partial z}(\mbox{\bf y}^{\ast}_{m}\mbox{\bf... ...\lambda_{m})\mbox{\bf y}^{\ast}_{m}\mbox{\bf H}\mbox{\bf y}_{n}\end{displaymath}$ (12)

Integrating through z from z_a to z_b, we have

$\begin{displaymath} \mbox{\bf y}^{\ast}_{m}\mbox{\bf J}\mbox{\bf y}_{n}\; \bigg\... ...z, \lambda_{m})\mbox{\bf H}(z)\mbox{\bf y}(z, \lambda_{n})\, dz\end{displaymath}$ (13)

If boundary conditions have been chosen so that no energy gets in or out at z_a and z_b, then the left-hand side vanishes. Since by hypothesis $\lambda_{n} \neq \lambda_{m}$ we must have the right-hand integral vanishing. This states the orthogonality of the two solutions (called the two modes) and the idea is the same as the orthogonality of eigenvectors of the Hermitian difference operator matrices. The orthogonality of these functions is frequently useful in theoretical and computational work.

Exercices:

Show that application of (9-5-8) to (9-5-7) leads to a definition of horizontal energy flux. You may wish to take k_x = k_r + ik_i and assume $\vert k_{r}\vert \gg \vert k_{i}\vert$ .

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