Solutions to Exercises

Abstract

From (1.12) at a fixed location s we can write $x = \sqrt {2{I_x}{\beta _x}} \cos {\phi _x},$where φ_x includes the initial phase φ₀ We then have

$$\begin{array}{*{20}{c}}{{{ \in }_{x}} = \frac{{\left\langle {{{x}^{2}}\left( s \right)} \right\rangle }}{{{{\beta }_{x}}\left( s \right)}}} \\{ = \int {d{{\phi }_{x}}d{{I}_{x}}2{{I}_{x}}{{{\cos }}^{2}}{{\phi }_{x}}\rho \left( {{{I}_{x}},{{\phi }_{x}}} \right)} } \\{ = \int {d{{\phi }_{x}}d{{I}_{x}}2{{I}_{x}}{{{\cos }}^{2}}{{\phi }_{x}}\rho \left( {{{I}_{x}}} \right)\frac{1}{{2\pi }}} } \\{ = \int {d{{I}_{x}}{{I}_{x}}\rho \left( {{{I}_{x}}} \right) = \left\langle {{{I}_{x}}} \right\rangle .} } \\\end{array}$$

(12.1)

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