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==== = -

+ C'\

ActualHeight, ActualWidth, Clip, Cursor, DataContext, DesiredSize, Dispatcher, DownloadProgress, Height, HorizontalAlignment, IsHitTestVisible, Language, Margin, MaxHeight, MaxWidth, MinHeight, MinWidth, Name, Opacity, OpacityMask, Parent, RenderSize,

C'\ C'\

+ ...

(4.1.27)

RenderTransform, RenderTransformOrigin, Resources, Source, Stretch, Style, Tag, Triggers, VerticalAlignment, Visibility, Width

Thus, the bilocal approximation accounts for those weakly connected diagrams that are formed by products of bilocal diagrams (i.e.,

c:::::. ).

(C(r, ro)) = C(O)(r, ro)

Arrange, ArrangeOverride, CaptureMouse, CheckAccess, Equals, Finalize, FindName, GetAnimationBaseValue, GetHashCode, GetType, GetValue, HitTest, InvalidateArrange, InvalidateMeasure, Measure, MeasureOverride, MemberwiseClone, ReleaseMouseCapture, SetBinding, SetValue, ToString, TransformToVisual, UpdateLayout

MIM2 C (O)(r, rl)C(O)(rl, r2) . (G(r2' ro))C(rl - r2)

(4.1.28)

Q(r) ~ C(r)C(O) (r)

Arrange, ArrangeOverride, CaptureMouse, CheckAccess, Equals, Finalize, FindName, GetAnimationBaseValue, GetHashCode, GetType, GetValue, HitTest, InvalidateArrange, InvalidateMeasure, Measure, MeasureOverride, MemberwiseClone, ReleaseMouseCapture, SetBinding, SetValue, ToString, TransformToVisual, UpdateLayout

(4.1.29)

(4.1.30)

where 5 is the variance of permittivity fluctuations and of k m ,

ActualHeight, ActualWidth, Clip, Cursor, DataContext, DesiredSize, Dispatcher, Fill, GeometryTransform, Height, HorizontalAlignment, IsHitTestVisible, Language, Margin, MaxHeight, MaxWidth, MinHeight, MinWidth, Name, Opacity, OpacityMask, Parent, RenderSize, RenderTransform, RenderTransformOrigin, Resources, Stretch, Stroke, StrokeDashArray, StrokeDashCap, StrokeDashOffset, StrokeEndLineCap, StrokeLineJoin, StrokeMiterLimit, StrokeStartLineCap, StrokeThickness, Style, Tag, Triggers, VerticalAlignment, Visibility, Width, X1, X2, Y1, Y2

Q(k)

(4.1.31) k 2a 2 + (1 - ik m a)2 Thus the effective propagation constant is using (4.1.23) and (4.1.31)

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m (4.1.32) K2 a2 + (1 - ik m a)2 It is important to keep both the leading real part and the leading imaginary part even though the leading imaginary part can be much smaller than the leading real part. The leading imaginary part represents attenuation due to scattering. If we assume K ~ k m , then 8k,4 a 2 8k,4 a 2 K 2 k2 + m = J...2 + m ' (4.1.33) m k2 a2 + (1 - ikm a)2 'm 1 - 2ikm a In ' For the case of a small correlation length of kma 1 we have K 2 ~ k n + 5k:~a2(1 + 2ik a) (4.1.34)

Arrange, ArrangeOverride, CaptureMouse, CheckAccess, Equals, Finalize, FindName, GetAnimationBaseValue, GetHashCode, GetType, GetValue, HitTest, InvalidateArrange, InvalidateMeasure, Measure, MeasureOverride, MemberwiseClone, ReleaseMouseCapture, SetBinding, SetValue, ToString, TransformToVisual, UpdateLayout

The first term in (4.1.34) can be regarded as the zeroth-order solution, and the second term in (4.1.34) can be regarded as the first-order solution. Thus, for (4.1.34) to be valid, the first-order solution has to be smaller than the zeroth-order solution, giving the condition

The principal root zl, in this case, is unity and instability occurs when lzil 2 1 (for i = 2,3, ,k, i # 1) and the true solution will eventually be swamped by this root as n increases If a method satisfies the above criteria, then it is said to be stable but the degree of stability requires further consideration Weak stability occurs where a method can be defined by the above as being stable, but because of the nature of the differential equation, the derivative part of (IV25) gives one or more roots which are greater than or equal to unity It has been shown by Dalquist [2] that a stable method which has the maximum order of accuracy is always weakly stable.

(4.1.35)

1, we may further simplify (4.1.34) to get 2 K ~ km + 8k;a + i2krn a)]

GotFocus, KeyDown, KeyUp, LayoutUpdated, Loaded, LostFocus, MouseEnter, MouseLeave, MouseLeftButtonDown, MouseLeftButtonUp, MouseMove, SizeChanged

(4.1.36)

We see that the randomness has introduced an effective loss term that causes the coherent wave to decay as it propagates in the random medium.

ActualHeight, ActualWidth, Clip, Cursor, DataContext, DesiredSize, Dispatcher, DisplayMemberPath, Height, HorizontalAlignment, HorizontalContentAlignment, IsHitTestVisible, IsTabStop, ItemContainerStyle, Items, ItemsHost, ItemsPanel, ItemsSource, ItemTemplate, Language, Margin, MaxHeight, MaxWidth, MinHeight, MinWidth, Name, Opacity, OpacityMask,

Q(k) =

=(0)

d1' e-ik.;p c(O) (r)C(r)

Padding, Parent, RenderSize, RenderTransform, RenderTransformOrigin, Resources, SelectedIndex, SelectedItem, SelectedItems, SelectionMode, Style, TabIndex, TabNavigation, Tag, Template, Triggers, VerticalAlignment, VerticalContentAlignment, Visibility, Width

(4.1.37)

=(0)

(r) -

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