Diffraction from a half-plane

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Problem 2.19

The general equation for determining the diffraction effect of a plane surface is given by equation (2.19b). Show that the diffraction effect of the half-plane in Figure 2.19a is given by the integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int\nolimits_{t=tr}^{+\infty } \phi \left(t\right)\mathrm{d}t} where


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi \left(t\right)&=\frac{\left(4chy_{0} /\pi V^{3} t\right)}{\left(t^{2} +t_{y}^{2} -t_{r}^{2} \right)(t^{2} -t_{r}^{2} )^{1/2} }, \quad t>t_{r}.\\ &=0,\qquad\qquad\qquad\qquad\qquad\quad\ \ t<t_{r}, \end{align} } (2.19a)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c} is a constant, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V} is the velocity (assumed to be constant), while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h,y_{0}, } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \xi } are defined in Figure 2.19a, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_{y} } , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_{r} } are two-way traveltimes along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \xi, y_{0} } , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r} .

Figure 2.19a  Calculating diffraction effect of a plane surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S} .

Background

The use of rays to describe wave propagation simplifies the phenomenon by ignoring diffraction (spreading of energy radiating from a virtual point source). Since a wave is reflected by all parts ofa surface, we can consider each point on the surface as a point source (Huygens's principle, see problem 3.1) and integrate over the surface to get the correct total effect. For a coincident source and receiver, the integral of the effects of point sources over the surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S} can be transformed into a line integral around the boundary of the surface (see Sheriff and Geldart, 1995, Section 2.8.2). When the origin is over the surface, the integration gives two terms, one term representing the reflection given by ray theory, the other the diffraction. When the origin is not over the surface, the reflection term is zero leaving only the diffraction term.

The diffraction response of a plane area to a unit impulse (see Sheriff and Geldart, 1955, Section 15.2.5) emitted by a source at the origin and recorded at the origin is obtained by integrating the quantity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi \left(t\right)} around the entire boundary of the area, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi \left(t\right)} is given by equation (2.19b) [see Sheriff and Geldart, 1995, equation (2.131)]:


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi \left(t\right)=\left(ch/\pi V^{2} t^{2} \right)\left(\frac{\mathrm{d}\theta }{\mathrm{d}t} \right), \end{align} } (2.19b)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi \left(t\right)} is the response of a unit element of the boundary, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t} is the two-way traveltime from the origin to the element of the boundry, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta} is the angle shown in Figure 2.19a.

Solution

In Figure 2.19a

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} OA&=\xi, \\ \xi ^{2} &=r^{2} +x^{2} =r^{2} +y_{0}^{2} \tan^{2} \theta, \end{align}}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y_{0} } is normal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle BD} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x=O'A} . The points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D} are in fact at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \pm \infty } ; thus, on integrating, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x} goes from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\infty } to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle +\infty } while 6 goes from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\pi /2} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle +\pi /2} . Dividing by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \left(V/2\right)} , we get the following relation between the traveltimes:


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} (2\xi /V)^{2} =t^{2} =t_{r}^{2} +t_{y}^{2} \tan ^{2} \theta. \end{align} } (2.19c)

Then


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\mathrm{d}t}{\mathrm{d}\theta } =\left(t_{y}^{2} /t\right)\tan\theta \sec^{2} \theta \end{align} } (2.19d)

and


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\mathrm{d}\theta }{\mathrm{d}t} =\left(t/t_{y}^{2} \right) \cot\theta \cos^{2} \theta. \end{align} } (2.19e)

From equation (2.19c) we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {\tan}^{2} \theta &=\left(t^{2} -t_{r}^{2} \right)/t_{y}^{2}, \\ {\cos}^{2} \theta &=1/\left(1+{\tan}^{2} \theta \right)\\ &=1/\left[1+\left(t^{2} -t_{r}^{2} \right)/t_{y}^{2} \right]\\ &=t_{y}^{2} /\left(l^{2} +l_{y}^{2} -t_{2}^{r} \right), \end{align}}

so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\mathrm{d}\theta }{\mathrm{d}t} &=\left(\frac{t}{t_{y}^{2} } \right)\frac{t_{y} }{(t^{2} -t_{r}^{2} )^{1/2} } \frac{t_{y}^{2} }{\left(t^{2} +t_{y}^{2} -t_{r}^{2} \right)} \\ &= \frac{t_y t}{(t^2 + t_y^2-t_r^2)(t^2-t_r^2)^{1/2}}. \end{align}}

Substituting this expression in equation (2.19b), we obtain the following result for the diffraction effect of a unit length of the boundary in Figure 2.19a:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi \left(t\right)=\frac{\left(ch/\pi V^{2} t^{2} \right)\left(t_{y} t\right)}{\left(t^{2} + t_{y}^{2} -t_{r}^{2} \right)(t^{2} -t_{r}^{2} )^{1/2} }. \end{align}}

Substituting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_{y} =2y_{0} /V} in the numerator, we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi \left(t\right)=\frac{\left(2chyo/\pi V^{3} t\right)}{\left(t^{2} + t_{y}^{2} -t_{r}^{2} \right)(t^{2} -t_{r}^{2} )^{1/2} }. \end{align}}

To get the total diffraction effect of the half-plane, we integrate this expression around the four sides of the half-plane. Three of the four sides are at infinity so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t} is infinite and the fraction, being proportional to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t^{-2} } , vanishes. Therefore the effect reduces to the integral along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle BD} . Because of symmetry, we can integrate along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle OD} and double the result. Thus, the diffraction function for a half-plane is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi \left(t\right)&=\frac{\left(4chyo/\pi V^{3} t\right)}{\left(t^{2} +t_{y}^{2} - t_{r}^{2} \right)(t^{2} -t_{r}^{2} )^{1/2} }, \quad t\ge t_{r}, \\ &=0,\qquad\qquad\qquad\qquad\qquad\quad\ \, t\le t_{r}. \end{align}}

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