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Seismic Problem


1.A seismic wave emanating from a shot point, when incident at angle 30 degrees at a point P on a refraction, gets refracted at an angle 60 degrees. If the wave is incident 52m away from P, it would be critically refracted.(Thanks to Neeraja-AU)

Find the

a) critical angle

2)depth

3) cross over distance.






a) critical angle



\frac{sini}{sinr}=\frac{V_{1}}{V_2}


\frac{sin30^o}{sin60^o}=\frac{V_{1}}{V_2}


\frac{V1}{V2}=\frac{1/2}{\sqrt3/2}=\frac{1}{\sqrt3}


\frac{sini}{sinr}=\frac{V_{1}}{V_2}


\frac{sini_c}{sin90^o}=\frac{1}{\sqrt3}


sini_c=\frac{1}{\sqrt{3}}


\cos i_c=\sqrt{(1-(\frac{1}{\surd3})^2)}

=\sqrt{1-\frac{1}{3}}

=\sqrt{\frac{3-1}{3}}

=\sqrt{\frac{2}{3}}


\tan i_c=\frac{sini_c}{cosi_c}=\frac{1/\sqrt{3}}{\sqrt{2}/\sqrt{3}}=\frac{1}{\sqrt{2}}


i_c=\tan^{-1}(\frac{1}{\sqrt{2}})



b) The depth of the refractor is about:



\triangle SPB


\tan30^o=\frac{x}{h}


\frac{1}{\sqrt{3}}=\frac{x}{h}


h=x\sqrt{3}


x=\frac{h}{\sqrt{3}}




\triangle SDC


\tan i_c=\frac{x+52}{h}


\frac{1}{\sqrt{2}}=\frac{x+52}{h}


\frac{h}{\sqrt{2}}={x+52}


\frac{h}{\sqrt{2}}=\frac{h}{\sqrt{3}}+52


h[\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}]=52


h[\frac{\sqrt{3}-\sqrt{2}}{\sqrt{6}}]=52


h=400m



c) cross over distance



z=\frac{x_{crs}}{2}\sqrt{\frac{V_2-V_1}{V_2+V_1}}


x_{crs}=2z\sqrt{\frac{V_2+V_1}{V_2-V_1}}


x_{crs}=2z\sqrt{\frac{V_2+V_1}{V_2-V_1}}

=2\times400\sqrt{\frac{1+\frac{V_1}{V_2}}{1-\frac{V_1}{V_2}}}

=800\sqrt{\frac{1+\frac{1}{\sqrt{3}}}{1-\frac{1}{\sqrt{3}}}}

=800\sqrt{\frac{\sqrt{3}+1}{\sqrt{3}-1}}

=800\times1.9311



xcrs=1540m

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