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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