1. A P-wave incident on the boundary at an angle of 30 degrees between media of velocities V1 and V2 . The angle of P-wave incidence ip and reflected P & S - waves are i’p ,i’s and refracted P & S – waves are rp and rs respectively .
Velocity of
reflected P-wave is (Vp1) = 4000m/s
Velocity of
reflected S-wave is (Vs1) = 2500m/s
Velocity of
refracted P-wave is (Vp2) = 5000m/s
Velocity of
refracted S-wave is (Vs2) = 3500m/s respectively.
Solution:
Snell's law:
$$\frac{sini_{p}}{V_{1p}}= \frac{sini^{!}_{p}}{V_{1p}} =\frac{sini^{!}_{s}}{V_{1s}}=\frac{sinr_{p}}{V_{2p}}=\frac{sinr_{s}}{V_{2s}}$$
Velocity of
reflected P-wave is (Vp1) = 4000m/s
Velocity of
reflected S-wave is (Vs1) = 2500m/s
Velocity of
refracted P-wave is (Vp2) = 5000m/s
Velocity of
refracted S-wave is (Vs2) = 3500m/s
$i^{!}_{p}$ - reflected P-wave
$i^{!}_{s}$ - reflected s-wave
$r_{p}$- refracted p-wave
$r_{s}$- refracted s-wave
Reflected P-wave
angle is:
$$\frac{sini_{p}}{V_{1p}}=
\frac{sini^{!}_{p}}{V_{1p}} $$
$$\frac{sin30^{0}}{4000}=
\frac{sini^{!}_{p}}{4000} $$
$$\frac{sini^{!}_{p}}{4000}=\frac{sin30^{0}}{4000}
$$
$${sini^{!}_{p}}=\frac{sin30^{0}}{4000}
\times4000 $$
$${sini^{!}_{p}}={sin30^{0}}
$$
$${i^{!}_{p}}={30^{0}}
$$
Refracted P-wave
angle is
$$\frac{sini_{p}}{V_{1p}}= \frac{sinr_{p}}{V_{2p}} $$
$$\frac{sin30^{0}}{4000}= \frac{sinr_{p}}{5000} $$
$$ \frac{sinr_{p}}{5000}=\frac{sin30^{0}}{4000} $$
$${sinr_{p}}=\frac{sin30^{0}}{4000} \times5000 $$
$${sinr_{p}}={sin30^{0}}\times1.25 $$
$${sinr_{p}}=0.5 \times1.25 $$
$${sinr_{p}}=0.625 $$
$$r=\arcsin{(0.625)} $$
$$r=38.68^{0} $$
Reflected S-wave
angle is:
$$\frac{sini_{p}}{V_{1p}}= \frac{sini^{!}_{s}}{V_{1s}} $$
$$\frac{sin30^{0}}{4000}= \frac{sini^{!}_{s}}{2500} $$
$$\frac{sini^{!}_{s}}{2500}=\frac{sin30^{0}}{4000} $$
$${sini^{!}_{s}}=\frac{sin30^{0}}{4000} \times2500 $$
$${sini^{!}_{s}}=0.31 $$
$$i_{s}^{!}=\arcsin{(0.31)} $$
$$i_{s}^{!}=18^{0} $$
Refracted S-wave
angle is
$$\frac{sini_{p}}{V_{1p}}= \frac{sinr_{s}}{V_{2s}} $$
$$\frac{sin30^{0}}{4000}= \frac{sinr_{s}}{3500} $$
$$ \frac{sinr_{s}}{3500}=\frac{sin30^{0}}{4000} $$
$${sinr_{s}}=\frac{sin30^{0}}{4000} \times3500 $$
$${sinr_{s}}={sin30^{0}}\times0.875 $$
$${sinr_{s}}=0.5 \times0.875 $$
$${sinr_{s}}=0.4375 $$
$$r_{s}=\arcsin{(0.4375)} $$
$$r_{s}=25.9^{0} $$
$${sinr_{s}}=\frac{sin30^{0}}{4000} \times3500 $$
$${sinr_{s}}={sin30^{0}}\times0.875 $$
$${sinr_{s}}=0.5 \times0.875 $$
$${sinr_{s}}=0.4375 $$
$$r_{s}=\arcsin{(0.4375)} $$
$$r_{s}=25.9^{0} $$
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