<p>P, or pressure, waves striking a discontinuity, or other density anomaly, that are reflected off that anomaly at an angle to their original direction of travel present as S, or shear, waves to the material they travel into. This is because the P wave has a three dimensional oscillation vector component in the movement of the material caught up in it. Some of the energy of which will be absorbed by the boundary interaction, some of which will be transmitted and some of which will be transformed. Some of the transformed vector is represented by the refraction of the transmitted P waves while the rest becomes S waves transmitted at an angle to the incident P waves.</p>

CC BY-SA 4.0
2024-08-17T01:39:08.830Z
2024-08-17T01:39:08.830Z
Ash
<p>Generally: you have to consider the continuity of the displacement and stress at the interface.</p>
<p>The following image is taken from <a href="https://www.amazon.de/Solid-Earth-Introduction-Global-Geophysics/dp/0521893070" rel="nofollow noreferrer">Fowler: The Solid Earth</a> and the derivation also follows their explanation</p>
<p><a href="https://i.sstatic.net/9BpXP6KN.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/9BpXP6KN.jpg" alt="p and s waves at an interface. Taken from Fowler: The solid Earth" /></a>
In the diagramme, Snell's law gives you <span class="math-container">$$ \frac{\sin i}{v_{p1}} = \frac{\sin e_1}{v_{p1}} = \frac{\sin f_1}{v_{s1}} = \frac{\sin f2}{v_{s2}} = \frac{\sin e2}{v_{p2}} =: p$$</span> where <span class="math-container">$p$</span> is often called 'ray parameter'.</p>
<p>In detail to get the importance of the different waves, you need at the interface a continuity in the displacement as well as the stress, both in horizontal as well as in vertical direction. The continuity of vertical and horizontal replacement can be expressed (<span class="math-container">$A$</span> and <span class="math-container">$B$</span> denoting the amplitudes of the p- and s-waves respectively) as
<span class="math-container">$$ A_1\cos e1 - B_1 \sin f_1 + A_2 \cos e_2 - B_2 \sin f_2 = A0 \cos i$$</span> and <span class="math-container">$$ A_1\sin e_1 + B_1\cos f_1 - A_2\sin e_2 - B_2\cos f_2 = -A_0\sin i$$</span>.</p>
<p>The continuity of the stress in the media reads as <span class="math-container">$$A_1Z_1\cos 2f_1 - B_1W_1\sin 2f_1 - A_2Z_2\cos 2f_2 + B_2W_2\sin 2f_2 = -A_0Z_1\cos 2f_1$$</span> and <span class="math-container">$$ A_1\gamma_1W_1\sin 2e_1 + B_1W_1\cos 2f_1 + A_2\gamma_2W_2\sin 2e_2 + B_2W_2\cos 2f_2 = A_0\gamma_1W_1\sin 2i$$</span> with the impedances of the p-waves <span class="math-container">$Z_1 = \rho_1v_{p1}$</span>, <span class="math-container">$Z_2 = \rho_2v_{p2}$</span>, the impedance of the s-waves <span class="math-container">$W_1 = \rho_1v_{1s}$</span> and <span class="math-container">$W_2 = \rho_2v_{s2}$</span> and the ratio of the p- and s-wave velocities <span class="math-container">$\gamma_1 = v_{1p} / v_{2p}$</span> and <span class="math-container">$\gamma_2 = v_{1s} / v_{2s}$</span>.</p>
<p>Thus for given material properties (e.g. the velocities and densities) you can thus calculate the amplitudes of the different reflected and refracted waves, given a known incident wave. For vertical incident you can e.g. simplify that as there is no shear wave generated (<span class="math-container">$B_1 = B_2 = 0$</span>) and you can then solve for the ratio of outgoing and ingoing amplitudes, the reflection coefficient <span class="math-container">$$ \frac{A_1}{A_0} = \frac{Z_2-Z_1}{Z_2+Z_1} = \frac{\rho_2v_{2p}-\rho_1v_{1p}}{\rho_2v_{2p}+\rho_1v_{1p}}$$</span> and the transmission coefficient <span class="math-container">$$\frac{A_2}{A_0} = \frac{2Z_1}{Z_2 + Z_1} = \frac{2\rho_1v_{1p}}{\rho_2v_{2p}+\rho_1v_{2p}}$$</span></p>

CC BY-SA 4.0
2024-08-19T14:51:55.687Z
2024-08-19T14:51:55.687Z
planetmaker
<p>This image (taken from a 1982 paper by L. Völgyesi and M. Moser) shows that seismic S-waves are generated at reflection and refraction of P-waves. Why?</p>
<p><a href="https://i.sstatic.net/wiA5ggRY.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/wiA5ggRY.png" alt="enter image description here" /></a></p>

<seismology>
Why reflection and refraction of seismic P-wave generate S-waves?
Leos Ondra
# Why reflection and refraction of seismic P-wave generate S-waves?

(Asked by: username_0 on 2024-08-16)

This image (taken from a 1982 paper by L. Völgyesi and M. Moser) shows that seismic S-waves are generated at reflection and refraction of P-waves. Why?

[![enter image description here](https://i.sstatic.net/wiA5ggRY.png)](https://i.sstatic.net/wiA5ggRY.png)

**Tags:** seismology


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## 2 answers

### Answer by username_1 on 2024-08-17. Score: 1

P, or pressure, waves striking a discontinuity, or other density anomaly, that are reflected off that anomaly at an angle to their original direction of travel present as S, or shear, waves to the material they travel into. This is because the P wave has a three dimensional oscillation vector component in the movement of the material caught up in it. Some of the energy of which will be absorbed by the boundary interaction, some of which will be transmitted and some of which will be transformed. Some of the transformed vector is represented by the refraction of the transmitted P waves while the rest becomes S waves transmitted at an angle to the incident P waves.


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### Answer by username_2 on 2024-08-19. Score: 1

Generally: you have to consider the continuity of the displacement and stress at the interface.

The following image is taken from [Fowler: The Solid Earth](https://www.amazon.de/Solid-Earth-Introduction-Global-Geophysics/dp/0521893070) and the derivation also follows their explanation

[![p and s waves at an interface. Taken from Fowler: The solid Earth](https://i.sstatic.net/9BpXP6KN.jpg)](https://i.sstatic.net/9BpXP6KN.jpg)
In the diagramme, Snell's law gives you $$ \frac{\sin i}{v\_{p1}} = \frac{\sin e\_1}{v\_{p1}} = \frac{\sin f\_1}{v\_{s1}} = \frac{\sin f2}{v\_{s2}} = \frac{\sin e2}{v\_{p2}} =: p$$ where $p$ is often called 'ray parameter'.

In detail to get the importance of the different waves, you need at the interface a continuity in the displacement as well as the stress, both in horizontal as well as in vertical direction. The continuity of vertical and horizontal replacement can be expressed ($A$ and $B$ denoting the amplitudes of the p- and s-waves respectively) as
$$ A\_1\cos e1 - B\_1 \sin f\_1 + A\_2 \cos e\_2 - B\_2 \sin f\_2 = A0 \cos i$$ and $$ A\_1\sin e\_1 + B\_1\cos f\_1 - A\_2\sin e\_2 - B\_2\cos f\_2 = -A\_0\sin i$$.

The continuity of the stress in the media reads as $$A\_1Z\_1\cos 2f\_1 - B\_1W\_1\sin 2f\_1 - A\_2Z\_2\cos 2f\_2 + B\_2W\_2\sin 2f\_2 = -A\_0Z\_1\cos 2f\_1$$ and $$ A\_1\gamma\_1W\_1\sin 2e\_1 + B\_1W\_1\cos 2f\_1 + A\_2\gamma\_2W\_2\sin 2e\_2 + B\_2W\_2\cos 2f\_2 = A\_0\gamma\_1W\_1\sin 2i$$ with the impedances of the p-waves $Z\_1 = \rho\_1v\_{p1}$, $Z\_2 = \rho\_2v\_{p2}$, the impedance of the s-waves $W\_1 = \rho\_1v\_{1s}$ and $W\_2 = \rho\_2v\_{s2}$ and the ratio of the p- and s-wave velocities $\gamma\_1 = v\_{1p} / v\_{2p}$ and $\gamma\_2 = v\_{1s} / v\_{2s}$.

Thus for given material properties (e.g. the velocities and densities) you can thus calculate the amplitudes of the different reflected and refracted waves, given a known incident wave. For vertical incident you can e.g. simplify that as there is no shear wave generated ($B\_1 = B\_2 = 0$) and you can then solve for the ratio of outgoing and ingoing amplitudes, the reflection coefficient $$ \frac{A\_1}{A\_0} = \frac{Z\_2-Z\_1}{Z\_2+Z\_1} = \frac{\rho\_2v\_{2p}-\rho\_1v\_{1p}}{\rho\_2v\_{2p}+\rho\_1v\_{1p}}$$ and the transmission coefficient $$\frac{A\_2}{A\_0} = \frac{2Z\_1}{Z\_2 + Z\_1} = \frac{2\rho\_1v\_{1p}}{\rho\_2v\_{2p}+\rho\_1v\_{2p}}$$


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