<p>A <em>Poisson solid</em> is a homogeneous isotropic medium whose linear elasticity can be modelled by a just single parameter, e.g. the <a href="https://en.wikipedia.org/wiki/Shear_modulus" rel="nofollow noreferrer">shear modulus</a> (<span class="math-container">$\mu$</span>), and which has a <a href="https://en.wikipedia.org/wiki/Poisson%27s_ratio" rel="nofollow noreferrer">Poisson's ratio</a>
<span class="math-container">$$\nu = \frac{1}{4}$$</span></p>
<p>The velocity of waves in a Poisson solid depends only on the shear modulus and mass density (<span class="math-container">$\rho$</span>) of the material:</p>
<ul>
<li><a href="https://en.wikipedia.org/wiki/P_wave" rel="nofollow noreferrer">P (primary/pressure) wave</a> speed: <span class="math-container">$\alpha=\sqrt{\frac{3\mu}{\rho}}$</span></li>
<li><a href="https://en.wikipedia.org/wiki/S_wave" rel="nofollow noreferrer">S (secondary/shear) wave</a>  speed: <span class="math-container">$\ \beta=\sqrt{\frac{\mu}{\rho}}$</span></li>
<li><a href="https://en.wikipedia.org/wiki/Rayleigh_wave" rel="nofollow noreferrer">Rayleigh surface wave</a> speed: <span class="math-container">$\quad\  c_R = \sqrt{0.8453}\beta =\sqrt{\frac{2.536\mu}{\rho}}$</span></li>
</ul>
<p>More information is given in <a href="https://physics.stackexchange.com/a/836156/145491">my answer</a> to <a href="https://physics.stackexchange.com/q/746421/145491">&quot;What is a <em>Poisson Solid</em>?&quot;</a> on Physics Stack Exchange.</p>

CC BY-SA 4.0
2024-12-08T10:48:44.763Z
Along with what's been posted here, keep in mind these 2 points: 1] these properties change at boundary/layer interfaces (in seismic terms "acoustic" or - because we are dealing with continuum mechanics here with elastic attributes - "shear impedance") and 2] think of all these equations and corresponding ideas in terms of "stress" (e.g. any seismic source) and "strain" (the earth convolving with or "responding to") that propagating seismic energy.
nate
CC BY-SA 4.0
2024-12-04T16:29:10.670Z
2024-12-04T16:29:10.670Z
David Bailey
<p>What is a Poisson solid and what affect does it have on p and s waves?</p>

CC BY-SA 4.0
2022-08-10T00:28:51.303Z
Does this definition help. [An isotropic elastic material for which the Lamé elastic constants λ and μ are equal. A Poisson solid has Poisson’s ratio σ=1/4](https://wiki.seg.org/wiki/Dictionary:Poisson_solid)?
Fred
CC BY-SA 4.0
2024-11-15T12:42:57.043Z
The definition is all over the internet, but what is so special about it? What made someone think "I need a special name for when the Lamé coefficients are equal."? Does it simplify the equations? Does it **only** simplify equations or is there some interesting property (or a property commonly seen in real solids) that would make one set the Lamé coefficients equal to each other?
Phineas Nicolson
<geology><geophysics>
What is a Poisson solid and what affect does it have on p and s waves?
LAG
# What is a Poisson solid and what affect does it have on p and s waves?

(Asked by: username_0 on 2022-08-09)

What is a Poisson solid and what affect does it have on p and s waves?

**Tags:** geology, geophysics

> **Comments:**
> **username_1**: Does this definition help. [An isotropic elastic material for which the Lamé elastic constants λ and μ are equal. A Poisson solid has Poisson’s ratio σ=1/4](https://wiki.seg.org/wiki/Dictionary:Poisson_solid)?  
> **username_2**: The definition is all over the internet, but what is so special about it? What made someone think "I need a special name for when the Lamé coefficients are equal."? Does it simplify the equations? Does it **only** simplify equations or is there some interesting property (or a property commonly seen in real solids) that would make one set the Lamé coefficients equal to each other?  

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## 1 answers

### Answer by username_3 on 2024-12-04. Score: 2

A *Poisson solid* is a homogeneous isotropic medium whose linear elasticity can be modelled by a just single parameter, e.g. the [shear modulus](https://en.wikipedia.org/wiki/Shear_modulus) ($\mu$), and which has a [Poisson's ratio](https://en.wikipedia.org/wiki/Poisson%27s_ratio)
$$\nu = \frac{1}{4}$$

The velocity of waves in a Poisson solid depends only on the shear modulus and mass density ($\rho$) of the material:

* [P (primary/pressure) wave](https://en.wikipedia.org/wiki/P_wave) speed: $\alpha=\sqrt{\frac{3\mu}{\rho}}$
* [S (secondary/shear) wave](https://en.wikipedia.org/wiki/S_wave) speed: $\ \beta=\sqrt{\frac{\mu}{\rho}}$
* [Rayleigh surface wave](https://en.wikipedia.org/wiki/Rayleigh_wave) speed: $\quad\ c\_R = \sqrt{0.8453}\beta =\sqrt{\frac{2.536\mu}{\rho}}$

More information is given in [my answer](https://physics.stackexchange.com/a/836156/145491) to ["What is a *Poisson Solid*?"](https://physics.stackexchange.com/q/746421/145491) on Physics Stack Exchange.

> **Comments:**
> **username_4**: Along with what's been posted here, keep in mind these 2 points: 1] these properties change at boundary/layer interfaces (in seismic terms "acoustic" or - because we are dealing with continuum mechanics here with elastic attributes - "shear impedance") and 2] think of all these equations and corresponding ideas in terms of "stress" (e.g. any seismic source) and "strain" (the earth convolving with or "responding to") that propagating seismic energy.  

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