海浪非常薄层-地球科学堆栈交换江南电子竞技平台江南体育网页版 最近30从www.hoelymoley.com 2023 - 07 - 09 - t13:12:08z //www.hoelymoley.com/feeds/question/19038 https://creativecommons.org/licenses/by-sa/4.0/rdf //www.hoelymoley.com/q/19038 4 海浪非常薄层 弗雷德里克 //www.hoelymoley.com/users/18769 2020 - 01 - 21 t11:11:36z 2020 - 03 - 09 - t05:09:39z

Elastic waves in the earth are described by the elastodynamic equations \begin{align} \rho \frac{\partial^2 u}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 u}{\partial x^2} + \mu \left( \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + (\lambda+\mu) \left( \frac{\partial^2 v}{\partial x \partial y} + \frac{\partial^2 w}{\partial x \partial z} \right) \\ \rho \frac{\partial^2 v}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 v}{\partial y^2} + \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial z^2} \right) + (\lambda+\mu) \left( \frac{\partial^2 u}{\partial x \partial y} + \frac{\partial^2 w}{\partial y \partial z} \right) \\ \rho \frac{\partial^2 w}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 w}{\partial z^2} + \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} \right) + (\lambda+\mu) \left( \frac{\partial^2 u}{\partial x \partial z} + \frac{\partial^2 v}{\partial y \partial z} \right) \end{align}

Is it possible to have a very thin and stiff layer in a surrounding soft material? If yes, is it possible to approximate the thin layer by a thin film with free boundaries and a uniform displacement over the thickness direction ($\partial / \partial z = 0$)? In this case, the equations of motion become \begin{align} \rho \frac{\partial^2 u}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 u}{\partial x^2} + \mu \frac{\partial^2 u}{\partial y^2} + (\lambda+\mu) \frac{\partial^2 v}{\partial x \partial y} \\ \rho \frac{\partial^2 v}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 v}{\partial y^2} + \mu \frac{\partial^2 v}{\partial x^2} + (\lambda+\mu)\frac{\partial^2 u}{\partial x \partial y} \\ \rho \frac{\partial^2 w}{\partial t^2} &= \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} \right) \end{align} Do these equations describe a realistic case? Does this state have a specific name (like plane stress or antiplane shear ...)?

//www.hoelymoley.com/questions/19038/-/19407 # 19407 6 回答dvoytan波在薄层 dvoytan //www.hoelymoley.com/users/18386 2020 - 03 - 09 - t05:09:39z 2020 - 03 - 09 - t05:09:39z < p >取消你提出的问题是,它是在全球实施PDE而不是本地(仅在薄层)。< / p >

Usually when we talk about vertically layered media we use a plane wave $\exp(i(\mathbf{k}\cdot\mathbf{x}-wt))$ trial solution (you can plug this in to confirm that it satisfies the PDE) and match boundary conditions at the interface of two layers to determine reflection and transmission coefficients. If you are not familiar with this, Aki and Richards (1980) is a good reference. See also the Zoeppritz equations https://en.wikipedia.org/wiki/Zoeppritz_equations

Introducing a very thin, stiff layer is equivalent to placing a 0 displacement (Dirichlet) boundary condition at the location of the layer. Physically, this models an infinite impedance reflector and will generate a total reflection. Waves on a string with both ends fixed are modeled in the same way.

Setting vertical derivatives in displacement to 0 is imposing a Neumann B.C. on the location of the thin layer. This is exactly how the free surface is modeled. Physically, this models vanishing vertical tractions at the boundary location.

So, if you want to model seismic wave propagation with an extremely stiff layer, then use the Dirichelet 0 displacement boundary condition. If the thin layer is only 'very stiff', use the reflection and transmission coefficients provided by the Zoeppritz equations. If you want tractions to disappear, use the Neumann boundary conditions. But, beware; this doesn't make physical sense if the thin layer is sandwiched between two other layers.

Hope this helps. -D

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