Homework 3

Theoretical part

1. In class, we derived the following acoustic wave equation for pressure $P(\mathbf{x},t)$:
   

   $${\frac{1}{V^2(\mathbf{x})}}\,{\frac{\partial^2 P}{\partial t... ...nabla\left(\frac{1}{\rho(\mathbf{x})}\right) \cdot \nabla P\;.$$ (1)

   
   
   Using the connection between the pressure and displacement, derive the acoustic wave equation for the displacement vector $\mathbf{u}(\mathbf{x},t)$:
   

   $$\rho(\mathbf{x})\,\frac{\partial^2 \mathbf{u}}{\partial t^2} = - \nabla P = \nabla \left(K(\mathbf{x})\nabla \cdot \mathbf{u}\right)$$ (2)

   $${\frac{1}{V^2(\mathbf{x})}}\,{\frac{\partial^2 \mathbf{u}}{\parti... ...mathbf{u}) + \frac{\nabla\cdot\mathbf{u}}{K(\mathbf{x})}\; \nabla K(\mathbf{x})$$ (3)

   
   
   Thanks to the Helmholtz's theorem, we can consider the displacement vector field to be curl free for the acoustic propagation:
   

   $${\frac{1}{V^2(\mathbf{x})}}\,{\frac{\partial^2 \mathbf{u}}{\parti... ...mathbf{u}) + \frac{\nabla\cdot\mathbf{u}}{K(\mathbf{x})}\; \nabla K(\mathbf{x})$$ (4)

   
   

2. Consider a geometrical wave representation in the vicinity of a wavefront
   

   $${\mathbf{u}(\mathbf{x},t)} = \mathbf{a}(\mathbf{x})\,f\left(t-T(\mathbf{x})\right)$$ (5)

   
   
   and derive partial differential equations for the traveltime function $T(\mathbf{x})$ and the vector amplitude $\mathbf{a}(\mathbf{x})$.
   

   $$\ddot{\mathbf{u}} = \mathbf{a}\;\ddot{f}$$ (6)

   $$\nabla\cdot\mathbf{u} = f\;(\nabla\cdot\mathbf{a}) - \dot{f}\;(\mathbf{a}\cdot\nabla T)$$ (7)

   $$\nabla^2\;\mathbf{u} = f\;\nabla^2\mathbf{a} - 2\dot{f}\;(\nabla ... ...thbf{a} + \ddot{f}\;(\nabla T)^2\,\mathbf{a} - \mathbf{a}\;\dot{f}\;\nabla^2\,T$$ (8)

   
   
   
   

   $$\left[S^2(\mathbf{x})-(\nabla T)^2\right]\mathbf{a}\;\ddot{f}(t-T(\mathbf{x}))$$ (9)

   $$+ \left[2(\nabla T \cdot \nabla)\mathbf{a}+\mathbf{a}\;\nabla^2\,T ... ...t\nabla T}{K(\mathbf{x})}\; \nabla K(\mathbf{x})\right]\dot{f}(t-T(\mathbf{x}))$$ (10)

   $$= \left[\nabla^2\mathbf{a}+ \frac{\nabla\cdot \mathbf{a}}{K(\mathbf{x})}\; \nabla K(\mathbf{x}) \right]{f}(t-T(\mathbf{x}))$$ (11)

   
   
   The term in front of $\ddot{f}$ is zero to enable discontinuities to propagate and is the eikonal equation. The term in front of $\dot{f}$ is zero, if we neglect the lower order term in $f$ in the framework of high frequencies (WKBJ), and this is the transport equation.
3. Assuming that the geometrical wave propagates in the direction of the traveltime gradient
   

   $${\mathbf{a}(\mathbf{x})} = A(\mathbf{x})\,V(\mathbf{x})\,\nabla T$$ (12)

   
   
   show that the amplitude equation can take the conservative form
   

   $$\nabla \cdot \left(\rho(\mathbf{x})\,V^2(\mathbf{x})\,A^2(\mathbf{x})\,\nabla T\right) = 0$$ (13)

   $$\nabla \cdot \left(K(\mathbf{x})\,A^2(\mathbf{x})\,\nabla T\right) = 0$$ (14)

   $$A^2(\mathbf{x})\left[\nabla K \cdot \nabla T\right] + 2K(\mathbf{... ...\nabla A \cdot \nabla T\right] + K(\mathbf{x})\;A^2(\mathbf{x})\;\nabla^2 T = 0$$ (15)

   
   
   On the other hand, with the transport equation:
   

   $$2(\nabla T \cdot \nabla)(A(\mathbf{x})\,\mathbf{u}_T) + A(\mathbf... ... + A(\mathbf{x}) \frac{S(\mathbf{x})}{K(\mathbf{x})}\; \nabla K(\mathbf{x}) = 0$$ (16)

   
   
   where $\mathbf{u}_T$ is the unitary vector along the traveltime gradient direction. By performing the dot product with the traveltime gradient vector and multiplying by $A(\mathbf{x})\;K(\mathbf{x})\;V(\mathbf{x})$:
   

   $$2\,K(\mathbf{x})\; A^2(\mathbf{x})\; V(\mathbf{x}) \left[(\nabla T \cdot \nabla)\;\mathbf{u}_T\right]\cdot\nabla T$$ (17)

   $$+ 2\,K(\mathbf{x})\; A(\mathbf{x})\left[\nabla A \cdot \nabla T\right]$$ (18)

   $$+ A^2(\mathbf{x})\,K(\mathbf{x})\;\nabla^2T$$ (19)

   $$+ A^2(\mathbf{x})\; \nabla K(\mathbf{x})\cdot\nabla T = 0$$ (20)

   
   
   The first term is zero since the gradient of the unitary vector along the traveltime gradient direction is zero.
4. The displacement amplitude continuation along a ray is given by equation
   

   $$\left\vert\mathbf{a}_1\right\vert = \left\vert\mathbf{a}_... ...eft(\frac{\rho_0\,V_0\,J_0}{\rho_1\,V_1\,J_1}\right)^{1/2}\;,$$ (21)

   
   
   thanks to the Gauss theorem, where $J_0$ and $J_1$ are the corresponding geometrical spreading factors.
5. (EXTRA CREDIT) Consider the elastic wave equation
   

   $$\rho\,\ddot{u}_i = C_{ijkl,j}\,u_{k,l} + C_{ijkl}\,u_{k,lj}\;$$ (22)

   
   
   in the case of an isotropic elasticity
   

   $$C_{ijkl} = \lambda\,\delta_{ij}\,\delta_{kl} + \mu\,(\delta_{ik}\,\delta_{jl} + \delta_{il}\,\delta_{jk})\;.$$ (23)

   
   
   Using the geometrical representation (5) with the P-wave polarization given by equation (12), show that the corresponding amplitude equations are similar to equations (13) and (21).

Homework 3