My main mode of study for ultrasonics was reviewing the homework assignments. The questions below are mostly based on the lecture notes and attempt to cover the most essential topics of the course.

- Elasticity
- Derivation of elastic wave equation
- Reflection and transmission
- Rayleigh waves
- Waves in plates
- Signal processing

Introductory topics covered in the course are not listed above because they can be found elsewhere on this site. Index notation is covered in the vector algebra and vector calculus section of the Math page. Basic wave phenomena are covered in the "What is a wave?" section of the Physical Acoustics page. The final sections of that page cover diffraction and arrays.

- What is the difference between \(\vec{X}\) and \(\vec{x}\)? Quantities written in terms of \(\vec{X}\) belong in the ____________ description while quantities written in terms of \(\vec{x}\) belong in the ____________ description. Which description is more conventionally used in acoustics, and which is more conventionally used in elasticity?
- Use Cartesian index notation, i.e., \(\vec{X} \mapsto X_k\), \(\vec{x} \mapsto x_k\), \(k = 1,2,3\). Define the motion and inverse motion.
- The displacement is defined as \(\vec{u} = \vec{x}- \vec{X}\). The velocity is the time derivative of the displacement, and the acceleration is the time derivative of the velocity. Evaluate the displacement, velocity, and acceleration in the Lagrangian and Eulerian desciptions.
- Noting that \[dx_k = \frac{\partial \hat{x}_k}{\partial X_m}\, dX_m\,,\] and defining \begin{align*} dS^2&= dX_k \, dX_k\\ ds^2&= dx_k \, dx_k = \frac{\partial \hat{x}_k}{\partial X_m}\, dX_m\, \frac{\partial \hat{x}_k}{\partial X_n}\, dX_n\,, \end{align*} show that \[ds^2-dS^2 = 2E_{mn} dX_m dX_n,\] where \[E_{mn} = \frac{1}{2}\bigg(\frac{\partial \hat{u}_m}{\partial X_n} + \frac{\partial \hat{u}_n}{\partial X_m} + \frac{\partial \hat{u}_k}{\partial X_m} \frac{\partial \hat{u}_k}{\partial X_n}\bigg)\,. \] What is \(E_{mn}\) called?
- To what does the stress-strain relation \(T_{km} = c_{kmij} E_{ij}\) reduce for isotropic linear elastic materials?

- The linearized differential form of Newton's second law in the absence of body forces reads \[\rho a_m = \frac{\partial T_{mk}}{\partial x_k}\,.\] In linear elasticity the left-hand side can be written as \(\rho a_m = \rho_0 \partial^2 u_m/\partial t^2\), while the stress tensor for a linear isotropic material, upon substitution of the linear strain tensor, becomes \[T_{mk} = \lambda \delta_{mk} \frac{\partial u_j}{\partial x_j} + \mu \bigg(\frac{\partial u_m}{\partial x_k} + \frac{\partial u_k}{\partial x_m} \bigg)\,.\] Obtain \[\rho_0 \frac{\partial^2 \vec{u}}{\partial t^2} = (\lambda +\mu) \vec{\nabla}(\vec{\nabla}\cdot \vec{u}) + \mu \nabla^2 \vec{u} \] by combining Newton's second law and the linear strain tensor.
- The vector Laplacian identity \(\nabla^2 \vec{A} = \vec{\nabla}(\vec{\nabla} \cdot \vec{A}) - \vec{\nabla}\times (\vec{\nabla}\times \vec{A})\) can be used to write the result of question (1) above as \[\rho_0 \frac{\partial^2 \vec{u}}{\partial t^2} = (\lambda +2 \mu) \vec{\nabla}(\vec{\nabla}\cdot \vec{u}) - \mu \vec{\nabla}\times (\vec{\nabla} \times \vec{u}) \] Apply the Helmholtz decomposition \(\vec{u} = \vec{\nabla} \phi + \vec{\nabla}\times \vec{\psi}\) to the above to obtain two wave equations, one corresponding to compressional waves, and the other corresponding to shear waves. Which one travels faster?

- What are the boundary conditions at the boundary of two media for normally incident compressional waves? Use knowledge from Acoustics I to immediately write the
*stress*- reflection and transmission coefficients. What are \(R\) and \(T\) for the*displacement*- reflection and transmission coefficients? Does the same relation hold for normally incident shear waves? - How are the
*displacement*reflection and transmission coefficients related to the*stress*reflection and transmission coefficients? - What are the boundary conditions for reflection and transmission between two elastic solids? Summarize qualitatively how to find the reflection and transmission coefficients.

- Consider an elastic half-space with a traction-free surface. Limit wave propagation to two dimensions, where \(x_3\) points downward, and \(x_1\) is oriented along the surface of the half-space: This "plane-strain" condition requires that there is no displacement in the \(x_2\) direction, i.e., \(u_2 = 0\). What are the potential functions \(\phi\) and \(\vec{\psi}\) that describe the displacement field \(\vec{u} = \vec{\nabla} \phi + \vec{\nabla}\times \vec{\psi}\)?
- Seek a solution to the compressional wave equation \(\nabla^2 \phi - \ddot{\phi}/c_L^2 = 0\) and the shear wave equation \(\nabla^2 \psi - \ddot{\psi}/c_T^2 = 0\) by assuming the form \(\phi = A(x_3)e^{j(\omega t - kx_1)}\) and \(\psi = B(x_3)e^{j(\omega t - kx_1)}\).
- The boundary conditions of a traction-free surface in 2D are that the normal and shear components of the stress vanish:
\begin{align*}
T_{13} &= 2\mu E_{13} = 0\\
T_{33} &= (\lambda + 2\mu)E_{33} + \lambda E_{11} = 0\,.
\end{align*}
Use the definition of the linear strain tensor to write these relations in terms of the displacements. Use the displacement potentials calculated previously to obtain a characteristic equation that satisfies these boundary conditions.
*Hint: Define \(r \equiv s^2 + 1 = 2- (c_R/c_T)^2\). Answer: \(r^2 - 4sq = 0\).* - Define \(\eta \equiv c_R/c_T\) and \(\zeta = c_T/c_L\). This gives
\begin{align*}
r &= 2 - (c_R/c_T)^2 = 2-\eta^2,\\
s &= \sqrt{1 - (c_R/c_T)^2} = \sqrt{1-\eta^2}\\
q &= \sqrt{1 - (c_R/c_L)^2} = \sqrt{1-\eta^2 \zeta^2}.
\end{align*}
Write the characteristic equation for Rayleigh waves at a traction-free surface, \(r^2 - 4sq = 0\), in terms of these parameters. How many roots does this equation have, and how many of them are physical?
*Answer: \[\eta^6 - 8\eta^4 + 8\eta^2 (3- 2\zeta^2) + 16(\zeta^2 - 1) = 0.\] (I am having a hard time showing this. Let me know if you know how it's done.)* - Are Rayleigh waves dispersive?
- In what case can a Rayleigh wave travel along a curved surface?

This section covers waves in plates of two types: horizontally polarized shear waves and Lamb waves. Note that Lamb waves were not discussed in depth in class, and the relevant homework assignments were numerical exercises. Their derivation is similar to that for Rayleigh waves.

- What is an SH wave? How are they different from Lamb waves?
- The wave equation for SH waves is very similar to the acoustic wave equation in 2D: \[\frac{\partial^2 u_3}{\partial x_1^2} + \frac{\partial^2 u_3}{\partial x_2^2} -\frac{1}{c_T^2}\frac{\partial^2 u_3}{\partial t^2} = 0\,.\] Assume the wave motion is of the form \(u_3(x_1,x_2,t) = f(x_2) e^{j(\omega t - kx_1)}\), where the coordinates are defined below. Obtain the function \(f(x_2)\), where \(q = \sqrt{k_T^2 - k^2}\). What kinds of modes does the application of the boundary conditions at \(h\) and \(-h\) give rise to?
- Given that \(q_n = n\pi/2h\), obtain the dispersion relation \(\Omega(K)\), where \(\Omega = 2h\omega/\pi c_T\) is the dimensionless frequency, and \(K = 2kh/\pi\) is the dimensionless wavenumber. Qualitatively describe the dispersion relation. Obtain the dimensionless group and phase speeds of the SH waves.
- Outline the derivation for Lamb waves between two layers, where the coordinates are as defined below. Limit the derivation to plane strain, i.e., \(u_2 = 0\), and do not attempt to satisfy the boundary conditions.