Question

The equation describing elastic waves in an isotropic medium is $$ (\lambda+2 \mu) \nabla \nabla \cdot \mathbf{a}-\mu \nabla \times(\nabla \times \mathbf{a})=\rho \frac{\partial^{2} \mathbf{a}}{\partial t^{2}} $$ where a is the displacement from equilibrium, \(\rho\) is the density, and \(\lambda\) and \(\mu\) the clastic constants of the medium. Find the lowest frequency of oscillation of an isotropic elastic sphere of radius \(R\), given that (a) \(a(\mathbf{x})\) is of the form \(f(r) \hat{e}\), and (b) the boundary condition at \(r=R\) is $$ \lambda \nabla \cdot \mathbf{a}+2 \mu \frac{\partial a_{\prime}}{\partial r}=0 $$ In order to get a definite number at the end, you may set \(\lambda=\mu\).

Short answer

\(\omega = \sqrt{ \frac{3 \mu}{\rho}} \)

Step-by-step solution

Write down the given equation

The given equation describing elastic waves in an isotropic medium is: \[(\lambda+2 \mu) abla abla \cdot \mathbf{a}-\mu abla \times(abla \times \mathbf{a})=\rho \frac{\partial^{2} \mathbf{a}}{\partial t^{2}}\]

Assume the solution form

Assume the displacement from equilibrium is of the form \( a(\mathbf{x}) = f(r) \hat{e}\).

Apply the divergence and curl

Calculate the divergence and curl of \(\mathbf{a}\). Since \(\mathbf{a} = f(r) \hat{e}\), the divergence and curl can be expressed as follows: \( abla \cdot \mathbf{a} = \frac{d (f(r))}{d r} \)\( abla \times \mathbf{a} = 0 \) Thus the given equation simplifies to: \[(\lambda+2 \mu) abla abla \cdot f(r) \hat{e} = \rho \frac{\partial^{2} f(r) \hat{e}}{\partial t^{2}} \] Since \(abla (abla \cdot f(r) \hat{e}) \) reduces to \( abla (\frac{d (f(r))}{d r}) \)

Simplify and Separate Variables

Rewrite the equation and separate variables: \[(\lambda + 2\mu) abla \left( \frac{d f(r)}{d r} \right) = \rho \frac{\partial^2 f(r)}{\partial t^2} \]Separating variables, we get: \((\lambda + 2\mu) \left( \frac{d^2 f(r)}{d r^2} + \frac{2}{r} \frac{d f(r)}{d r} \right) = \frac{\rho}{\omega^2} f(r)\)

Apply boundary conditions

The boundary condition at \(r = R\) is: \[ \lambda abla \cdot \mathbf{a} + 2\mu \frac{\partial a_{\prime}}{\partial r} = 0 \]Applying the solution form and simplifying, we get: \[ \lambda \frac{d f(r)}{d r} + 2\mu \frac{d f(r)}{d r} = 0 \rightarrow (\lambda + 2\mu) \frac{d f(r)}{d r} = 0 \]

Simplify using given conditions

Given that \(\lambda = \mu\), then the previous equation can be simplified to: \((2\mu + 2\mu) \frac{d f(r)}{d r} = 0 \rightarrow 4\mu \frac{d f(r)}{d r} = 0\)Since \(\mu eq 0\), we have: \( \frac{d f(r)}{d r} = 0 \)

Solve the frequency of oscillation equation

Using the resulting boundary conditions, solve the frequency equation: \[(\lambda + 2\mu) \left( \frac{d^2 f(r)}{d r^2} + \frac{2}{r} \frac{d f(r)}{d r} \right) = \frac{\rho}{\omega^2} f(r) \to (3 \mu) \frac{d^2 f(r)}{d r^2} = \frac{\rho}{\omega^2} f(r)\]Given the boundary condition \(\frac{d f(r)}{d r} = 0 \), we solve for \(\omega\).This leads to the frequency relation: \[\omega = \sqrt{ \frac{3 \mu}{\rho}} \]

Key concepts

isotropic elasticity

Isotropic elasticity is a fundamental concept when dealing with materials that have the same properties in all directions, which means their behavior under stress and strain is uniform regardless of orientation. This property simplifies many equations and calculations.
In isotropic materials, two elastic constants are key: the Lamé constants, \( \lambda\ \) and \( \mu\ \). Here, \( \lambda\ \) is the first Lamé parameter, and \( \mu\ \) is the shear modulus. These constants define the material's response to stress and help in understanding the propagation of elastic waves within it.
Understanding isotropic elasticity is crucial for analyzing elastic waves because it allows the reduction of complex vector equations into simpler scalar forms due to the uniformity of the material's properties. This also involves using the concept of displacement fields, where every point in the material moves uniformly under deformation.

wave equation

The wave equation in an isotropic medium describes how elastic waves propagate through the material. It links the displacement field \( \mathbf{a}\ \) with the material constants and density. For isotropic media, the wave equation is given as:
\[ (\lambda+2 \mu) abla abla \cdot \mathbf{a} - \mu abla \times (abla \times \mathbf{a}) = \rho \frac{\partial^{2} \mathbf{a}}{\partial t^{2}} \]
Here, \( \mathbf{a}\ \) is the displacement from equilibrium, \( \lambda\ \) and \( \mu\ \) are the Lamé constants, and \( \rho\ \) is the density of the medium. This equation shows how internal forces within the medium, such as stress and strain, interact with inertial forces, leading to oscillations.
The wave equation can be tricky because it involves both divergence and curl operators. These help to capture both compressional (longitudinal) waves and shear (transverse) waves, which behave differently in the material.

boundary conditions

Boundary conditions are crucial in solving wave equations because they define how the wave behaves at the limits of the medium. For an isotropic elastic sphere, we need to satisfy certain conditions at the boundary to find a meaningful solution.
In our case, the boundary condition given at radius \( \mathbf{r = R}\ \) is:
\[ \lambda abla \cdot \mathbf{a} + 2 \mu \frac{\partial a'}{\partial r} = 0 \]
This condition ensures the displacement and strain at the surface match physical constraints, like no net force. It simplifies the solution process by providing additional equations to solve alongside the wave equation.
Boundary conditions can often be complex and require applying boundary values and ensuring they meet physical and mathematical constraints. They are essential for obtaining specific solutions, like the oscillation frequencies of the sphere.

oscillation frequency

The oscillation frequency refers to how often the material oscillates back and forth per unit time. It is a critical aspect of wave phenomena in elastic media and depends on the medium's properties, including its density and elastic constants.
For our isotropic elastic sphere, we find the lowest frequency of oscillation by analyzing the boundary conditions and solving the simplified wave equation. By setting \( \lambda = \mu\ \), the equation for frequency becomes simpler:
\[ \omega = \sqrt{ \frac{3 \mu}{\rho}} \]
Here, \( \omega\ \) is the angular frequency, \( \mu\ \) is the shear modulus, and \( \rho\ \) is the density. This relation shows how material properties directly affect the frequency of oscillation.
Finding oscillation frequencies is essential for understanding the resonance and stability of objects under stress, which has applications in fields like materials science, engineering, and geophysics.