1-D discrete systems

Let's first consider a discrete 1-D elastic medium such as that represented in Figure . Two characteristics distinguish this medium from the continuous 1-D medium whose behavior is described by the wave equation. First, energy is localized in discrete points in space. Second, the internal forces are dissociated from mass, which implies that traction is not continuous since each mass is considered as a rigid body.

 

springmodel
Figure 1 a spring-mass model for 1D-wave propagation in a discrete system.

Newton's law applied to the ith mass of Figure  leads to the following equation:  

$$m_i {d^2 u_i(t) \over d t^2} = k_{i} u_{i-1}(t) - (k_i + k_{i+1}) u_i(t) + k_{i+1} u_{i+1}(t),$$ (1)

where u_i = y_i - y⁰_i is the displacement of mass i relative to its equilibrium position y⁰_i at time zero. The reference positions remain the same, while the equilibrium positions are a function of time.

The system of equations defined by equation (1) with i=1,N can be expressed as

 

$${d^2 {\bf u}(t) \over d t^2} = {\bf A} {\bf u}(t),$$ (2)

where

$${\bf A} = \left( \begin{array} {cccccccc} {-k_1-k_2 \over m_... ... 0 & {-k_N \over m_N} & {k_N \over m_N} \\ \end{array}\right).$$

This is a second-order system of N ordinary differential equations in time for the N unknown functions ${\bf u_i(t)}$. To find the solution, the pair of initial conditions u(0) and $d{\bf u(0)}/dt$must be specified. For the continuous case, the wave equation has also a second-order space differentiation, and thus requires two boundary conditions. Here, these two conditions correspond to the first and last equations.

Fourier transforming equation (2) over time results in  

$$- \omega^2 {\bf u}(\omega) = {\bf A} {\bf u}(\omega),$$ (3)

which represents an eigenvalue decomposition problem. For each eigenvalue ($\lambda_i = -\omega^2_i$) there will be two frequencies: $\omega^+_i = +\sqrt{-\lambda}$ and $\omega^-_i = -\sqrt{-\lambda}$.

Let ${\bf E}$ be the matrix formed by the eigenvectors ${\bf e_i}$ of ${\bf A}$ and ${\bf \Lambda}$ the diagonal matrix with the respective eigenvalues. If ${\bf A}$ is Hermitian, then the set of eigenvectors ${\bf e_i}$ form a complete orthonormal basis for ${\bf u}$.If the masses are identical ${\bf A}$ is real-symmetric and the general solution ${\bf u(\omega)}$ is given by  

$${\bf u}(\omega) = \sum_{i=1}^N a_i {\bf e_i} \delta(\omega-\omega_i) + b_i {\bf e_i} \delta(\omega+\omega_i),$$ (4)

which in the time domain corresponds to

$${\bf u}(t) = \sum_{i=1}^N \; {\bf e_i} \; \{ a_i \; {\it e}^{-\omega_i t} + b_i \; {\it e}^{+\omega_i t} \}$$

or, after some manipulation,

 

$$\begin{array} {ccccc} {\bf u}(t) & = & {\bf E} & \left[ \co... ...{array} {c} {\bf c} \\ {\bf d} \end{array} \right),\end{array}$$ (5)

where ${\bf \Omega} = {\bf (-\Lambda)}^{1 \over 2}$. Therefore, $\cos({\bf \Omega}t)$ and $\sin({\bf \Omega}t)$ are N by N diagonal matrices with $\cos (\omega_i t)$ and $\sin (\omega_i t),$ respectively, in the diagonals, and ${\bf c}$ and ${\bf d}$ are constant vectors of dimension N to be determined by the initial conditions.

Applying the initial conditions to equation (5) and recalling that ${\bf E}$ is orthonormal, we find the equation for the unknowns c and d:

$$\begin{array} {ccccc} \left( \begin{array} {c} {\bf c} \\ {... ... {\bf u}(0) \\ \dot{\bf u}(0) \end{array} \right). \end{array}$$

Substituting this result in equation (5) and in the equivalent relation for ${\bf \dot{u}}(t)$ results in  

$$\begin{array} {cccccc} \left( \begin{array} {c}{\bf u}(t) \\... ...{\bf u}(0) \\ {\bf \dot{u}}(0) \end{array} \right).\end{array}$$ (6)

Figure  compares the exact, analytical solution of the wave equation (-a) with the discrete-model solution (-b) from equation (6). The initial conditions are ${\bf \dot{u}}(0) = 0$and ${\bf u}(0) = (0,...,0,1,0,...,0)$. There are two important differences between the two cases: the dispersive character of the propagating wave-packet in the discrete case, and the continuing irradiation of energy from the impulsive source position, which is also only present in the discrete case. In both cases the spectrum is characterized by a discrete set of frequencies because the model is spatially bounded, but, while in the continuous case the set is infinite, in the discrete case only N components are present which implies an upper frequency limit.

 

s1d1
Figure 2 A comparison of (a) a continuous medium solution and (b) a discrete medium solution for the 1-D case.

A clear picture of the dispersion process can be obtained from the eigenvalue-eigenvector structure. Each eigenvector ${\bf e_i}$corresponds to a spatial harmonic associated with the eigenfrequency $\omega_i$. If we Fourier transform these eigenvectors and rescale the vertical axis by $2\pi$, the resulting matrix ${\bf \tilde{E}}$ will represent the spatial spectra of the model. To get the dispersion relation it is necessary to stretch the horizontal axis using the eigenfrequencies $\omega_i$, so that the spectra will be a function of $\omega$ instead of i. Figure  compares the $\kappa(\omega)$ spectrum obtained with this process with the dispersion relation predicted by the wave equation (continuous line) and with the function $\kappa = \omega_{max} \sin(\kappa\pi/2\kappa_{max})$ (dashed line) predicted by the discretized wave equation (Claerbout, 1985). The dispersion relation shown in this figure was generated from the eigen-spectrum of a three-layer model. The relation between the maximum spatial and temporal frequencies $\omega_{max}=2 v \kappa_{max}/\pi$ is valid not only for the discrete case but also for the continuous case. The meaning of such relation is that the minimum time interval for transmission of energy between adjacent points is equal to half the fundamental period $\sqrt{k/m}$. The difference is that $\kappa_{max}=\infty$ for the continuous case while $\kappa_{max}=\pi/\delta l$ for a discrete system with distance $\delta l$between masses.

The intrinsically dispersive character of wave propagation in a discrete medium can also be related to the entropic behavior of a discrete system. In a continuous medium, the constraints imposed by the continuity of stresses and displacements represent a hampering of the process of the increasing of entropy, while in a discrete system, the extra degree of freedom (no continuity constraints) allows the system to evolve into less ordered states of movement.

 

kofw
Figure 3 Fourier transformed eigenvectors as a function of the corresponding eigenfrequencies. The straight lines correspond to the constant velocities of each layer, as predicted by wave theory, while the dashed lines are $\omega_{max} \sin(\kappa\pi/2\kappa_{Nyquist})$.