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Higher Order Terms

The complete, linear, time-invariant generalization of the lossy, stiff string is described by the differential equation

$\displaystyle \sum_{k=0}^\infty \alpha_k \frac{\partial^k y(t,x)}{\partial t^k} = \sum_{l=0}^\infty \beta_l \frac{\partial^l y(t,x)}{\partial x^l}. \protect$ (C.33)

which, on setting $ y(t,x) = e^{st+vx}$, (or taking the 2D Laplace transform with zero initial conditions), yields the algebraic equation,

$\displaystyle \sum_{k=0}^\infty \alpha_k s^k = \sum_{l=0}^\infty \beta_l v^l$ (C.34)

Solving for $ v$ in terms of $ s$ is, of course, nontrivial in general. However, in specific cases, we can determine the appropriate attenuation per sample $ G(\omega)$ and wave propagation speed $ c(\omega)$ by numerical means. For example, starting at $ s=0$, we normally also have $ v=0$ (corresponding to the absence of static deformation in the medium). Stepping $ s$ forward by a small differential $ j{{\Delta}}\omega $, the left-hand side can be approximated by $ \alpha_0+\alpha_1j{{\Delta}}\omega $. Requiring the generalized wave velocity $ s/v(s)$ to be continuous, a physically reasonable assumption, the right-hand side can be approximated by $ \beta_0+\beta_1 \Delta v$, and the solution is easy. As $ s$ steps forward, higher order terms become important one by one on both sides of the equation. Each new term in $ v$ spawns a new solution for $ v$ in terms of $ s$, since the order of the polynomial in $ v$ is incremented. It appears possible that homotopy continuation methods [316] can be used to keep track of the branching solutions of $ v$ as a function of $ s$. For each solution $ v(s)$, let $ {v_r}(\omega)$ denote the real part of $ v(j\omega)$ and let $ {v_i}(\omega)$ denote the imaginary part. Then the eigensolution family can be seen in the form $ \exp{\left\{j\omega t\pm v(j\omega)x\right\}}=\exp{\left\{\pm{v_r}(\omega)x\right\}}\cdot\exp{\left\{j\omega\left(t \pm {v_i}(\omega)x/\omega\right)\right\}}$. Defining $ c(\omega)\isdeftext \omega/{v_i}(\omega)$, and sampling according to $ x\to x_m\isdeftext mX$ and $ t\to t_n\isdeftext nT(\omega)$, with $ X\isdeftext c(\omega)T(\omega)$ as before, (the spatial sampling period is taken to be frequency invariant, while the temporal sampling interval is modulated versus frequency using allpass filters), the left- and right-going sampled eigensolutions become
$\displaystyle e^{j\omega t_n\pm v(j\omega)x_m}$ $\displaystyle =$ $\displaystyle e^{\pm{v_r}(\omega)x_m}\cdot e^{ j\omega\left(t_n\pm x_m/c(\omega)\right)}$ (C.35)
  $\displaystyle =$ $\displaystyle G^m(\omega)\cdot e^{ j\omega\left(n \pm m\right)T(\omega)}$  

where $ G(\omega)\isdef e^{\pm{v_r}(\omega)X}$. Thus, a general map of $ v$ versus $ s$, corresponding to a partial differential equation of any order in the form (C.33), can be translated, in principle, into an accurate, local, linear, time-invariant, discrete-time simulation. The boundary conditions and initial state determine the initial mixture of the various solution branches as usual. We see that a large class of wave equations with constant coefficients, of any order, admits a decaying, dispersive, traveling-wave type solution. Even-order time derivatives give rise to frequency-dependent dispersion and odd-order time derivatives correspond to frequency-dependent losses. The corresponding digital simulation of an arbitrarily long (undriven and unobserved) section of medium can be simplified via commutativity to at most two pure delays and at most two linear, time-invariant filters. Every linear, time-invariant filter can be expressed as a zero-phase filter in series with an allpass filter. The zero-phase part can be interpreted as implementing a frequency-dependent gain (damping in a digital waveguide), and the allpass part can be seen as frequency-dependent delay (dispersion in a digital waveguide).
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