The Dispersive 1D Wave EquationIn the ideal vibrating string, the only restoring force for transverse displacement comes from the string tension (§C.1 above); specifically, the transverse restoring force is equal the net transverse component of the axial string tension. Consider in place of the ideal string a bundle of ideal strings, such as a stranded cable. When the cable is bent, there is now a new restoring force arising from some of the fibers being compressed and others being stretched by the bending. This force sums with that due to string tension. Thus, stiffness in a vibrating string introduces a new restoring force proportional to bending angle. It is important to note that string stiffness is a linear phenomenon resulting from the finite diameter of the string.
In typical treatments,C.3bending stiffness adds a new term to the wave equation that is proportional to the fourth spatial derivative of string displacement:
where the moment constant is the product of Young's modulus (the ``relative-displacement spring constant per unit cross-sectional area,'' discussed in §B.5.1) and the area moment of inertia (§B.4.8); as derived in §B.4.9, a cylindrical string of radius has area moment of inertia equal to . This wave equation works well enough for small amounts of bending stiffness, but it is clearly missing some terms because it predicts that deforming the string into a parabolic shape will incur no restoring force due to stiffness. See §6.9 for further discussion of wave equations for stiff strings. To solve the stiff wave equation Eq.(C.32), we may set to get
Substituting for in terms of in gives the general eigensolution
(for frequency-dependent wave velocity).That is, each delay element becomes an allpass filter which approximates the required delay versus frequency. A diagram appears in Fig.C.8, where denotes the allpass filter which provides a rational approximation to . allpass filter is given by 
(Phase Delay)Minimizing the Chebyshev norm of the phase-delay error,
(Group Delay)The group delay of a filter gives the delay experienced by the amplitude envelope of a narrow frequency band centered at , while the phase delay applies to the ``carrier'' at , or a sinusoidal component at frequency . As a result, for proper tuning of overtones, phase delay is what matters, while for precisely estimating (or controlling) the decay time in a lossy waveguide, group delay gives the effective filter delay ``seen'' by the exponential decay envelope. See §9.4.1 for designing allpass filters with a prescribed delay versus frequency. To model stiff strings, the allpass filter must supply a phase delay which decreases as frequency increases. A good approximation may require a fairly high-order filter, adding significantly to the cost of simulation. (For low-pitched piano strings, order 8 allpass filters work well perceptually .) To a large extent, the allpass order required for a given error tolerance increases as the number of lumped frequency-dependent delays is increased. Therefore, increased dispersion consolidation is accompanied by larger required allpass filters, unlike the case of resistive losses. The function piano_dispersion_filter in the Faust distribution (in effect.lib) designs and implements an allpass filter modeling the dispersion due to stiffness in a piano string [154,170,368]. stiff string is described by the differential equation
which, on setting , (or taking the 2D Laplace transform with zero initial conditions), yields the algebraic equation,
Solving for in terms of is, of course, nontrivial in general. However, in specific cases, we can determine the appropriate attenuation per sample and wave propagation speed by numerical means. For example, starting at , we normally also have (corresponding to the absence of static deformation in the medium). Stepping forward by a small differential , the left-hand side can be approximated by . Requiring the generalized wave velocity to be continuous, a physically reasonable assumption, the right-hand side can be approximated by , and the solution is easy. As steps forward, higher order terms become important one by one on both sides of the equation. Each new term in spawns a new solution for in terms of , since the order of the polynomial in is incremented. It appears possible that homotopy continuation methods  can be used to keep track of the branching solutions of as a function of . For each solution , let denote the real part of and let denote the imaginary part. Then the eigensolution family can be seen in the form . Defining , and sampling according to and , with 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
where . Thus, a general map of versus , 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).
Alternative Wave Variables
A Lossy 1D Wave Equation