## Finite-Difference Schemes

*Finite-Difference Schemes* (FDSs) aim to solve differential
equations by means of finite differences. For example, as discussed
in §C.2, if denotes the displacement in meters of a vibrating
string at time seconds and position meters, we may approximate
the first- and second-order partial derivatives by

where denotes the time sampling interval and denotes the spatial sampling interval. Other types of finite-difference schemes were derived in Chapter 7 (§7.3.1), including a look at frequency-domain properties. These finite-difference approximations to the partial derivatives may be used to compute solutions of differential equations on a discrete grid:

where is a positive real constant (which turns out to be wave propagation speed). Then, as derived in §C.2, setting and substituting the finite-difference approximations into the ideal wave equation leads to the relation

*i.e.*, for all and ). Solving for in terms of displacement samples at earlier times yields an

*explicit finite-difference scheme*for string displacement:

The FDS is called

*explicit*because it was possible to solve for the state at time as a function of the state at earlier times (and any other positions ). This allows it to be implemented as a time recursion (or ``digital filter'') which computes a solution at time from solution samples at earlier times (and any spatial positions). When an explicit FDS is not possible (

*e.g.*, a non-causal filter is derived), the discretized differential equation is said to define an

*implicit*FDS. An implicit FDS can often be converted to an explicit FDS by a rotation of coordinates [55,481].

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Convergence

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Non-Cylindrical Acoustic Tubes