### Convergence in Audio Applications

Because the range of human hearing is bounded (nominally between 20 and 20 kHz), spectral components of a signal outside this range are not audible. Therefore, when the solution to a differential equation is to be considered an audio signal, there are frequency regions over which convergence is not a requirement.

Instead of pointwise convergence, we may ask for the following two properties:

*Superposition*holds.- Convergence occurs
*within the frequency band of human hearing*.

*bandlimited interpolator*design (see §4.4).

In many cases, such as in digital waveguide modeling of vibrating
strings, we can do better than convergence. We can construct finite
difference schemes which agree with the corresponding continuous
solutions *exactly* at the sample points. (See §C.4.1.)

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Finite Difference Time Domain (FDTD) Scheme

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