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*.

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University