Modeling the Instrument Bell

Another example, is the continually changing cross-section of an instrument bell.

Figure 15: The profile of a trombone bell.

The tombone bell is well described by the so-called Bessel horn,

$$a(x) = b(x + x_0)^{-\gamma},$$ (3)

where $x_0$ is the position of the mouth of the horn, $x$ is the distance from the horn mouth, and $a(x)$ is the radius over the length of the bell.

For more efficient computation, the effects of the bell may be lumped to a single reflection and transmission function.