A cubic interpolation function: folkloric copypasta or clever trade-off?
A reasonable, analytically derived cubic interpolation method will give a straight line if the inputs are on a straight line. With $a = 0$, $b = 1$, $c = 2$, $d = 3$ you get from cubic_bourke $A = d - c - a + b = 3 - 2 - 0 + 1 = 2$ as the coefficient for $x^3$ whereas for a straight line you would have $A = 0$ (and $B = 0$ as the coefficient for $x^2$). So the method doesn't give a straight line for straight-line input. Instead it will, as you say, wobble. In the frequency response that will manifest itself as a low multiplicity of the roots at multiples of the sampling frequency.
The impulse response is not symmetrical. Normally splines have a symmetrical impulse response. The asymmetry will result in phase non-linearity:
Figure 1. Impulse response of cubic_bourke.
The magnitude frequency response shows a rather nice treble response for audio purposes, but gives a lot of aliasing, worse than linear interpolation:
Figure 2. Magnitude frequency rseponse of cubic_bourke (red), linear interpolation (blue) and cubic Hermite spline (green). Half the input sampling frequency is at $\omega = \pi$.
Cubic Hermite (Catmull-Rom) will give better treble response up to 15 kHz for a 44.1 kHz input sampling frequency, and lower aliasing, than cubic_bourke.
So I say scrap it.