Lets say that I take chunks of signal S at a frequency F and I apply curve C accelaration of each chunk indepently so that each chunk starts whith the original speed and then fastens up (or down). Isn't that a phaser?
Taking two signals from the same source, if one deviates in speed while the other stays constant, the peaks and troughs of the waveform interact with each other either cancelling or reinforcing each other in various ways. This can sometimes sound like you are listening through a tunnel or a drainpipe.
Phasing is similar to flanging in that you are using one direct and one modified version of the same source audio.
However, with phasing, the peaks and troughs of the modified audio are artificially shifted out of phase compared to the original sound without changing the speed. This produces a similar sound to flanging but is usually much softer.
I think you want a frequency-domain effect like frequency modulation.
I'm sure you know what frequency modulation is, well, if the modulator signal is a sawtooth wave, then you would get periodic rising and (ideally) instant resetting of the carrier frequency. The period would be equal to the
modulator frequency/1. So a 1 Hz modulator frequency would reset the rising of the carrier frequency every second - this would be your "chunk". You can change the slope of the sawtooth wave to get an increasing or decreasing acceleration curve. The amplitude of the sawtooth would be known as deviation in Hz. i.e. Linear deviation from the carrier frequency.
Technically, a sawtooth wave would also lower the frequency of the carrier wave by
deviation/2 Hz. So in order to begin and end the chunks at
carrier frequency, you would need to raise the modulator's DC offset by
So then, with a carrier of 50 Hz, a modulator of 1 Hz with a deviation of 10 Hz, the carrier signal would rise for one second to 60 Hz then fall back to 50 Hz.
Here is a quick example of what I'm talking about, firstly using a parabolic sine wave as the carrier, then a (slightly down-shifted) song. (You may be able to hear the "speed" changes more clearly in the song):
As a side note, Phasers are similar in a way, but they create (usually 1-~10) additional versions of the carrier signal, inverting the phase of some and slightly detuning them with a Low Frequency Oscillator (the modulator) to get a moving-effect as the multiple carrier signals fall in and out of phase with one another. I think it's too complicated for what you talk about.
This is an example of a phaser I recorded a while back:
As far as I can tell, simple FM is what you're referring to. I'm not a mathematician, so t'=kt² scrapes over my head :)
Note: I'm assuming by "speed", you mean cycles per second, i.e. frequency (frequency is inversely proportional to wavelength, which is the distance from one cycle to the next, so as the wave gets shorter, the frequency rises). When we talk about logarithmic frequency changes, we usually use the term "pitch" instead.