sine wave that slowly ramps up frequency from f1 to f2 for a given time

Solution 1:

if you want angular frequency (w=2 pi f) to vary linearly with time then dw/dt = a and w = w0 + (wn-w0)*t/tn (where t goes from 0 to tn, w goes from w0 to wn). phase is the integral of that, so phase = w0 t + (wn-w0)*t^2/(2tn) (as oli says):

void sweep(double f_start, double f_end, double interval, int n_steps) {
    for (int i = 0; i < n_steps; ++i) {
        double delta = i / (float)n_steps;
        double t = interval * delta;
        double phase = 2 * PI * t * (f_start + (f_end - f_start) * delta / 2);
        while (phase > 2 * PI) phase -= 2 * PI; // optional
        printf("%f %f %f", t, phase * 180 / PI, 3 * sin(phase));
    }
}

(where interval is tn and delta is t/tn).

here's the output for the equivalent python code (1-10Hz over 5 seconds):

from math import pi, sin

def sweep(f_start, f_end, interval, n_steps):
    for i in range(n_steps):
        delta = i / float(n_steps)
        t = interval * delta
        phase = 2 * pi * t * (f_start + (f_end - f_start) * delta / 2)
        print t, phase * 180 / pi, 3 * sin(phase)

sweep(1, 10, 5, 1000)

ps incidentally, if you're listening to this (or looking at it - anything that involves human perception) i suspect you don't want a linear increase, but an exponential one. but that's a different question...

Solution 2:

How do I change frequency smoothly for a given time period?

A smooth sinusoid requires continuous phase. Phase is the integral of frequency, so if you have a linear function for frequency (i.e. a constant-rate increase from f1 to f2), then phase will be a quadratic function of time.

You can figure out the maths with pen and paper, or I can tell you that the resulting waveform is called a linear chirp.

Should I be looking into Fourier transformations?

The Fourier transform of a linear chirp is itself a linear chirp, so probably no.

Solution 3:

It should be fairly simple. Rather than thinking of varying the frequency, think of making an object spin faster and faster. The angular distance it has traveled might be X after N seconds, but will be more that 2X (maybe 4X) after 2N seconds. So come up with a formula for the angular distance (eg, alpha = k1 * T + k2 * T**2) and take the sine of that angular distance to find the value of the waveform at any time T.