Here I demonstrate how much modulator modulates the carrier in the Heisenberg's phase modulation. When the modulator's value is 1 (the value is in the range (-1,1)), no phase shift of the carrier occurs. When the modulator's value is -1, some phase shift occurs depending on how the amount of modulation. The amount of phase shift is a linear function of the value of the modulator when the amount of modulation is constant.

I set out to find the phase shift when modulator = -1 as a function of amount of modulation. I used the sawtooth down as the modulator because it is linear and starts at ~1, and a sine wave at 0 Hz as the carrier so that the phase is always (phase + phase shift). Note that there is an imperfection at the jump of the sawtooth wave because it's band limited.

At modulation amount = 0, nothing is heard (no phase shift).

At modulation amount = 50, a sine wave is heard at the modulator's frequency. This means that for every wave of the modulator, the carrier is shifted 1 wavelength, or 2π. In other words, a modulator value of -1 shifts the carrier 2π.

When the amount of modulation is 100, a sine wave is heard at 4 times the modulator's frequency (Carrier shift is 4 wavelengths, or 8π

when modulator = -1).

I extrapolated from these two points that at a modulation amount of sqrt(3)*100/2≈86.603, the carrier would be shifted 3 wavelengths, 6π, and at sqrt(2)*100/2≈70.711, it would be shifted 2 wavelengths, 4π. I was correct so I could be reasonably certain that the phase shift when the modulator is -1 is

s = 2π*(2x/100)^2

where "s" is the phase shift in radians and x is the amount of modulation. Alternatively,

λ = (2x/100)^2

where λ is the number of wavelengths of the carrier.

If you modulate the carrier using two modulators, then the phase shifts add (2 modulators at 100 modulation can potentially shift 16π, or 8 wavelengths).