You want to map the interval $(0,24)$ to the interval $(0,2\pi)$ - a full cycle -; the function to do so is

$$2\pi  \frac{\mathrm{hour}}{24}$$

You then need *two* terms in your linear model (recall that an equivalent non-linear parametrization uses phase & amplitude):

$$\beta_1  \sin\left(2\pi\frac{\mathrm{hour}}{24}\right) + \beta_2 \cos\left(2\pi \frac{\mathrm{hour}}{24}\right)$$

Noon & midnight aren't constrained to result in equal predictor values because the phase is estimated from your data. Noon might be at the peak and midnight at the trough of the wave.