Why A half-wave rectifier passes only one polarity of the AC input, so a single diode is all that is needed. It conducts on just one half of each input cycle, so the output ripple frequency equals the AC line frequency (60 Hz in the US), and the filter capacitor must hold up the voltage for the entire missing half cycle. That makes half-wave supplies poorer at delivering current and harder to filter than full-wave designs.
Watch out The ripple statement is backwards: a full-wave rectifier uses both halves of the cycle and so produces 120 Hz ripple, twice the half-wave's 60 Hz. Doubling the peak input voltage describes a voltage doubler circuit, not a plain half-wave rectifier.
Half-wave = one diode, 60 Hz ripple. Full-wave = two or more diodes, 120 Hz ripple.