1/2 the voltage = 1/4 the power. That pesky inverse-square law again! E
It's not the inverse square law, and it's not half the voltage.
The inverse square law is about electromagnetic radiation. Power in a resistor is proportional to voltage squared, there is no inverse anywher.
Anyhow, the voltge is not halved, depending on how you average it, but there are several ways of calculating the average voltage.
You could simply measure the average voltage with a DC meter. It would show zero with a full AC waveform, and quite a lot with a half wave rectified waveform, so that would be increased.
If you measure AC volts r.m.s. (route mean square), which is how mains AC voltages are normally measured, the voltage will be reduced to 70.7% of what it had been. RMS voltage and current are often used because it allows simple calculation of power.
The positive peak voltage would be unchanged when the diode is added.
The negative peak voltage would be the same magnitude as the positive peak without the diode but would be zero with the diode.
The peak to peak voltage would be halved when thd diode is added.
Also, how voltmeters, especially cheap ones, measure non-sinusoidal voltages is anybody's guess. A true RMS voltmeter would show a half wave rectified voltage as being 70.7% of the votlage.
Anyhow, the average power is halved, because the same power is only consumed but for only half the time. It really is as simple as that. There is no square law for running stuff for a fraction of the time. If you use an device for half of the time, it uses half of the energy. I would guess that you have noticed that if you drive half as far, your car uses about half the amount of petrol, not a quarter....
If you had a square wave, on for half of the time, and off for half of the time, it would show half the voltage on a DC voltmeter. However, if that square wave were loaded with a resistor, the power would not be the same as it would if the voltage were steady and half of the peak.
Say 10 V, 100 Ω
Full power = 1 W
square wave, 50% duty cycle. Average voltage is 5 V
When on, 1 W. When off 0 W, so the average is 0.5 W
Half voltage, 5 V,
power is 0.25 W.