Energy is proportional to voltage squared. If we make the assumption that a variable voltage is being fed to a constant load-resistor, then with voltage, current would increase, and current would get multiplied by voltage again, to result in energy.
Sound energy is proportional to sound pressure squared. With increasing sound pressure, minute displacement / compression of air results, which causes displacement to rise, and displacement times pressure is again – energy.
The decibel scale is in energy units, not pressure units. Therefore, if a voltage increases by the square root of two, and if that voltage is fed to a constant load, then energy doubles, which is loosely expressed as a 3db relationship. A doubling of voltages would result in a quadrupling of energy units, which is loosely described as a 6db relationship.
Something similar happens to digitally sampled sound. The amplitudes of the samples correspond roughly to the Statistical concept of Standard Deviation, while the Statistical concept of Variance, corresponds to signal-energy. Variance equals Standard Deviation squared…
I should add that this applies to small-signal processing, but not to industrial power-transmission. In the latter case, the load resistances are intentionally made to scale with voltages, because the efficiency-gains that stem from voltage-increases, only stem from keeping current-levels under control. Thus, in the latter case, higher amounts of power are transmitted, but without involving higher levels of current. And so here, voltages tend to relate to power units more-or-less linearly.
Similarly, in any type of power-conversion system, the output current needs to be inversely proportional to its voltage, in order for the system to remain energy efficient.
This is also why for a simple light-dimmer, used in home lighting, almost never a variable resistor gets used by itself. If one was used, then it being in series with the load, and the amount of current flowing through it being the same at the higher and medium voltages, would cause a simplistic potentiometer to absorb a major fraction of the energy.
Light-dimmers contain active circuits – as well as sometimes a low-current potentiometer – to keep them efficient.
The main concept of this posting is only true, if the load is assumed to be a constant, abstract resistance, which can also be taken out of the equation in certain cases – such as with digital sound-samples.
In the case of a capacitor, if its capacitance stays the same, the stored energy is proportional to its voltage squared.
This follows from the fact that its voltage is also proportional to its charge, and if we integrate a linear function, we obtain a second-degree function. Energy is also equal to charge times voltage.