# How is the energy conservation principle preserved in the case mentioned below?

Suppose there are 2 sinusoidal signals $\cos(\omega t)$ and $\cos(\omega t)$, which implies that their individual power is $1/2$ unit and total power is $1$ unit. When added, the signal $2\cos(\omega t)$, however, has power $2$ unit.

Obviously,

$$\rm Energy_{signal} = Power_{signal}\times Time_{period}$$

So, how does the principle of energy conservation hold in this case?

## 1 Answer

Power is homogeneous to a second degree quantity. When the data is doubled, the power is quadrupled.

The energy here (classicaly) is defined in an Hilbert space as a squared $L_2$ norm, deriving from some inner product $\langle\cdot,\cdot\rangle$, assuming a real Hilbert space, $\|x\|^2 = \langle x,x\rangle$. Hence: $$\|x+y\|^2 = \langle x+y,x+y\rangle = \|x\|^2 + 2\langle x,y\rangle+ \|y\|^2 \,.$$

Now set $y=x$, you gracefully get $\|x+x\|^2 = 4\|x\|^2$.

• True. Power is proportional to the square of amplitude. I understand that. My question is regarding how energy is conserved in this case, That is, we had 2 individual cosine waves before. After passing them through a summer, the energy is more than the sum of their energies before passing them through the summer. How is this possible? – Curiosity Mar 7 '18 at 19:14
• @Curiosity The adder is consuming power and delivering it in the form of the new signal. – MBaz Mar 7 '18 at 19:20
• Energy is not a simple conservative additive quantity. Imagine two signals $\cos \omega t$ and $-\cos \omega t$. Their sum will be zero energy. Why are you ressorting to an energy conservation principle here? – Laurent Duval Mar 7 '18 at 19:22
• Well, yes, that's my question. I mean, that's the part I don't get. Why should energy not be conserved here? – Curiosity Mar 7 '18 at 19:25
• I mean, what is the process that consumes the energy? – Curiosity Mar 7 '18 at 19:26