What does the ~ symbol, placed above a vector, mean

Question

On the wikipedia page for the Poynting Vector, under the section 'time averaged Poynting vector', the ~ symbol is used above some of the vectors. What does this mean?

Thank you.

Answer 1

I've not seen that decoration given any standardized meaning.

Based on the text description and how it's actually used there, I believe the following is what is meant.

The (time-varying) vector $\mathbf{E}$ is assumed to be sinusoidal function of $t$ -- it is a common trick to replace sinusoidal waves with complex exponentials: e.g.

$$ \cos x = \mathop{\text{Re}} \left( e^{ix} \right)$$ $$ \sin x = \mathop{\text{Im}}\left(e^{ix}\right) = \mathop{\text{Re}}\left( -i e^{ix} \right)$$

The (time-varying) vector $\tilde{\mathbf{E}}$ appears to be defined, along with the constant (with respect to time), complex vector $\mathbf{E}_c$ and real number $\omega$, by the two properties:

• $\mathbf{E} = \text{Re}(\tilde{\mathbf{E}})$
• $\mathbf{E} = \mathbf{E}_c e^{i \omega t}$

Answer 2

It usually stands for unit vector:

$$\tilde{v} = \frac{\vec{v}}{|\vec{v}|}.$$

The Poynting vector is a very useful concept in electromagnetism and it's usually denoted by $\vec{S}$.

Cheers!