# One-Tilde: A New Symbol for Wavy Math

For years I’ve been using a symbol in any math involving waves.  Normally, in physics we use radians for angles, and for describing the phase of a wave along time or space.  A typical equation involving vibrations or waves will contain bits like this:

$e ^ {i \omega t}$

and this:

$e ^ { i k x}$

Omega is the angular frequency in radians per second. Angular frequency is great for differential equations, and summing infinite series for numerical computations.

In electronics engineering, radio, music, signal processing, and almost all areas of applied mathematics, we like to count cycles. Hertz, which is cycles per second. Wavelengths, the size of one full cycle.  Convert according to $\omega=2\pi f$.  We write plenty of things like this:

$e ^ {2 \pi i f t}$

and

$latex e ^ {2 \pi i x \lambda}$

You find $2\pi i$ factors all over the place.   Frequency in cycles is much easier for everyday application, for measurement, and for labeling electronic parts.

Sometimes I want to stick with cycle frequency, but prefer not to write $2\pi i$ over and over.   So I invented:

${\rlap {1}{\sim}} = e^{2\pi i}$

Here it is, as an image in case my LaTeX markup doesn’t work right:

We all know that

$e^{2\pi i} = 1$

so maybe I could use just a plain one?  No, because $1^p = 1$; unity to any power doesn’t do anything interesting. The tilde over the one means that, for noninteger $p$, don’t take the principal value but the “next” value.  It works like this:

${{\rlap {1}{\sim}}}^{f t} = (e^{2\pi i})^{f t} = e^{(2\pi if t)} = e^{i\omega t}$

We can write a Fourier transform like this:

$S(t) = \int_{-\infty}^{\infty} A(f){\rlap {1}{\sim}}^{ft} dt$

and reverse it as:

$A(t) = \int_{-\infty}^\infty S(f) {{\rlap {1}{\sim}}}^{-ft} dt$

I find it nice that there’s no $2\pi$ factors appearing anywhere.

When adding waves in the study of X-ray diffraction in crystals, or understanding image formation from radio dish arrays, or toying with harmonics in audio processing, the one-tilde keeps things neat and clean.

## One-Tilde in LaTeX

This works, allows for some adjusting, but leaves out some fine points of good typography.

\newcommand\onetilde{{%
\ooalign{\raisebox{.2ex}{$\sim$}\cr
\hidewidth$1$\hidewidth}}}

$\psi(t) = \onetilde^{ft}$



This is simple, but ugly since there’s no kerning for good taste:

{\rlap {1}{\sim}}

## One-Tilde in Lout

I don’t know yet….