What does the $e^{iomega t}$ notation mean in Feynman's water wave thought experiment?

Question

Excerpt from the Feynman Lectures, Volume III, Quantum Behavior:

You will remember that the quantitative relationship between $I_1$, $I_2$, and $I{_1}{_2}$ can be expressed in the following way: The instantaneous height of the water wave at the detector for the wave from hole 1 can be written as (the real part of) $hat{h}_1e^{iomega t}$, where the “amplitude” h1 is, in general, a complex number.

Here's Figure 1-2:

Nowhere it is explained what the $e$, $i$, $omega$ and $t$ symbols represent. What are they and how are they related to Figure 1-2?

Paul Razvan Berg · Accepted Answer

Notation

$e$ is Euler's number
$i$ is the unit imaginary number (recall that complex numbers have a real part and an imaginary part)
$omega$ is the Greek letter Omega
$t$ is simply a unit of time
$omega t$ is the angular frequency of the waveform measured in radians per second

Explanation
The reason why the instantaneous height of the water wave is ~$hat{h}_1e^{iomega t}$ is because waves can be considered to be scaled rotations, and scaled rotations can be computed with complex numbers. Refer to this answer by user CR Drost for a deep dive into the mathematics behind Feynman's formula.

What does the $e^{iomega t}$ notation mean in Feynman's water wave thought experiment?

One Answer

Notation

Explanation

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