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1.4: Plane waves

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    49301

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    We can combine time and spatial phase oscillations to make a traveling wave. For example

    \[ \psi(x,t) = e^{i(k_{0}x-\omega_{0}t)} \nonumber \]

    We define the intensity of the wave as

    \[ |\psi|^{2} = \psi^{*} \psi \nonumber \]

    Where \(\psi^{*}\) is the complex conjugate of \(\psi\). Since the intensity of this wave is uniform everywhere \(|\psi|^{2} =1\) it is known as a plane wave.

    A plane wave has at least four dimensions (real amplitude, imaginary amplitude, x, and t) so it is not so easy to plot. Instead, in Figure 1.4.1 we plot planes of a given phase. These planes move through space at the phase velocity, \(v_{p}\), of the wave. For example, consider the plane corresponding to \(\phi=0\).

    \[ k_{0}x-\omega_{0}t=0 \nonumber \]

    Now,

    \[ v_{p} = \frac{dx}{dt} = \frac{\omega_{0}}{k_{0}} \nonumber \]

    Screenshot 2021-04-13 at 21.50.13.png
    Figure \(\PageIndex{1}\): In a plane wave planes of constant phase move through space at the phase velocity.

    This page titled 1.4: Plane waves is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Marc Baldo (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.