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Physics LibreTexts

5.1: Waves in Spacetime

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We now look at the characteristics of waves in spacetime. Recall that a sine wave moving to the right in one space dimension can be represented by

A(x,t)=A0sin(kxωt)

where A0 is the (constant) amplitude of the wave, k is the wavenumber, and ω is the angular frequency, and that the quantity ϕ=kxωt is called the phase of the wave. For a plane wave in three space dimensions, the wave is represented in a similar way,

A(x,t)=A0sin(kxωt)

where k is now the position vector and k is the wave vector. The magnitude of the wave vector, |k|=k is just the wavenumber of the wave and the direction of this vector indicates the direction the wave is moving. The phase of the wave in this case is ϕ=kxωt.

clipboard_ea43bfe8ab3db86b0f032bfa85ebe3303.png
Figure 5.1.1:: Sketch of wave fronts for a wave in spacetime. The large arrow is the associated wave four-vector, which has slope ω/ck. The slope of the wave fronts is the inverse, ck/ω . . The phase speed of the wave is greater than c in this example. (Can you tell why?)

In the case of a one-dimensional wave moving to the right ϕ=kxωt. A wave front has constant phase ϕ, so solving this equation for t and multiplying by c, the speed of light in a vacuum, gives us an equation for the world line of a wave front:

ct=ckxωcϕω=cxupcϕω( wave front )

The slope of the world line in a spacetime diagram is the coefficient of x, or c/up, where up=ω/k is the phase speed. The world lines of the wave fronts of a wave are illustrated in Figure 5.1.1:.


This page titled 5.1: Waves in Spacetime is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by David J. Raymond (The New Mexico Tech Press) via source content that was edited to the style and standards of the LibreTexts platform.

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