While going through Halliday Resnick Walker's Principles of Physics, I came across electric field in a travelling electromagnetic wave described as $$E=\sin(kx-\omega t).$$ A similar treatment was used for magnetic field. However isn't the given equation of a travelling wave valid for a wave with constant amplitude? But as the distance from the source increases the intensity should decrease.

Any help would be appreciated.


2 Answers 2


You are right, the expression $E=sin(kx-\omega t)$ describes a wave with constant amplitude. This expression describes a planewave (https://en.wikipedia.org/wiki/Plane_wave) going in the x direction. It is the simplest kind of wave and is extremely useful as a conceptual object when we make approximation. You are also right that the amplitude of a wave should decrease with distance from the source, this is called geometrical damping.

If you consider a point source in 3D, the waves will be emitted spherically and their amplitude will decrease as $1/r$: https://en.wikipedia.org/wiki/Wave_equation#Monochromatic_spherical_wave So the wavefront of is a sphere. Now imagine that you look at a point very far from the source, or equivalently at a very small portion of the wavefront. In this case you will not see the curvature of the wavefront and it appears flat, and if you consider propagation over distance small compared to the distance from the source, you will not see the decrease in amplitude. In these cases it makes sense to use the planewave expression that you give, as a very good approximation.


Absolutely! A plane wave can be expressed as you have above without additional factors modelling the attenuation of the wave amplitude as it propagates along $x$.

Note: Since a plane-wave can also have some specific phase with respect to its starting point (or anywhere else really) it is useful to model your wave as an exponential $A_0e^{i(kx-\omega t)}$ where $A_0$ is a complex exponential as well and incorporates the phase (ie. $A_0 = Ae^{i\phi}$)

The decrease in amplitude $A_0$ can be modeled by adding another exponential which describes the wave amplitude attenuation in the media the wave is propagating in. The total form then looks something like this: $$E = A_0 e^{-\alpha x}e^{i(kx-\omega t)}$$ Where $\alpha$ tells you how much of the wave amplitude is attenuated after you travel $x$ units of distance in the medium. Often a useful parameter is the "skin depth" which defines how deep the wave penetrates into a lossy media until it is attenuated in amplitude by $e^{-1}$. It's clear from the above equation that the skin depth $\delta$ is calculated as $\frac{1}{\alpha}$.

It might also be useful to note that if the wave is traveling in lossless media then the wave amplitude is not attenuated at all.


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