Fundamentals: Plane waves in homogeneous media

[dougoldenburg]

I suggest we emphasize the apps and their usefulness. This deserves discussion since this section can serve as a template for other sections. The goal is to get engagement first and then have people go into derivations.

For this section we have:
Introduction (current one is ok)
Simulation: Plane Wave App.
Existing material in Setup
An introduction about the app and its functionality
Learning Exercises (these can address each of the items in the introduction that focuses on what will be learned)

Skin Depth: A plane wave attenuates as it propagates into the earth. Use the default frequency of 10Hz and conductivity sigma=1 for the initial simulation.
Plot the amplitude with depth and estimate the skin depth.
Evaluate the skin depth using the formula xxx
How does the skin depth change if f=100Hz and sigma=0.1
How does the skin depth change f=1 and sigma=1
experiment…..

Real and Imaginary Components: As the plane wave propagates into the earth there is a real (in-phase) and imaginary (quadrature or out-of-phase) component. By default the App shows these signals at time t=0, so at the surface the Ex field is purely real and the Hy field is purely imaginary. Why is this true?
Consider a depth z=XXX meters. Evaluate the real and imaginary components of the field at that point and combine them into an amplitude value. How does your number compare to that from App. Use the attenuation formula (XXX) to compute the theoretical amplitude at that depth.
Evaluate the phase using the Real and Imaginary parts of the electric field at z=xxx meters. How does your number compare with that provided by the app? Notice that the phase curve exhibits some strange behavior. Does this seem physically real and what is happening?

Wavelength:
There a number of plots from which you can estimate the wavelength of the EM fields. Which one seems the most precise (and why)?. Compare the number with the theoretical value (eq xxx lambda=2pi delta)

Phase Velocity
The App provides a snapshot of the waves in the earth at time t=0. As time progresses the waves propagate downward. If you follow an individual peak or trough with time you can estimate the phase velocity. Do this by adjusting the time slider. Pick a particular peak at two times (say 0.02 and 0.04 seconds) and estimate the phase velocity. Compare your result with the theoretical result given by equation xxxx

Impedance
The impedance Z=E/H is a complex number and it is viewable in the App along with the E and H fields. The fields vary with depth and with time but the impedance has a constant amplitude and phase. Use the theoretical relationships (eq xxx) to validate the numbers in the graph. (other questions ??)

Apparent resistivity
Impedances can be converted to apparent resistivities and phases. Use your impedance values and eq xxxx to compute the apparent resistivity. Note that it doesn’t matter at what depth the measurements were obtained.

The remainder of the current section (derivations for downward propagating fields) can go down one level.

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Google doc
https://docs.google.com/document/d/1D3WeClMCvoZoqPWKmh6kehZN_bZqzhXwir0Lb5aNlfQ/edit#

[lheagy]

http://em.geosci.xyz/content/maxwell1_fundamentals/plane_waves_in_homogeneous_media/index.html