Introduction to Partial Differential Equations
Exercise Sheet 13
Ross Ogilvie 25th November,
2024
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How should you modify D’Alembert’s formula for this situation?
Solve this for the initial data , and . (5 points)
Suppose that is a solution to the following modified wave equation:
| () |
where are constants.
is a solution of exactly when
or is linear. Solutions of with this form are called plane waves. (2 points)
Interpret this equation in terms of direction and speed. (3 points)
In physics, electrical and magnetic fields are modelled as time-dependent vector fields, which mathematically are smooth functions . Through a series of experiments in the 18th and 19th centuries, the existence and properties of these fields were discovered. Importantly, it was discovered that the two phenomena were connected (both magnets and static electricity had been known since antiquity). In 1861 James Clerk Maxwell published a series of papers summarising electromagnetic theory, including a collection of 20 differential equations. Over time these were further reduced to the following four (by Heaviside 1884 using vector notation), called Maxwell’s Equations:
As is usual, the operator acts on the spatial coordinates , and the denotes the cross product of . The constants , the electrical permittivity, and , the magnetic permeability, are approximately and (V=Volt, s=Seconds, A=Ampere and m=Metre) in a vacuum. Electrical charges are included via the charge density and electric currents are the movements of charges, for a velocity field .
The two equations with divergence were formulated by Gauss, based on known inverse-square force laws, the curl of the electric field is due to Faraday, and the curl of the magnetic field is due to Ampère. The last term in Ampère’s law that has the time-derivative of the electrical field was an addition of Maxwell. With this correction, he was able to derive the equations for electromagnetic waves, as you will now do.