Introduction to Partial Differential Equations
Exercise Sheet 13

Ross Ogilvie      25th November, 2024
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38.
Distinct characteristics
(a)
Show that the a smooth function u = u(ζ,η) : × is a solution to ζηu = 0 exactly when it is of the form u(ζ,η) = F(ζ) + G(η), for smooth functions F,G : .
(2 points)
(b)
Under the parameterisation ζ = x + t,η = x t, show that u obeys the one dimensional wave equation (t2 x2)u = 0 exactly when it solves the PDE in (a). (3 points)
(c)
From parts (a) and (b), derive D’Alembert’s formula. (2 points)
39.
Faster!

How should you modify D’Alembert’s formula for this situation?

{ t2u a2x2u = 0 u(x,0) = g(x) tu(x,0) = h(x),

Solve this for the initial data a = 2, g(x) = sin(x) and h(x) = 1. (5 points)

40.
Plane Waves

Suppose that u : n × is a solution to the following modified wave equation:

2u t2 j=1nc j2 2u xj2 = 0,
()

where c1,,cn > 0 are constants.

(a)
Let α n be a unit vector α = 1, μ and F : a twice continuously differentiable function. Show that
u(x,t) := F(α x 𝜇𝑡)

is a solution of (∗) exactly when

μ2 = j=1nα j2c j2

or F is linear. Solutions of (∗) with this form are called plane waves. (2 points)

(b)
For the solutions in (a), examine whether the following property holds for all x n and t :
u(x,t) = u(x 𝜇𝑡𝛼,0).

Interpret this equation in terms of direction and speed. (3 points)

41.
Electromagnetic Waves

In physics, electrical and magnetic fields are modelled as time-dependent vector fields, which mathematically are smooth functions E,B : 3 × 3. 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:

E = 1 𝜀0ρ × E = ∂𝐵 ∂𝑡 B = 0 × B = μ0J + 𝜀0μ0∂𝐸 ∂𝑡 .

As is usual, the operator acts on the spatial coordinates x, and the × denotes the cross product of 3. The constants 𝜀0, the electrical permittivity, and μ0, the magnetic permeability, are approximately 𝜀0 8,854 1012 As Vm and μ0 1,257 106 Vs Am (V=Volt, s=Seconds, A=Ampere and m=Metre) in a vacuum. Electrical charges are included via the charge density ρ and electric currents are the movements of charges, J := 𝑣𝜌 for a velocity field v.

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.

(a)
Let E und B be solutions to Maxwell’s equations in the absence of electric charges, ρ = 0,J = 0. Show that they each satisfy a modified wave equation (Question 40). You may use without proof the identity × (× f) = ( f) f for smooth functions f : 3 3.
(3 points)
(b)
Predict the speed of these waves. (2 Bonus Points)
(c)
Argue that Ampère’s law in its original form × B = μ0J violates the conservation of charge ρ under some conditions. Refer to Exercise Sheet 5 for the definition of a conservation law. Thereby derive Maxwell’s additional term. (3 Bonus Points)