# PDEs and Waves

penalty.

1. An example of the use of Green’s functions in 3 dimensions. Green’s functions are often applied to ﬁnd a potential. In this context, the Green’s function represents the potential of a point charge (for electrical potential) or of a point mass (for gravitational potential). The following question is about a gravitational potential V (x)=V (x,y,z) at a point (x,y,z) due to a spherical mass. Along the way you will prove a classic result from graviational theory. Consider the force of gravity acting on a point mass m, located at x =(x,y,z) due to a point mass µ located at ξ =( ξ,η,ζ). According to Newton’s Law of Gravitation, this force F is given by F = −! γmµ |x − ξ|2″ x − ξ |x − ξ| , where γ is the universal gravitational constant and x−ξ is the displacement vector between the two masses. The force F can be written in terms of a potential v(x;ξ) wherev(x;ξ) is the potential at (x,y,z) due to the mass atξ =( ξ,η,ζ): F m = −∇xv(x;ξ) where the subscript x indicates that the partial derivatives are taken with respect to the x, y and z coordinates. Here

v = −

γµ |x − ξ| and so, by comparing V with the expression of the Green’s function solution to the inhomogeneous solution to the Laplace equation in inﬁnite space (which we derived in lectures) we see that: ∇x 2v =4πµγδ(x − ξ).

Now let’s assume that we have a distribution of mass, density ρ = ρ(ξ) throughout a domain Ω where ξ ∈ Ω (rather than a point mass at only one point x = ξ). Then the potential V (x,y,z) at (x,y,z) due to the distributed mass is given by: V (x,y,z)=−γ#Ω ρ(ξ) |x − ξ| dξ.

Copyright c ⃝ 2015 The University of Sydney 1

(a) Assume the domain Ω represents a spherical mass, radius R with a constant density ρ for 0 ≤ r ≤ R. (i) Explain why we can centre the spherical mass at the origin and take the point where we evaluate the potential to be (0,0,Z) without loss of generality. (ii) By writing the integral over Ω in spherical polars and writing down an explicit expression in terms of r, Z and φ for |x − ξ|, show that when 0 <R<Zthen V (0,0,Z)=µ# R 0 # π 0 # 2π 0 ρsinφ $Z2 + r2 − 2rZcosφ drdφdθ. (iii) Evaluate this integral and show that

V (0,0,Z)=

4πρµ 3Z

.

Explain why this result is often interpreted as ”the gravitational potential outside a shpere of uniform density is the same as if all the mass in the sphere was concentrated at its centre”. (b) What happens when 0 <Z<R; that is, when we take the potential at a point inside a spherical mass of constant density ρ?

Adapted from Kevorkian, Partial Diﬀerential Equations. Analytic Solution Techniques. Second edition.

2. The Klein-Gordon equation is a linearised version of the Sine-Gordon equation for small u(x,t): Autt − Kuxx + Tu=0 where A, K and T are constants. (a) Find all travelling wave solutions to this equation. (b) If u(x,t) is, indeed, small then u must remain bounded. Which wavespeeds c admit a travelling wave solution which is bounded? Sketch some representative bounded travelling wave solutions. (c) Is the Klein-Gordon equation dispersive? In particular, do wave train solutions with high frequency travel faster, slower or at the same speed as solutions with low frequency? (d) Show that there is a cutoﬀ frequency ω0 such that solutions with frequency ω ≤ ω0 are not permitted.

Adapted from Knobel An Introduction to the Mathematical Theory of Waves.

2

3. Find a travelling pulse solution for the modiﬁed KdV eqution:

∂u ∂t

+ u2

∂u ∂x

+

∂3u ∂x3

=0 .

That is, ﬁnd explicitly a solution of the form u(x,t)=f(x − ct) wheref(z), f′(z) and f′′(z) all go to zero as z →±∞. Does this solution exhibit the same or similar properties to the unmodiﬁed KdV equation? Explain your answer. Adapted from Knobel An Introduction to the Mathematical Theory of Waves.

4. (a) Consider the equation

ut + c(x,t)ux =0 . Show that along a characteristic curve x = x(t) that

d dt

(u(x(t),t)) = 0

where

dx dt

= c(x,t).

(b) Consider the following initial value problem:

ut = txux =0,u (x,0) =

1 1+x2

,

where −∞ <x<∞ and t ≥ 0.

(i) Solve the equation

dx dt

= c(x,t) to ﬁnd the explicit solution for the family of characteristic curves. Hence show that the characteristic curve with x = x0 when t = 0 has equation x = x0et2/2. Plot several characteristics in the xt-plane for t ∈ [0,2] and x ∈ [−5,5]. (ii) Write down the solution u(x,t) obtained using the method of characteristics.

Adapted from Knobel An Introduction to the Mathematical Theory of Waves.

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