4.1 Supposeu(t,x)is a solution of the wave equation (4.5) forx∈R. LetPbe a parallelogram in the(t,x)plane whose sides are characteristic lines. Show that the value ofuat each vertex ofPis determined by the values at the other three vertices.
4.2 The wave equation (4.5) is an appropriate model for the longitudinal vibrations of a spring. In this applicationu(t,x)represents displacement parallel to the spring.
Suppose that spring has length and is free at the ends. This corresponds to the Neumann boundary conditions
∂u
∂x(t,0)= ∂u
∂x(t, )=0, for allt ≥0.
Assume the initial conditions aregandhas in (4.16), which also satisfy Neumann boundary conditions on [0, ]. Determine the appropriate extensions of g and h from[0, ]toRso that the solutionu(t,x)given by (4.8) will satisfy the Neumann boundary problem for allt.
4.3 In the derivation in Sect.4.1, suppose we include the effect of gravity by adding a term−ρgΔxto the discrete equation of motion (4.2), whereg>0 is the constant of gravitational acceleration. The wave equation is then modified to
∂2u
∂t2 −c2∂2u
∂x2 = −g. (4.48)
Assume thatx ∈ [0, ], withusatisfying Dirichlet boundary conditions at the end- points.
(a) Find anequilibriumsolutionu0(x)for (4.48), that satisfies the boundary condi- tions but does not depend on time.
(b) Show that ifu1is a solution of the original wave equation (4.5), also with Dirichlet boundary conditions, thenu =u0+u1solves (4.48).
(c) Given the initial conditionsu(x,0)=0, ∂∂ut(x,0)=0, find the corresponding initial conditions foru1. Then apply Theorem4.5to findu1 and hence solve foru.
4.4 In Example4.8, let the forcing term be
f(t,x)=cos(ωt)sin(ωkx), withω>0 and
ωk:=kπ .
Find the solution u(t,x)given initial conditionsg = h = 0. Include both cases ω=ωkandω=ωk.
4.5 The telegraph equation is a variant of the wave equation that describes the propagation of electrical signals in a one-dimensional cable:
∂2u
∂t2 +a∂u
∂t +bu−c2∂2u
∂x2 =0,
where u(t,x) is the line voltage, c is the propagation speed, and a,b > 0 are determined by electrical properties of the cable (resistance, inductance, etc.). Show that the substitution
u(t,x)=e−at/2w(t,x)
reduces the telegraph equation to an ordinary wave equation forw, providedaand bsatisfy a certain condition. Find the general solution in this case. (This result has important practical applications, in that the electrical properties of long cables can be “tuned” to eliminate distortion.)
4.6 An alternative approach to the one-dimensional wave equation is to recast the PDE as a pair of ODE. Consider the wave equation with forcing term,
∂2u
∂t2 −c2∂2u
∂x2 = f.
(a) Define a vector-valued functionv=(v1, v2)with components
v1:=∂u
∂t, v2:= ∂u
∂x. Show thatvsatisfies a vector equation
∂v
∂t −A· ∂v
∂x =b. (4.49)
whereb:=(f,0)andAis the matrix A:=
0 c2 1 0
.
(b) The vector equation (4.49) can be solved by diagonalizingA. Check that if we set
T :=
1 c 1−c
, then
T AT−1= c 0
0−c
.
Then show under that the substitution w:=Tv,
(4.49) reduces to a pair of linear conservation equations for the components of
w: ∂w1
∂t −c∂w∂x1 = f,
∂w2
∂t +c∂w∂x2 = f. (4.50)
(c) Translate the initial conditions
u(0,x)=g(x), ∂u
∂t(0,x)=h(x),
into initial conditions forw1andw2, and then solve (4.50) using the method of characteristics.
(d) Combine the solutions forw1andw2to computev1=∂u/∂t, and then integrate to solve foru. Your answer should be a combination of of the d’Alembert formula (4.8) and the Duhamel formula (4.22).
4.7 The evolution of a quantum-mechanical wave functionu(t,x)is governed by theSchrödinger equation:
∂u
∂t −iu =0 (4.51)
(ignoring the physical constants). Suppose that u(t,x)is a solution of (4.51) for t ∈ [0,∞)andx ∈Rn, with initial condition
u(0,x)=g(x).
Assume that
Rn|g|2dnx<∞.
(a) Show that for allt ≥0,
Rn|u(t,x)|2dnx=
Rn|g|2dnx.
(In quantum mechanics|u|2is interpreted as a probability density, so this identity is conservation of total probability.)
(b) Show that a solution of Schrödinger’s equation is uniquely determined by the initial conditiong.
4.8 InRnconsider the wave equation
∂2u
∂t2 −c2u =0. (4.52)
Theplane wavesolutions have the form
u(t,x)=ei(k·x−ωt), (4.53) whereω∈Randk∈Rnare constants.
(a) Find the condition onω=ω(k)for whichusolves (4.52).
(b) For fixedt,θ ∈ R, show that
x ∈Rn; u(t,x)=eiθ
is a set of planes per- pendicular tok. Show that these planes propagate, astincreases, in a direction parallel tokwith speed given byc. (Hence the term “plane” wave.)
4.9 TheKlein-Gordon equationinRnis a variant of the wave equation that appears in relativistic quantum mechanics,
∂2u
∂t2 −u+m2u =0, (4.54)
wheremis the mass of a particle.
(a) Find a formula forω =ω(k,m)under which this equation admits plane wave solutions of the form (4.53).
(b) Show that we can define a conserved energyEfor this equation by adding a term proportional tou2to the integrand in (4.46).
Separation of Variables
Some PDE can be split into pieces that involve distinct variables. For example, the equation
∂u
∂t −a(t)b(x)u=0 could be written as
1 a(t)
∂u
∂t =b(x)u,
provideda(t)=0. This puts all of thetderivatives andt-dependent coefficients on the left and all of the terms involvingxon the right.
Splitting an equation this way is called separation of variables. For PDE that admit separation, it is natural to look for product solutions whose factors depend on the separate variables, e.g.,u(t,x)=v(t)φ(x). The full PDE then reduces to a pair of equations for the factors. In some cases, one or both of the reduced equations is an ODE that can be solved explicitly.
This idea is most commonly applied to evolution equations such as the heat or wave equations. The classical versions of these PDE have constant coefficients, and separation of variables can thus be used to split the time variable from the spatial variables. This reduces the evolution equation to a simple temporal ODE and a spatial PDE problem.
Separation among the spatial variables is sometimes possible as well, but this requires symmetry in the equation that is also shared by the domain. For example, we can separate variables for the Laplacian on rectangular or circular domains inR2. But if the domain is irregular or the differential operator has variable coefficients, then separation is generally not possible.
Despite these limitations, separation of variables plays a significant role the devel- opment of PDE theory. Explicit solutions can still yield valuable information even if they are very special cases.
© Springer International Publishing AG 2016
D. Borthwick,Introduction to Partial Differential Equations, Universitext, DOI 10.1007/978-3-319-48936-0_5
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Fig. 5.1 Frequency decomposition for the sound of a violin string