246 Chapter 3 The Wave Equation
Show that this is correct. You will need Leibniz’s rule (see the Appendix) to
differentiate the integral.3.7 Comments and References
Thewaveequationisoneoftheoldestequations of mathematical physics.
Euler, Bernoulli, and d’Alembert all solved the problem of the vibrating string
about 1750, using either separation of variables or what we called d’Alembert’s
method. This latter is, in fact, a very special case of the method of characteris-
tics, in essence a way of identifying new independent variables having special
significance. Street’sAnalysis and Solution of Partial Differential Equationshas
a chapter on characteristics, including their use in numerical solutions. Wan’s
Mathematical Models and Their Analysisgivesapplicationstotrafficflowand
also discusses other wave phenomena. (See the Bibliography.)
Because many physical phenomena described by the wave equation are
part of our everyday experience — the sounds of musical instruments, for
instance — they are often featured in popular expositions of mathematical
physics. The book of Davis and Hersch (The Mathematical Experience)ex-
plains standing waves (product solutions) and superposition in an elemen-
tary way. Of course, many other phenomena are described by the wave equa-
tion. Among the most important for modern life are electrical and mag-
netic waves, which are solutions of special cases of the Maxwell field equa-
tions. These and other kinds of waves (including water waves) are studied in
Main’sVibrations and Waves in Physics; both exposition and figures are first
rate.
The potential differenceVbetween the interior and exterior of a nerve axon
can be modeled approximately by the Fitzhugh–Nagumo equations,
∂V
∂t =∂^2 V
∂x^2 +V−1
3 V
(^3) −R,
∂R
∂t
=k(V+a−bR).
HereRrepresents a restoring effect anda,b,andkare constants. At first glance,
one would expectVto behave like the solution of a heat equation. But a trav-
eling wave solution,
V(x,t)=F(x−ct), R(x,t)=G(x−ct),
of these equations can be found that shows many important features of nerve
impulses. This system and many other exciting biological applications of
mathematics are reported by Murray’s excellent book,Mathematical Biology.