- REAL ANALYSIS 497
For a string fastened at two points, say (0,0) and (L, 0) and vibrating so that
its displacement above or below the point (x, 0) at time t is y(x, t), mathematicians
agreed that the best compromise between realism and comprehensibility to describe
this motion was the one-dimensional wave equation, which d'Alembert studied in
1747,^5 publishing the results in 1749:
D'Alembert pointed out that the solution must be of the form
y(x,t) = 9(t + x) + T(t-x),where for simplicity he assumed that c = 1. The equation alone does not determine
the function, of course, since the vibrations depend on the initial position and
velocity of the string. Accordingly, d'Alembert followed up with a prescribed initial
position f(x) = y(x,0) and velocity v(x) = f^|t_ 0 - He considered first the case
when the initial position is identically zero, for which the function Ö must be an
even function of period 2L, then the more general case.
The following year Euler took up this problem and commented on d' Alembert's
solution. He observed that the initial position could be any shape at all, "either
regular or irregular and mechanical." D'Alembert found that claim hard to accept.
After all, the functions Ö and à had to have periodicity and parity properties. How
else could they be defined except as power series containing only odd or only even
powers? Euler and d'Alembert were not interpreting the word "function" in the
same way. Euler was even willing to consider initial positions f(x) with corners (a
"plucked" string), whereas d'Alembert insisted that f(x) must have two derivatives
simply to satisfy the equation.
Three years later Daniel Bernoulli tried to straighten this matter out, giving a
solution in the form
which he did not actually write out. Here the coefficients an were to be chosen so
that the initial condition was satisfied, that is,Observing that he had an infinite set of coefficients at his disposal for "fitting"
the function, Bernoulli claimed that "any" function f(x) had such a representa-
tion. Bernoulli's solution was the first of many instances in which the classical
partial differential equations of mathematical physics—the wave, heat, and poten-
tial equations—were studied by separating variables and superposing the resulting
solutions. The technique was ultimately to lead to what are called Sturm-Liouville
problems, which we shall mention again below.
Before leaving the wave equation, we must mention one more important crossing
between real and complex analysis in connection with it. In studying the action(^5) Thirty years earlier Brook Taylor (1685-1731) had analyzed the problem geometrically and
concluded that the normal acceleration at each point would be proportional to the normal cur-
vature at that point. That statement is effectively the same as this equation, and was quoted by
d'Alembert.
d^2 y = 2 d2V
dt2 c dx^2 '
n=l