In part one, the emphasis was on the meaning of
periodic reflective boundary conditions in the case of the one-dimensional wave equation. This is also called a rigid or fixed boundary condition. Other possible b.c.'s for the string are the free and the circular conditions. (The circular b.c. for a string is apparently the same as the more general periodic b.c., which I mistakenly said in earlier versions of these posts is the name for what is really a reflective b.c.) In Part 3, the final part, the emphasis will be on solutions to the
wave equation for reflective boundaries, but first it may be helpful to be reminded of
what the “most general solution” is to the 1-D wave equation with no boundary
or initial conditions prescribed.
This generic solution is most easily found by using a "difference of two squares" method to factor the equation. I first saw this method used in some notes written by Prof. Richard Rolleigh for a second-semester classical mechanics class I took at Hendrix College. (Yeah, two semesters of classical mechanics were required, and ditto for classical E&M.) The overall subject of Dr. Rolleigh's 14-page handout is "Classical Mechanics of Continuous Media." The key idea is that the 1-D wave equation,
∂ 2y/∂x2 — (1/c2)
∂2y/∂t2 = 0, (1-4)
(see the middle part of Wikipedia's Wave Equation entry) can be factored into
(∂ /∂x + (1/c) ∂ /∂t)( ∂ /∂x – (1/c) ∂ /∂t)) y(x,t) = 0.
The factored form of the equation makes it simpler to look for the necessary relation between x and t that makes the left-hand side identically equal to zero: in the first factor we have the requirement that the first partial of y with respect to x equals (1/c) times the first partial of y with respect to t , and in the second factor that the first partial of y with respect to x is (-1/c) times the first partial with respect to t. Since the factors commute with each other, an expression satisfying either requirement is a solution.
The simplest relations between x and t that satisfy these requirements are x-ct and x+ct. (Below we’ll see, courtesy of Dudley Towne, that these are the only relations between x and t that fit the 1-D wave equation as a function of position. We’ll also see [Exercise 2] how these relations can be re-written as a more useful time dependence.) We are thus looking for any functions f(x-ct) and g(x+ct). To see for yourself how this works, try each of these in the factored form of the equation. Hint: you should write these as f(u) and g(v), where u=x-ct and v=x+ct, and then do the chain rule partial derivatives.
One more standard thing needs to be pointed out before I reproduce three pages from Chapter 1 of Towne’s book: The
functional form f(x-ct) represents a waveform travelling to
the right on the string, and the form g(x+ct) represents a (possibly
different) waveform traveling to the left. To show this for f(x-ct)
we can move time ahead by Δt, so that distance is changed by Δx = cΔt,
giving
f(x+cΔt, t+Δt) = f(x+cΔt – c(t+Δt))
= f(x+cΔt – ct - cΔt)) =f(x-ct)).
The waveform’s vertical displacement is
unchanged, meaning the waveform has moved to the right at speed c. (How would
you do the same calculation for g(x+ct)?)
Now for the Towne pages, wherein we see how the sum (superposition!) of f(x,t) and g(x,t) given above provides the necessary form of the one dimensional wave equation solution. Prior to this page, Towne has shown f(x-ct) to be a solution, by doing what I mentioned above as a chain rule hint for using f(u).
…pressed by the general solution. We
are not as fortunate in the case of the two-and three-dimensional wave
equations, for which such a convenient form of general solution does not exist."
Towne’s general description of the motion of the string, and of the variables, constants, and assumptions involved in the 1-D wave equation, are also worth reproducing here, and the same goes for his
first two textbook problems in Chapter 1, which I’ll give as exercises for the
interested reader:
The
motion of the string can be specified by a function y(x,t) which is a
function of the two independent variables x and t. Thus, for
example, the graph of y versus x determined by the equation y
= y(x, t1) depicts the shape of the string at fixed time t1,
and the graph of y versus t determined by the equation y = y(x1,t)
specifies the transverse displacement of the single particle located at x1
as a function of the time. The latter graph is sometimes referred to as the history
of the particle at x1. … Assume that the string in the
undisturbed configuration has a uniform linear mass density, σ,
and is under uniform tension T. Also assume that any changes in either
of these quantities which occur during the motion of the string are sufficiently
small so that they may be neglected. … The equation [1-4] is a valid
representation of the physical conditions in the system only so long as the
inclination of the string remains everywhere small. [Meaning the slope is
small, so ∂y /∂x << 1. Not really what's shown in the drawings above.] … The particles of the string are moving in a
transverse direction, whereas the waveform propagates along the string. The “propagation”
is one of form, but not of substance.
Exercises
1. Which of the following are solutions
to the one-dimensional wave equation for transverse waves on a string? [For
use in (c), the wave speed, or propagation speed of the waveform in the
x direction, is given by c = (T/σ)1/2.]
a) x2-2xct+c2t2
b) 10(x2-
c2t2)
c) σx2 + Tt2
d) {sin[(x-ct)3]}1/3
e) 2x-3ct
f) 10(sin x)(cos ct)
2. If h(u)
is an arbitrary twice-differentiable function of u, show by direct
calculation that y(x,t) = h(t + x/c) satisfies the one-dimensional wave
equation. What relation does this solution have to the general solution written
as y(x,t) = f(x-ct) + g(x+ct)?
