So far, I’ve discussed the
meaning of periodic reflective boundary conditions for the 1-D wave equation and looked at
the general solution, or what might be called the necessary generic form of the
solution, to this equation as it applies to a small disturbance (y-direction small-amplitude
motion) on a taut string having linear density σ and tension τ. This
solution is written as
y(x,t)
= f(x-ct) + g(x+ct)
or (see end of previous post)
y(x,t) = f(t-x/c) + g(t+x/c),
where c =
(τ/σ)1/2 is the speed of the disturbance, which is
actually two disturbances when both terms in either of these equations are
nonzero—the f-disturbance travelling to the right and the g-disturbance
travelling to the left.
I’ll first look at how we get from the generic traveling wave solution to a traveling sine wave solution for the infinitely long string, then look at the form the generic solution takes for the semi-infinite string (where there’s a boundary at one end), and finally look at what happens when the traveling wave (the “incident wave”) on a semi-infinite string is a sine wave. Part 3 (the next, and last, post on this subject) will be a return to the string fixed at both ends and a look at what happens to the generic form of the solution and the sine wave solution in that case. The beginning of Schrödinger’s wave mechanics in 1926 will also be discussed in Part 3.
(The titles of Erwin's first four wave mechanics papers were "Quantization as an Eigenvalue Problem" Parts 1, 2, 3, 4. Since many people, myself included, have eigenproblems understanding the proper eigencontext of the prefix "eigen," I will include some eigencomments in my Part 3.)
The generic form of the 1-D wave equation solution above is derived without consideration of initial conditions or boundary conditions and has a unique speed associated with it, but no single frequency or wavelength like we might expect from a wave. But, as you probably realize, if it did have a frequency and wavelength in place of speed—if we just tried to substitute λν = c or ω/k = c into the generic solution—it wouldn’t be a generic solution anymore because f and g would have to be pure sine waves having their own frequencies ν or ω, and wave numbers k, with f and g necessarily expressible, for instance, as Acos(kx ± ωt + φ), where A, k, ω, and φ are constants, although we can have a superposition of sine waves of different amplitudes and frequencies.
Dudley Towne says in Section
1.7 of Wave Phenomena, where he’s still considering waves on a string of
infinite length:
Contrary to the impression which may be
created by the fact that waves of sinusoidal form are the most frequently cited examples of waves, it is to be noted that neither wavelength nor periodicity
is an essential characteristic of a wave. An initial waveform of any desired
shape determines an allowable solution to the wave equation.
After making this cautionary statement, Towne gives several reasons why sinusoidal waves are fundamental in the study of wave phenomena: 1) a sine wave is “one of the simplest analytic functions which is bounded on an infinite interval,” 2) it represents a pure tone in sound and a “spectral color” of light, and 3) it “achieves an overwhelming importance through Fourier’s theorem.” Dudley also mentions dispersion: “In some contexts involving the phenomenon of dispersion, the wave equation is not satisfied except for waves of sinusoidal form, and then only when a wave propagation velocity appropriate to the given frequency is substituted.”
Sinusoidal
Progressive Wave on an Infinite String
For the 1-D wave equation,
how do we get to a traveling sine wave solution, y(x,t) = Acos(kx±ωt + φ),
from the generic traveling wave solution, y(x,t) = f(x-ct) + g(x+ct)?
The same way we always get from the general to the particular when dealing with
differential equations: initial conditions and boundary conditions. For the
infinite string, there are no boundaries, so we just have initial conditions. And
for partial differential equations like the 1-D wave equation, it’s
really initial functions we have to deal with, a subject I’ll mention
again further down in this post (see the paragraph below that starts with
“Now”).
Right now, I just want to particularize the generic solution
to the case of a traveling sine wave. That means we start with an initial
sinusoidal disturbance on our infinitely long string, courtesy of Towne’s Section
1.7, “Description of a Progressive Sinusoidal Wave,” where he says
An initial waveform which is sinusoidal is described by
the function
y(x,
0) =
Re {yeikx}
where k and y are
given constants. If the complex amplitude y is
written in the form
y = ymeiφ,
the real amplitude ym determines
the maximum displacement of the string from the x-axis and the phase φ
determines the position of the curve with respect to translation parallel to
the x-axis. … When the quantity (kx) increases by 2π, the
function y(x,0) goes through one cycle of its values. The corresponding
increment in x is, of course, the wavelength, λ. Thus k(x + λ) = kx + 2π, or kλ = 2π, or k = 2π/λ.
The parameter k is referred to as the wave number and may be thought
of as the number of waves contained in a distance of 2π meters.
Towne chooses φ = 0,
which makes the initial waveform a cosine function—but let’s keep the
solution in exponential form for the moment. Next, we declare the initial
waveform to be in motion in the positive x direction, making g(x+ct) in
the generic solution of the wave equation identically equal to zero. The
travelling wave solution is now
y(x,t)
= f(x-ct) = Re {ymeik(x-ct)}.
Towne says, “Note that if we
focus on a particular particle of the string, say the particle [located at] x = x1, the
motion of this particle as a function of time is sinusoidal.” In a footnote, he
says “in any progressive wave the curve which describes the history of a single
particle is of the same geometric form as the wave profile of the string.” Then
he continues on about the motion being sinusoidal, but I won’t quote him from
this paragraph, except to say this is where he brings angular frequency into
the argument: as the wave travels, the particles of the string undergo simple
harmonic motion, with angular frequency ω = kc. Making
this substitution in the above equation, we get
y(x,t)
= Re {ymei(kx-ωt)} = ymcos(kx-ωt).
Thus, starting from the
generic form y(x,t) = f(x-ct) + g(x+ct), we’ve arrived at our desired expression for a
traveling sinusoidal wave.
Now I’ll mention the use of “initial
functions,” as discussed in Towne’s last section in Chapter One, “Initial
Conditions Applied to the Case of a String of Infinite Length,” which, by the way, he says
“may be omitted without loss of continuity.” (Dr. Rolleigh didn't omit this subject in his notes.) I’ll omit the derivations but
state the main idea. Since the wave equation is a 2nd order partial
differential equation, knowing the initial positions and initial velocities of
all the particles on the string—the initial wave profile and the initial
velocity profile—allows the functions f and g to
be uniquely determined. As Towne says at the beginning of Chapter Three (Boundary
Value Problems, where we’re about to go now), “the solution is uniquely
determined for an infinite string by the requirements that y(x,t)
and dy/dt reduce
at t=0 to given functions.”
Sinusoidal Progressive Wave on Semi-Infinite String
The semi-infinite string extends
from minus infinity on the x-axis to x=0, where it’s tied or
fixed or anchored to a solid wall—the boundary. The 1-D wave equation must be
satisfied within the interval – ∞ < x < 0. Towne chooses the form
of the generic solution to be
y(x,t)
= f(t-x/c) + g(t+x/c) (3-1)
“to simplify algebraic
manipulations,” he says. I’ll just use two pages from the book to show the
general waveform case and the sinusoidal waveform case. The boundary condition
is y(0,t) = 0.
y(x,t) to vanish, regardless of the value of t. The fixed end is a node, and the spacing between successive nodes is a half-wavelength. Unlike the progressive wave, the sinusoidal waveform does not undergo translation parallel to the x-axis. This type of motion is referred to as a standing wave.
d) The amplitude of the simple harmonic motion of an individual particle depends on its location. The particles halfway between the nodes have the largest amplitude for their motion. These halfway positions are referred to as antinodes.
Before
moving on to the string having boundaries at both ends, I’d like to note a
couple of things about the semi-infinite string.
First,
as you can surmise from the picture of the sine wave(s) above, there is no
restriction on the frequency (or wavelength) the standing wave can have.
This is shown by the standing wave solution itself,
y(x,t)
= 2ymsin(kx) sin(ωt),
where k and ω can take on a continuous range of values. (And now that we do have a sinusoidal wave, we necessarily have the relation ω/k = c = (τ/σ)1/2.) This range of continuous values is in contrast to waves propagating in a “confined region,” such as on a string fixed at both ends, where the allowed frequencies are determined by the dimensions of the region, such as the length of the string. This latter case has a discretely-infinite set of frequencies that are characteristic of the dimensions of the limited region--a periodicity in the frequency space and k-space. In the former case, there is a continuously-infinite number of possible sinusoidal standing wave frequencies that can be formed.
Secondly, the standing wave solution is a product of a function only of x and a function only of t, something I’ll be returning to in Part 3—and something probably familiar to you from the usual “separation of variables” technique. I don’t like to invoke this rather abstruse technique unnecessarily, and neither does Dudley Towne, so I’ll follow his Wave Phenomena Chapter 15 discussion in Part 3.
