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W© are goïng to apply complex numbers to our analysis of physical phenomena
by the following trick. We have examples of things that oscillate; the oscillation
may have a driving force which is a certain constant times cos œý. Now such a force,
E = Fpocosut, can be written as the real part of a complex number #' = Fpe”“t
because e?“ — cosuf + ?sin ý. The reason we do this is that it is easier to work
with an exponential function than with a cosine. So the whole trick is to represent
our oscillatory functions as the real parts of certain complex functions. “The
complex number #' that we have so defned is not a real physical force, because
no force in physies is really complex; actual forces have no imaginary part, only
a real part. We shall, however, speak of the “force” Fpe”“t, but of course the
actual force 1s the real par‡ of that expression.
Let us take another example. Suppose we want to represent a force which
is a cosine wave that is out of phase with a delayed phase A. 'This, of course,
would be the real part of Fuef=^), but exponentials being what they are, we
may wribe e/@~Ä) = e?“fe—/A, 'Thus we see that the algebra of exponentials is
much easier than that of sines and cosines; this is the reason we choose to use
complex numbers. We shall often write
E= Fục lêct = ft, (23.1)
We write a little caret (2) over the #' to remind ourselves that this quantity is a
complex number: here the number 1s
là — Fạc ?S,
Now let us solve an equation, using complex numbers, to see whether we can
work out a problem for some real case. For example, let us try to solve
da + kử = T = ro COS (UẺ, (23.2)
d2 ?m 1m 1n
where #! is the force which drives the oscillator and z is the displacement. Now,
absurd though it may seem, let us suppose that z and #! are actually complex
numbers, for a mathematical purpose only. That is to say, ø has a real part and
an Imaginary part times ?, and ?#! has a real part and an imaginary part times ¿.
Now ïf we had a solution of (23.2) with complex numbers, and substituted the
complex numbers in the equation, we would get
———— + —————~—=_————-
đị2 m m
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Ti nn. na _ đu. HỘ
d2 m d2 m m— 1n
Now, since if two complex numbers are equal, their real parts must be equal and
their imaginary parts must be equal, we deduce that #he redl part oƑ + satlisfies
the cquation tuïth the real part oƒ the forcc. We must emphasize, however, that
this separation into a real part and an imaginary part is not valid in general,
but is valid only for equations which are znear, that is, for equations in which +
appears in every term only in the frst power or the zeroth power. Eor instance,
if there were in the equation a term À#2, then when we substitute #„ -L 7;, we
would get A(z„ + ¿z;)2, but when separated into real and imaginary parts this
would yield A(#2 — z?) as the real part and 22Aø„ø; as the imaginary part. So we
see that the real part of the equation would not involve just A#2, but also —Àz‡.
In this case we get a diferent equation than the one we wanted to solve, with z¿,
the completely artificial thing we introduced in our analysis, mixed in.
Let us now try our new method for the problem of the forced oscillator, that
we already know how to solve. We want to solve Đq. (23.2) as before, but we say
that we are going to try to solve
d2x„ kxz Êc*“t
PP + mm (23.3)
where “is a complex number. Of course # will also be complex, but remember
the rule: take the real part to fnd out what is really going on. So we try %O
solve (23.3) for the forced solution; we shall discuss other solutions later. The
forced solution has the same frequency as the applied force, and has some
amplitude of oscillation and some phase, and so i§ can be represented also
by some complex number ê whose magnitude represents the swing of z and
whose phase represents the time delay in the same way as for the force. Now
a wonderful feature of an exponential function is that d(©c”2®)/dt = iuâc!*t,
'When we diferentiate an exponential function, we bring down the exponenft as
a simple multiplier. 'Phe second derivative does the same thing, it brings down
another ?œ, and so It is very simple to write Immediately, by inspection, what the
equation is for Ê: every time we see a diferentiation, we simply multiply by 2œ.
(Differentiation is now as easy as multiplication! 'This idea oŸ using exponentials
in linear diferential equations is almost as great as the invention of logarithms,
--- Trang 409 ---
in which multiplication is replaced by addition. Here diferentiation is replaced
by multiplication.) Thus our equation becomes
()22 + (kê/m) = Ê/m. (23.4)
(We have cancelled the common factor e”“f,) See how simple it is! Diferential
equations are immediately converted, by sight, into mere algebraic equations; we
virtually have the solution by sight, that
4= —_— ` —
(k/m) — 0°)
since ()” = —w#. This maybe slightly simplifed by substituting k/m = u,
which gives
â = ÊJ/m(uậ — œ`). (23.5)
This, of course, is the solution we had before; for since m(œổ — (2) is a real
number, the phase angles of P and of ê are the same (or perhaps 1802 apart, If
@2 > œ[), as advertised previously. The magnitude of ê, which measures how far
it oscillates, is related to the size of the by the faetor 1/m(œ[ — œ2), and this
factor becomes enormous when œ is nearly equal to œo. So we get a very strong
response when we apply the right frequency œ (ïf we hold a pendulum on the
end of a string and shake it at just the right Ífrequency, we can make it swing
very high).
23-2 The forced oscillator with damping
That, then, ¡is how we analyze oscillatory motion with the more elegant
mathematical technique. But the elegance of the technique is not at all exhibited
in such a problem that can be solved easily by other methods. It is only exhibited
when one applies it to more difficult problems. Let us therefore solve another,
more dificult problem, which furthermore adds a relatively realistic feature to
the previous one. Equation (23.5) tells us that ¡f the frequency œ were exactly
cqual to œọ, we would have an infinite response. Actually, of course, no such
Infinite response occurs because some other things, like friction, which we have
so far ignored, limits the response. Let us therefore add to Eq. (23.2) a friction
Ordinarily such a problem is very difcult because of the character and
complexity of the frictional term. There are, however, many circumstances In
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