## 23Resonance

### 23–1Complex numbers and harmonic motion

In the present chapter we shall continue our discussion of theharmonic oscillator and, in particular, the forced harmonicoscillator, using a new technique in the analysis. In the precedingchapter we introduced the idea of complex numbers, which have real andimaginary parts and which can be represented on a diagram in which theordinate represents the imaginary part and the abscissa represents thereal part. If $a$ is a complex number, we may write it as $a = a_r +ia_i$, where the subscript$r$ means the real part of$a$, and thesubscript$i$ means the imaginary part of$a$. Referring toFig.23–1, we see that we may also write a complex number$a= x + iy$ in the form $x + iy = re^{i\theta}$, where $r^2 = x^2 + y^2 =(x + iy)(x - iy) = aa\cconj$. (The complex conjugate of$a$, written$a\cconj$, is obtained by reversing the sign of$i$ in$a$.) So we shallrepresent a complex number in either of two forms, a real plus animaginary part, or a magnitude$r$ and a phase angle$\theta$,so-called. Given $r$ and$\theta$, $x$ and$y$ are clearly $r\cos\theta$and$r\sin\theta$ and, in reverse, given a complex number$x + iy$, $r =\sqrt{x^2 + y^2}$ and$\tan\theta= y/x$, the ratio of the imaginary tothe real part.

Fig. 23–1.A complex number may be represented by a point in the“complex plane.”

We are going to apply complex numbers to our analysis of physicalphenomena by the following trick. We have examples of things thatoscillate; the oscillation may have a driving force which is a certainconstant times $\cos\omega t$. Now such a force, $F = F_0\cos\omega t$, can be written as the real part of a complex number$F =F_0e^{i\omega t}$ because $e^{i\omega t} = \cos\omega t + i\sin\omega t$. The reason we do this is that it is easier to work with anexponential function than with a cosine. So the whole trick is torepresent our oscillatory functions as the real parts of certaincomplex functions. The complex number$F$ that we have so defined isnot a real physical force, because no force in physics is reallycomplex; actual forces have no imaginary part, only a real part. Weshall, however, speak of the “force”$F_0e^{i\omega t}$, but ofcourse the actual force is the *real part* of that expression.

Let us take another example. Suppose we want to represent a forcewhich is a cosine wave that is out of phase with a delayedphase$\Delta $. This, of course, would be the real part of$F_0e^{i(\omegat-\Delta)}$, but exponentials being what they are, we may write$e^{i(\omega t-\Delta)}= e^{i\omega t}e^{-i\Delta}$. Thus we see thatthe algebra of exponentials is much easier than that of sines andcosines; this is the reason we choose to use complex numbers. We shalloften write\begin{equation}\label{Eq:I:23:1}F=F_0e^{-i\Delta}e^{i\omega t}=\hat{F}e^{i\omega t}.\end{equation}We write a little caret ($\hat{\enspace}$) over the $F$ to remindourselves that this quantity is a complex number: here the number is\begin{equation*}\hat{F}=F_0e^{-i\Delta}.\end{equation*}

Now let us solve an equation, using complex numbers, to see whether wecan work out a problem for some real case. For example, let us try tosolve\begin{equation}\label{Eq:I:23:2}\frac{d^2x}{dt^2}+\frac{kx}{m}=\frac{F}{m}=\frac{F_0}{m}\cos\omega t,\end{equation}where $F$ is the force which drives the oscillator and $x$ is thedisplacement. Now, absurd though it may seem, let us suppose that $x$and$F$ are actually complex numbers, for a mathematical purposeonly. That is to say, $x$ has a real part and an imaginary part times$i$, and $F$ has a real part and an imaginary part times $i$. Now ifwe had a solution of(23.2) with complex numbers, andsubstituted the complex numbers in the equation, we would get\begin{equation*}\frac{d^2(x_r+ix_i)}{dt^2}+\frac{k(x_r+ix_i)}{m}=\frac{F_r+iF_i}{m}\end{equation*}or\begin{equation*}\frac{d^2x_r}{dt^2}+\frac{kx_r}{m}+i\biggl(\frac{d^2x_i}{dt^2}+\frac{kx_i}{m}\biggr)=\frac{F_r}{m}+\frac{iF_i}{m}.\end{equation*}Now, since if two complex numbers are equal, their real parts must beequal *and* their imaginary parts must be equal, we deduce that*the real part of$x$ satisfies the equation with the real partof the force*. We must emphasize, however, that this separation into areal part and an imaginary part is *not* valid in general, but isvalid only for equations which are *linear*, that is, forequations in which $x$ appears in every term only in the first poweror the zeroth power. For instance, if there were in the equation aterm$\lambda x^2$, then when we substitute $x_r + ix_i$, we would get$\lambda(x_r + ix_i)^2$, but when separated into real and imaginaryparts this would yield $\lambda(x_r^2 - x_i^2)$ as the real part and$2i\lambda x_rx_i$ as the imaginary part. So we see that the real partof the equation would not involve just $\lambda x_r^2$, but also$-\lambda x_i^2$. In this case we get a different equation than theone we wanted to solve, with $x_i$, the completely artificial thing weintroduced 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 solveEq.(23.2) as before, but we say that we are going to try tosolve\begin{equation}\label{Eq:I:23:3}\frac{d^2x}{dt^2}+\frac{kx}{m}=\frac{\hat{F}e^{i\omega t}}{m},\end{equation}where $\hat{F}e^{i\omega t}$ is a complex number. Of course $x$ willalso be complex, but remember the rule: take the real part to find outwhat is really going on. So we try to solve(23.3) for theforced solution; we shall discuss other solutions later. The forcedsolution has the same frequency as the applied force, and has someamplitude of oscillation and some phase, and so it can be representedalso by some complex number$\hat{x}$ whose magnitude represents theswing of$x$ and whose phase represents the time delay in the same wayas for the force. Now a wonderful feature of an exponential function $x=\hat{x}e^{i\omega t}$is that $dx/dt = i\omega x$. When we differentiate an exponential function, we bring down theexponent as a simple multiplier. The second derivative does the samething, it brings down another $i\omega$, and so it is very simple towrite immediately, by inspection, what the equation is for $x$:every time we see a differentiation, we simply multiplyby$i\omega$. (Differentiation is now as easy as multiplication! Thisidea of using exponentials in linear differential equations is almostas great as the invention of logarithms, in which multiplication isreplaced by addition. Here differentiation is replaced bymultiplication.) Thus our equation becomes\begin{equation}\label{Eq:I:23:4}(i\omega)^2\hat{x}+(k\hat{x}/m)=\hat{F}/m.\end{equation}(We have cancelled the common factor$e^{i\omega t}$.) See how simpleit is! Differential equations are immediately converted, by sight,into mere algebraic equations; we virtually have the solution bysight, that\begin{equation*}\hat{x}=\frac{\hat{F}/m}{(k/m)-\omega^2},\end{equation*}since $(i\omega)^2 =-\omega^2$. This may be slightly simplified bysubstituting $k/m = \omega_0^2$, which gives\begin{equation}\label{Eq:I:23:5}\hat{x}=\hat{F}/m(\omega_0^2-\omega^2).\end{equation}This, of course, is the solution we had before; for since$m(\omega_0^2 - \omega^2)$ is a real number, the phase angles of$\hat{F}$and of$\hat{x}$ are the same (or perhaps $180^\circ$ apart,if $\omega^2 > \omega_0^2$), as advertised previously. The magnitudeof$\hat{x}$, which measures how far it oscillates, is related to thesize of the$\hat{F}$ by the factor$1/m(\omega_0^2-\omega^2)$, andthis factor becomes enormous when $\omega$ is nearly equalto$\omega_0$. So we get a very strong response when we apply the rightfrequency$\omega$ (if we hold a pendulum on the end of a string andshake it at just the right frequency, we can make it swing very high).

### 23–2The forced oscillator with damping

That, then, is how we analyze oscillatory motion with the more elegantmathematical technique. But the elegance of the technique is not atall exhibited in such a problem that can be solved easily by othermethods. It is only exhibited when one applies it to more difficultproblems. Let us therefore solve another, more difficult problem,which furthermore adds a relatively realistic feature to the previousone. Equation(23.5) tells us that if thefrequency$\omega$ were exactly equal to$\omega_0$, we would have an infiniteresponse. Actually, of course, no such infinite response occursbecause some other things, like friction, which we have so farignored, limits the response. Let us therefore add toEq.(23.2) a friction term.

Ordinarily such a problem is very difficult because of the characterand complexity of the frictional term. There are, however, manycirc*mstances in which the frictional force is *proportional tothe speed* with which the object moves. An example of such friction isthe friction for slow motion of an object in oil or a thickliquid. There is no force when it is just standing still, but thefaster it moves the faster the oil has to go past the object, and thegreater is the resistance. So we shall assume that there is, inaddition to the terms in(23.2), another term, a resistanceforce proportional to the velocity: $F_f =-c\,dx/dt$. It will beconvenient, in our mathematical analysis, to write the constant$c$ as$m$ times$\gamma$ to simplify the equation a little. This is just thesame trick we use with $k$ when we replace it by$m\omega_0^2$, justto simplify the algebra. Thus our equation will be\begin{equation}\label{Eq:I:23:6}m(d^2x/dt^2)+c(dx/dt)+kx=F\end{equation}or, writing $c = m\gamma$ and$k = m\omega_0^2$ and dividing out themass$m$,\begin{equation*}\label{Eq:I:23:6a}(d^2x/dt^2)+\gamma(dx/dt)+\omega_0^2x=F/m.\tag{23.6a}\end{equation*}

Now we have the equation in the most convenient form to solve. If$\gamma$ is very small, that represents very little friction; if$\gamma$ is very large, there is a tremendous amount of friction. Howdo we solve this new linear differential equation? Suppose that thedriving force is equal to$F_0\cos\,(\omega t+\Delta)$; we could putthis into(23.6a) and try to solve it, but we shall insteadsolve it by our new method. Thus we write $F$ as the real partof$\hat{F}e^{i\omega t}$ and $x$ as the real part of$\hat{x}e^{i\omegat}$, and substitute these into Eq.(23.6a). It is noteven necessary to do the actual substituting, for we can see byinspection that the equation would become\begin{equation}\label{Eq:I:23:7}[(i\omega)^2\hat{x}+\gamma(i\omega)\hat{x}+\omega_0^2\hat{x}]e^{i\omega t}=(\hat{F}/m)e^{i\omega t}.\end{equation}[As a matter of fact, if we tried to solve Eq.(23.6a) byour old straightforward way, we would really appreciate the magic ofthe “complex” method.] If we divide by$e^{i\omega t}$ on bothsides, then we can obtain the response$\hat{x}$ to the givenforce$\hat{F}$; it is\begin{equation}\label{Eq:I:23:8}\hat{x}=\hat{F}/m(\omega_0^2-\omega^2+i\gamma\omega).\end{equation}

Thus again $\hat{x}$ is given by$\hat{F}$ times a certainfactor. There is no technical name for this factor, no particularletter for it, but we may call it $R$ for discussion purposes:\begin{equation}R=\frac{1}{m(\omega_0^2-\omega^2+i\gamma\omega)}\notag\end{equation}and\begin{equation}\label{Eq:I:23:9}\hat{x}=\hat{F}R.\end{equation}(Although the letters $\gamma$ and$\omega_0$ are in very common use,this $R$ has no particular name.) This factor$R$ can either bewritten as$p+iq$, or as a certain magnitude$\rho$ times$e^{i\theta}$. If it is written as a certain magnitude times$e^{i\theta}$, let us see what it means. Now $\hat{F}=F_0e^{i\Delta}$, and the actual force$F$ is the real part of$F_0e^{i\Delta}e^{i\omega t}$, that is, $F_0 \cos\,(\omega t +\Delta)$. Next, Eq.(23.9) tells us that $\hat{x}$ isequal to$\hat{F}R$. So, writing $R = \rho e^{i\theta}$ as anothername for $R$, we get\begin{equation*}\hat{x}=R\hat{F}=\rho e^{i\theta}F_0e^{i\Delta}=\rho F_0e^{i(\theta+\Delta)}.\end{equation*}Finally, going even further back, we see that the physical$x$, whichis the real part of the complex$\hat{x}e^{i\omega t}$, is equal tothe real part of$\rho F_0e^{i(\theta+\Delta)}e^{i\omega t}$. But$\rho$ and$F_0$ are real, and the real partof$e^{i(\theta+\Delta+\omega t)}$ is simply $\cos\,(\omega t + \Delta +\theta)$. Thus\begin{equation}\label{Eq:I:23:10}x=\rho F_0\cos\,(\omega t+\Delta+\theta).\end{equation}This tells us that the amplitude of the response is the magnitude ofthe force$F$ multiplied by a certain magnification factor, $\rho$;this gives us the “amount” of oscillation. It also tells us,however, that $x$ is not oscillating in phase with the force, whichhas the phase$\Delta$, but is shifted by an extraamount$\theta$. Therefore $\rho$ and$\theta$ represent the size of theresponse and the phase shift of the response.

Now let us work out what $\rho$ is. If we have a complex number, thesquare of the magnitude is equal to the number times its complexconjugate; thus\begin{equation}\begin{aligned}\rho^2&=\frac{1}{m^2(\omega_0^2-\omega^2+i\gamma\omega)(\omega_0^2-\omega^2-i\gamma\omega)}\\[1ex]&=\frac{1}{m^2[(\omega_0^2-\omega^2)^2+\gamma^2\omega^2]}.\end{aligned}\label{Eq:I:23:11}\end{equation}

In addition, the phase angle $\theta$ is easy to find, for if we write\begin{equation}1/R=1/\rho e^{i\theta}=(1/\rho)e^{-i\theta}=m(\omega_0^2-\omega^2+i\gamma\omega),\notag\end{equation}we see that\begin{equation}\label{Eq:I:23:12}\tan\theta=-\gamma\omega/(\omega_0^2-\omega^2).\end{equation}It is minus because $\tan (-\theta) =-\tan\theta$. A negative valuefor $\theta$ results for all $\omega$, and this corresponds to thedisplacement$x$ lagging the force$F$.

Fig. 23–2.Plot of $\rho^2$ versus $\omega$.

Fig. 23–3.Plot of $\theta$ versus $\omega$.

Figure23–2 shows how $\rho^2$ varies as a function offrequency ($\rho^2$ is physically more interesting than $\rho$,because $\rho^2$ is proportional to the square of the amplitude, ormore or less to the *energy* that is developed in the oscillatorby the force). We see that if $\gamma$ is very small, then$1/(\omega_0^2 - \omega^2)^2$ is the most important term, and theresponse tries to go up toward infinity when $\omega$ equals$\omega_0$. Now the “infinity” is not actually infinite because if$\omega=\omega_0$, then $1/\gamma^2\omega^2$ is still there. The phaseshift varies as shown in Fig.23–3.

In certain circ*mstances we get a slightly different formulathan(23.8), also called a “resonance” formula, and onemight think that it represents a different phenomenon, but it does not.The reason is that if $\gamma$ is very small the most interesting partof the curve is near $\omega = \omega_0$, and we mayreplace(23.8) by an approximate formula which is veryaccurate if $\gamma$ is small and $\omega$ is near $\omega_0$. Since$\omega_0^2 - \omega^2 = (\omega_0-\omega)(\omega_0 + \omega)$, if$\omega$ is near $\omega_0$ this is nearly the same as$2\omega_0(\omega_0 - \omega)$ and $\gamma\omega$ is nearly the same as$\gamma\omega_0$. Using these in(23.8), we see that$\omega_0^2-\omega^2 + i\gamma\omega \approx2\omega_0(\omega_0-\omega+i\gamma/2)$, so that\begin{equation}\begin{gathered}\hat{x}\approx\hat{F}/2m\omega_0(\omega_0-\omega+i\gamma/2)\\[.5ex]\text{ if }\gamma\ll\omega_0\text{ and }\omega\approx\omega_0.\end{gathered}\label{Eq:I:23:13}\end{equation}It is easy to find the corresponding formula for $\rho^2$. It is\begin{equation*}\rho^2\approx1/4m^2\omega_0^2[(\omega_0-\omega)^2+\gamma^2/4].\end{equation*}

We shall leave it to the student to show the following: if we call themaximum height of the curve of $\rho^2$ vs.$\omega$ one unit, andwe ask for the width$\Delta\omega$ of the curve, at one half themaximum height, the full width at half the maximum height of the curveis$\Delta\omega=\gamma$, supposing that $\gamma$ is small. Theresonance is sharper and sharper as the frictional effects are madesmaller and smaller.

As another measure of the width, some people use a quantity$Q$ whichis defined as$Q = \omega_0/\gamma$. The narrower the resonance, thehigher the $Q$: $Q= 1000$ means a resonance whose width is only$1000$th of the frequency scale. The $Q$ of the resonance curve shownin Fig.23–2 is$5$.

The importance of the resonance phenomenon is that it occurs in manyother circ*mstances, and so the rest of this chapter will describesome of these other circ*mstances.

### 23–3Electrical resonance

The simplest and broadest technical applications of resonance are inelectricity. In the electrical world there are a number of objectswhich can be connected to make electric circuits. These *passivecircuit elements*, as they are often called, are of three main types,although each one has a little bit of the other two mixed in. Beforedescribing them in greater detail, let us note that the whole idea ofour mechanical oscillator being a mass on the end of a spring is onlyan approximation. All the mass is not actually at the “mass”; someof the mass is in the inertia of the spring. Similarly, all of thespring is not at the “spring”; the mass itself has a littleelasticity, and although it may appear so, it is not *absolutely*rigid, and as it goes up and down, it flexes ever so slightly underthe action of the spring pulling it. The same thing is true inelectricity. There is an approximation in which we can lump thingsinto “circuit elements” which are assumed to have pure, idealcharacteristics. It is not the proper time to discuss thatapproximation here, we shall simply assume that it is true in thecirc*mstances.

Fig. 23–4.The three passive circuit elements.

The three main kinds of circuit elements are the following. The first iscalled a *capacitor* (Fig.23–4); anexample is two plane metallic plates spaced a very small distance apartby an insulating material. When the plates are charged there is acertain voltage difference, that is, a certain difference in potential,between them. The same difference of potential appears between theterminals $A$ and$B$, because if there were any difference along theconnecting wire, electricity would flow right away. So there is acertain voltage difference$V$ between the plates if there is a certainelectric charge $+q$ and$-q$ on them, respectively. Between the platesthere will be a certain electric field; we have even found a formula forit (Chapters 13 and14):\begin{equation}\label{Eq:I:23:14}V=\sigma d/\epsO=qd/\epsO A,\end{equation}where $d$ is the spacing and $A$ is the area of the plates. Note thatthe potential difference is a linear function of the charge. If we donot have parallel plates, but insulated electrodes which are of anyother shape, the difference in potential is still preciselyproportional to the charge, but the constant of proportionality maynot be so easy to compute. However, all we need to know is that thepotential difference across a capacitor *is proportional to thecharge: $V = q/C$*; the proportionality constant is$1/C$, where $C$is the *capacitance* of the object.

The second kind of circuit element is called a *resistor*;itoffers resistance to the flow of electrical current. It turns out thatmetallic wires and many other substances resist the flow ofelectricity in this manner: if there is a voltage difference across apiece of some substance, there exists an electric current$I= dq/dt$that is proportional to the electric voltage difference:\begin{equation}\label{Eq:I:23:15}V=RI=R\,dq/dt\end{equation}The proportionality coefficient is called the*resistance*$R$. This relationship may alreadybe familiar to you; it is Ohm’s law.

If we think of the charge$q$ on a capacitor as being analogous to thedisplacement$x$ of a mechanical system, we see that the current, $I =dq/dt$, is analogous to velocity, $1/C$ is analogous to a springconstant$k$, and $R$ is analogous to the resistive coefficient$c=m\gamma$ in Eq.(23.6). Now it is very interesting thatthere exists another circuit element which is the analog of *mass*!This is a coil which builds up a magnetic field within itself when thereis a current in it. A *changing* magnetic field develops in thecoil a voltage that is proportional to$dI/dt$ (this is how atransformer works, in fact). The magnetic field is proportional to acurrent, and the induced voltage (so-called) in such a coil isproportional to the rate of change of the current:\begin{equation}\label{Eq:I:23:16}V=L\,dI/dt=L\,d^2q/dt^2.\end{equation}The coefficient$L$ is the *self-inductance*, andis analogous to the mass in a mechanical oscillating circuit.

Fig. 23–5.An oscillatory electrical circuit with resistance,inductance, and capacitance.

Suppose we make a circuit in which we have connected the three circuitelements in series (Fig.23–5); then the voltage acrossthe whole thing from $1$ to$2$ is the work done in carrying a chargethrough, and it consists of the sum of several pieces: across theinductor, $V_L = L\,d^2q/dt^2$; across the resistance, $V_R =R\,dq/dt$; across the capacitor, $V_C = q/C$. The sum of these isequal to the applied voltage, $V$:\begin{equation}\label{Eq:I:23:17}L\,d^2q/dt^2+R\,dq/dt+q/C=V(t).\end{equation}Now we see that this equation is exactly the same as the mechanicalequation(23.6), and of course it can be solved in exactlythe same manner. We suppose that $V(t)$ is oscillatory: we are drivingthe circuit with a generator with a pure sine wave oscillation. Thenwe can write our $V(t)$ as a complex$\hat{V}$ with the understandingthat it must be ultimately multiplied by$e^{i\omega t}$, and the realpart taken in order to find the true $V$. Likewise, the charge$q$ canthus be analyzed, and then in exactly the same manner as inEq.(23.8) we write the corresponding equation: the secondderivative of$q$ is $(i\omega)^2q$; the first derivativeis$(i\omega)q$. Thus Eq.(23.17) translates to\begin{equation*}\biggl[L(i\omega)^2+R(i\omega)+\frac{1}{C}\biggr]\hat{q}=\hat{V}\end{equation*}or\begin{equation*}\hat{q}=\frac{\hat{V}}{L(i\omega)^2+R(i\omega)+\dfrac{1}{C}}\end{equation*}which we can write in the form\begin{equation}\label{Eq:I:23:18}\hat{q}=\hat{V}/L(\omega_0^2-\omega^2+i\gamma\omega),\end{equation}where $\omega_0^2 = 1/LC$ and $\gamma= R/L$. It is exactly the samedenominator as we had in the mechanical case, with exactly the sameresonance properties! The correspondence between the electrical andmechanical cases is outlined in Table23–1.

Table 23–1

General characteristic | Mechanical property | Electrical property |

indep. variable | time $(t)$ | time $(t)$ |

dep. variable | position $(x)$ | charge $(q)$ |

inertia | mass $(m)$ | inductance $(L)$ |

resistance | drag coeff. $(c=\gamma m)$ | resistance $(R=\gamma L)$ |

stiffness | stiffness $(k)$ | (capacitance)$^{-1}$ $(1/C)$ |

resonant frequency | $\omega_0^2=k/m$ | $\omega_0^2=1/LC$ |

period | $t_0=2\pi\sqrt{m/k}$ | $t_0=2\pi\sqrt{LC}$ |

figure of merit | $Q=\omega_0/\gamma$ | $Q=\omega_0L/R$ |

We must mention a small technical point. In the electrical literature,a different notation is used. (From one field to another, the subjectis not really any different, but the way of writing the notations isoften different.) First, $j$ is commonly used instead of$i$ inelectrical engineering, to denote$\sqrt{-1}$. (After all, $i$ must bethe current!) Also, the engineers would rather have a relationshipbetween $\hat{V}$ and$\hat{I}$ than between $\hat{V}$ and$\hat{q}$,just because they are more used to it that way. Thus, since $I=dq/dt = i\omega q$, we can just substitute$\hat{I}/i\omega$ for $\hat{q}$ and get\begin{equation}\label{Eq:I:23:19}\hat{V}=(i\omega L+R+1/i\omega C)\hat{I}=\hat{Z}\hat{I}.\end{equation}Another way is to rewrite Eq.(23.17), so that it looksmore familiar; one often sees it written this way:\begin{equation}\label{Eq:I:23:20}L\,dI/dt+RI+(1/C)\int^tI\,dt=V(t).\end{equation}At any rate, we find the relation(23.19) betweenvoltage$\hat{V}$ and current$\hat{I}$ which is just the sameas(23.18) except divided by$i\omega$, and that producesEq.(23.19). The quantity$R + i\omega L + 1/i\omega C$ is acomplex number, and is used so much in electrical engineering that ithas a name: it is called the *complex impedance*, $\hat{Z}$. Thus we can write$\hat{V}=\hat{Z}\hat{I}$. The reason that the engineers like to do thisis that they learned something when they were young: $V = RI$ forresistances, when they only knew about resistances and dc. Nowthey have become more educated and have ac circuits, so theywant the equation to look the same. Thus they write$\hat{V}=\hat{Z}\hat{I}$, the only difference being that the resistanceis replaced by a more complicated thing, a complex quantity. So theyinsist that they cannot use what everyone else in the world uses forimaginary numbers, they have to use a$j$ for that; it is a miracle thatthey did not insist also that the letter$Z$ be an$R$! (Then they getinto trouble when they talk about current densities, for which they alsouse$j$. The difficulties of science are to a large extent thedifficulties of notations, the units, and all the other artificialitieswhich are invented by man, not by nature.)

### 23–4Resonance in nature

Although we have discussed the electrical case in detail, we couldalso bring up case after case in many fields, and show exactly how theresonance equation is the same. There are many circ*mstances in naturein which something is “oscillating” and in which the resonancephenomenon occurs. We said that in an earlier chapter; let us nowdemonstrate it. If we walk around our study, pulling books off theshelves and simply looking through them to find an example of a curvethat corresponds to Fig.23–2 and comes from the sameequation, what do we find? Just to demonstrate the wide range obtainedby taking the smallest possible sample, it takes only five or sixbooks to produce quite a series of phenomena which show resonances.

Fig. 23–6.Response of the atmosphere to external excitation. $a$ is therequired response if the atmospheric $S_2$-tide is of gravitationalorigin; peak amplification is$100:1$. $b$ is derived from observedmagnification and phase of$M_2$-tide. [Munk and MacDonald,“Rotation of the Earth,” Cambridge University Press (1960)]

The first two are from mechanics, the first on a large scale: theatmosphere of the whole earth. If the atmosphere, which we supposesurrounds the earth evenly on all sides, is pulled to one side by themoon or, rather, squashed prolate into a double tide, and if we couldthen let it go, it would go sloshing up and down; it is anoscillator. This oscillator is *driven* by the moon, which iseffectively revolving about the earth; any one component of the force,say in the $x$-direction, has a cosine component, and so the responseof the earth’s atmosphere to the tidal pull of the moon is that of anoscillator. The expected response of the atmosphere is shown inFig.23–6, curve$b$ (curve$a$ is another theoretical curveunder discussion in the book from which this is taken out of context).Now one might think that we only have one point on this resonance curve,since we only have the one frequency, corresponding to the rotation ofthe earth under the moon, which occurs at a period of$12.42$hours—$12$hours for the earth (the tide is a double bump), plus alittle more because the moon is going around. But from the *size*of the atmospheric tides, and from the *phase*, the amount ofdelay, we can get both $\rho$ and$\theta$. From those we can get$\omega_0$ and$\gamma$, and thus draw the entire curve! This is anexample of very poor science. From two numbers we obtain two numbers,and from those two numbers we draw a beautiful curve, which of coursegoes through the very point that determined the curve! It is of no use*unless we can measure something else*, and in the case ofgeophysics that is often very difficult. But in this particular casethere is another thing which we can show theoretically must have thesame timing as the natural frequency$\omega_0$: that is, if someonedisturbed the atmosphere, it would oscillate with thefrequency$\omega_0$. Now there *was* such a sharp disturbance in1883; theKrakatoa volcano exploded and half the island blew off, and it made sucha terrific explosion in the atmosphere that the period of oscillation ofthe atmosphere could be measured. It came out to $10\tfrac{1}{2}$hours.The $\omega_0$ obtained from Fig.23–6 comes out$10$hours and $20$minutes, so there we have at least one check on thereality of our understanding of the atmospheric tides.

Next we go to the small scale of mechanical oscillation. This time wetake a sodium chloride crystal, which has sodium ions and chlorineions next to each other, as we described in an early chapter. Theseions are electrically charged, alternately plus and minus. Now thereis an interesting oscillation possible. Suppose that we could driveall the plus charges to the right and all the negative charges to theleft, and let go; they would then oscillate back and forth, the sodiumlattice against the chlorine lattice. How can we ever drive such athing? That is easy, for if we apply an electric field on the crystal,it will push the plus charge one way and the minus charge the otherway! So, by having an external electric field we can perhaps get thecrystal to oscillate. The frequency of the electric field needed is sohigh, however, that it corresponds to *infraredradiation!*So we try to find a resonance curve by measuring the absorption ofinfrared light by sodium chloride. Such a curve is shown inFig.23–7. The abscissa is not frequency, but is given interms of wavelength, but that is just a technical matter, of course,since for a wave there is a definite relation between frequency andwavelength; so it is really a frequency scale, and a certain frequencycorresponds to the resonant frequency.

Fig. 23–7.Transmission of infrared radiation through a thin (0.17$\mu$m) sodiumchloride film. [After R.B. Barnes, Z.Physik75, 723 (1932). Kittel, Introduction to Solid StatePhysics, Wiley, 1956.]

But what about the width? What determines the width? There are manycases in which the width that is seen on the curve is not really thenatural width$\gamma$ that one would have theoretically. There aretwo reasons why there can be a wider curve than the theoreticalcurve. If the objects do not all have the same frequency, as mighthappen if the crystal were strained in certain regions, so that inthose regions the oscillation frequency were slightly different thanin other regions, then what we have is many resonance curves on top ofeach other; so we apparently get a wider curve. The other kind ofwidth is simply this: perhaps we cannot measure the frequencyprecisely enough—if we open the slit of the spectrometer fairlywide, so although we thought we had only one frequency, we actuallyhad a certain range$\Delta\omega$, then we may not have the resolvingpower needed to see a narrow curve. Offhand, we cannot say whether thewidth in Fig.23–7 is natural, or whether it is due toinhom*ogeneities in the crystal or the finite width of the slit of thespectrometer.

Fig. 23–8.Magnetic energy loss in paramagnetic organic compound as function ofapplied magnetic field intensity. [Holden et al., Phys.Rev.75, 1614 (1949)]

Now we turn to a more esoteric example, and that is the swinging of amagnet. If we have a magnet, with north and south poles, in a constantmagnetic field, the N end of the magnet will be pulled one way and theS end the other way, and there will in general be a torque on it, soit will vibrate about its equilibrium position, like a compassneedle. However, the magnets we are talking about are*atoms*. These atoms have an angular momentum, the torque doesnot produce a simple motion in the direction of the field, butinstead, of course, a *precession*. Now, looked at from the side,any one component is “swinging,” and we can disturb or drive thatswinging and measure an absorption. The curve in Fig.23–8represents a typical such resonance curve. What has been done here isslightly different technically. The frequency of the lateral field thatis used to drive this swinging is always kept the same, while we wouldhave expected that the investigators would vary that and plot the curve.They could have done it that way, but technically it was easier for themto leave the frequency$\omega$ fixed, and change the strength of theconstant magnetic field, which corresponds to changing $\omega_0$ in ourformula. They have plotted the resonance curve against $\omega_0$.Anyway, this is a typical resonance with a certain $\omega_0$and$\gamma$.

Fig. 23–9.The intensity of gamma-radiation from lithium as a function of theenergy of the bombarding protons. The dashed curve is a theoretical onecalculated for protons with an angular momentum$\ell = 0$. [Bonner and Evans,Phys. Rev.73, 666 (1948)]

Now we go still further. Our next example has to do with atomicnuclei. The motions of protons and neutrons in nuclei are oscillatoryin certain ways, and we can demonstrate this by the followingexperiment. We bombard a lithium atom with protons, and we discoverthat a certain reaction, producing $\gamma$-rays, actually has a verysharp maximum typical of resonance. We note in Fig.23–9,however, one difference from other cases: the horizontal scale is not afrequency, it is an *energy!* The reason is that in quantummechanics what we think of classically as the energy will turn out to bereally related to a frequency of a wave amplitude. When we analyzesomething which in simple large-scale physics has to do with afrequency, we find that when we do quantum-mechanical experiments withatomic matter, we get the corresponding curve as a function of energy.In fact, this curve is a demonstration of this relationship, in a sense.It shows that frequency and energy have some deep interrelationship,which of course they do.

Fig. 23–10.[Courtesy of Dr.R. Mössbauer]

Now we turn to another example which also involves a nuclear energylevel, but now a much, much narrower one. The $\omega_0$ inFig.23–10 corresponds to an energy of $100{,}000$electronvolts, while the width$\gamma$ is approximately $10^{-5}$electronvolt; in other words, this has a$Q$ of$10^{10}$! When this curve wasmeasured it was the largest$Q$ of any oscillator that had ever beenmeasured. It was measured by Dr.Mössbauer, and it was the basis of his Nobel prize. Thehorizontal scale here is velocity, because the technique for obtainingthe slightly different frequencies was to use the Dopplereffect, by moving the source relative tothe absorber. One can see how delicate the experiment is when we realizethat the speed involved is a few centimeters per second! On the actualscale of the figure, zero frequency would correspond to a point about$10^{10}$cm to the left—slightly off the paper!

Fig. 23–11.Momentum dependence of the cross section for the reactions(a)$\text{K}^- + \text{p} \to \Lambda + \pi^+ + \pi^-$ and (b)$\text{K}^- +\text{p} \to \overline{\text{K}}{}^0 + \text{n}$. The lower curves in (a)and(b) represent the presumed nonresonant backgrounds, while the upper curvescontain in addition the superposed resonance. [Ferro-Luzzi et al.,Phys. Rev. Lett.8, 28 (1962)]

Finally, if we look in an issue of the *Physical Review*, saythat of January1, 1962, will we find a resonance curve? Every issuehas a resonance curve, and Fig.23–11 is the resonancecurve for this one. This resonance curve turns out to be veryinteresting. It is the resonance found in a certain reaction amongstrange particles, a reaction in which a K$^-$ and a protoninteract. The resonance is detected by seeing how many of some kindsof particles come out, and depending on what and how many come out,one gets different curves, but of the same shape and with the peak atthe same energy. We thus determine that there is a resonance at acertain energy for the K$^-$ meson. That presumably means that thereis some kind of a state, or condition, corresponding to thisresonance, which can be attained by putting together a K$^-$ and aproton. This is a new particle, or resonance. Today we do not knowwhether to call a bump like this a “particle” or simply aresonance. When there is a very *sharp* resonance, it correspondsto a very *definite energy*, just as though there were a particleof that energy present in nature. When the resonance gets wider, thenwe do not know whether to say there is a particle which does not lastvery long, or simply a resonance in the reaction probability. In thesecond chapter, this point is made about the particles, but when thesecond chapter was written this resonance was not known, so our chartshould now have still another particle in it!