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**J. Acoust Soc. Am. 89 (2), February 1991 **

Copyright © ASA 1991

( Received 9 March 1990; accepted for publication 17 October 1990 )

Using the approach first pioneered by Raman, the Helmholtz motion of a bowed string is discussed as a special case of "two-velocity motions," in which a given point (at which the bow is located ) alternates, in the course of a cycle, between two constant velocities. The fact that the bow typically presents a negative resistance to the string during the "slipping" part of the cycle is adduced as a reason for the "duty cycle," that is, the fraction of the period that corresponds to slipping, to try to become as short as possible. It is shown that, for a string without dissipation or stiffness, this duty cycle can be arbitrarily low for general bow positions; data obtained with the "digital bow" illustrate this behavior. It is shown theoretically, and confirmed with computer simulations, that instabilities arising from the negative slipping resistance cannot be eliminated by assigning a finite positive value to the sticking resistance. The apparent stability of Helmholtz motion observed in real playing situations remains a puzzle.

The fact that at each location on the string, including an imagined "bowing point," the Helmholtz motion corresponds to an alternation between two constant string velocities makes it easy to conceptualize it as an embodiment of the kind of"stick-slip" phenomenon that we associate with the dry friction between string and rosined bow. In particular, one usually thinks of the interval of smaller velocity (which lasts for a longer time) as "the string sticking to the bow;" the larger, oppositely directed velocity then corresponds to "the string slipping back," only to be "caught" by the bow once again.

Considerable advances have added to our understanding of bowed-string motion in recent years;^{[2]-[4]} among them,
computer simulations^{[3]-[4]} have shown how, under various
realistic conditions, Helmholtz motion, or some close
approximation to it, indeed arises when a string that is originally
at rest has a bow applied to it. In fact, such simulations
have provided considerable fine detail on likely realistic
regimes of motion. Yet, for some purposes, such approaches give
too much information in that the elementry "stick-slip"
concept is not always easily applied to their understanding.
Our purpose in this paper is to develop a somewhat different
conceptual approach, which, in our opinion, can do much
to clarify the discussion of why Helmholtz motion is to
be expected and under what circumstances it is stable. The
point
of view we use has some overlap with older work of Raman,^{[5]}
but is developed considerably further.

Our frictional characteristic is drawn so that its "sticking" portion is not vertical but has a finite slope; that is, we do not assume that "sticking" automatically implies that the relative velocity between string and bow hair is identically zero. In fact, an infinite slope is not required for further development of the theory, but can always be inserted later as a special case. Specifically, "the digital bow" (which supplied some experimental data to be quoted below) has an adjusta- ble frictional characteristic whose slope is ordinarily not infinite.

Physically, on the other hand, the concept of"sticking" does imply that the force-velocity characteristic has an infinite slope

FIG. I. Generic frictional characteristic, showing the (negative of) the force exerted by the bow on the string as a function of the string's velocity Q

_{a}a and Q_{g}are the point corresponding lo sticking ("adhérence")and slipping ("glissage"). The line Q_{a}Q_{g}is horiontal and is divided by the force axis in the ratio /(1-) for Helmholtz motion or, in the case of a general two-velocity motion, in the ratio /(1-); R_{a}and R_{g}are the two "resistances" that is, the slope at the operating points.

Thus, for eample. it was pointed out by McIntyre^{[6]} that one can use the bow to pull the string to one side and hold it there for a practically indefinite length of time. Yet, there are a number of factors that. in practice, may
produce an "effective slope" which is not infinite.
One is the torsional motion of the string;^{[7]-[8]} another is the dynamical vibration of the bow system itself, which may make the
instantaneous velocity of the bow hair deviate from the
nominally constant velocity of an idealized player's wrist.
In the latter connection, Cremer^{[8]} argues that effects due to
longitudinal (stretching) vibration of the bow hair themselves
are probably negligible; nonetheless, other motions of the
bow system, such as bending of the stick, cannot be as easily
ruled out.

One should note that the assumption of a dispersionless string is equivalent to the absence of any stiffness, that is, to any contribution to the potential energy that depends on the curvature of the string. Without such an assumption, any configuration that contains a kink would imply an infinite potential energy and would thus be ruled out immediately.

Under these assumptions. the Helmholtz motion (as defined in the first paragraph of this paper) is a possible solution of the free string equation, that is, a possible motion of the string without a bow^{[1]-[5]-[2]}. Having established this
fact. we must ask whether, and how, it is also a solution of
the equations that govern the system when the bow is applied.

Let *L* be the total length of the string, and *L* and (1-*L*) the distances from the bowing point to either string end. Since the Helmholtz motion requires the
string at the bowing point to move with one constant velocity for a
fraction of a period, then with a different (but again
constant) velocity for the remaining 1- of a period; and
since we have assumed a frictional characteristic that unequivocally specifies the force as a function of the velocity; it follows that the bow will exert one constant force on the string for a fraction and another constant force for the remaining fraction 1- of the period. In fact, these two constant forces must be equal to each other; otherwise, the bow would do a net amount of work on the string in the course of a period, which is impossible if the motion is steady and there is no dissipation. On this basis, we can say that the two points
marked Q_{a} and Q_{g} in Fig. 1, which characterize the two
states normally called "sticking" and "slipping,"
must lie on the same horizontal line. (Since "sticking"
and "slipping" have the same initial letter, we use the subscripts
a and g derived from the corresponding French words "adhérence" and "glissement." )

At the same time, we know that the two velocities of
the string at the bowing point must have magnitudes inversely
proportional to the time spent in the respective states,
so as to make the net displacement in the course of a complete
period vanish. For Helmholtz motion, this means that
the two velocities must be in the ratio -/(1-). Given the frictional characteristic and the speed and location
of the bow, the two requirements that Q_{a} and Q_{g} lie on the same horizontal line, and have abscissae whose ratio is as
given above, in most cases determine the location of these
two points.

We ask next whether the Helmholtz motion specified
by this construction is dynamically possible; for, although
it is known to be a solution of the *free* string equation,
the string is now no longer free because the bow is applying a
force to it.
Since, however, this force is constant in time. its
effects can be taken care of by superposing upon the Helmholtz motion
a stationary configuration consisting of two straight
line segments with a *fixed* kink at the bow location, and having
a slope discontinuity corresponding to the static displacement
of the same string under the same constant localized
transverse force. Because linearity of the string has been
assumed, the superposition of this stationary kink upon the moving
Helmholtz kink now constitutes a solution for a string
which is free except for the application of the (time-independent
) bow force^{[9]}.

Indeed, as was already shown by Raman,^{[5]} there do exist
other possible motions of a free string for which the velocity
at a given location alternates between two fixed values
(although the Helmholtz motion is the only one for which
such a statement can be made about every point of the string). To
clarify this, we ask the following question: Given an
arbitrary function of time, under what circumstances can
it correspond to the velocity of a given point of the string
when it is moving freely

We know, of course, that any vibration of the free string
can be expanded in normal modes; and since we have assumed that there is no dispersion and that the endpoints are
fixed, the period of the *n*th normal mode will be /n,
where is the string's fundamental period. Clearly, any superposition of such modes will still have an overall periodicity
,
from which it follows that a function which purports
to represent the time dependence of velocity of a given point
must itself have that periodicity. This is an obvious necessary
condition.

It this condition is satisfied, we can expand the given
function as a Fourier series; we then interpret each
coefficient as the strength of the corresponding normal mode
at the given string position. In this way an arbitrary
periodic function can be seen as all expansion in the normal
modes, and hence a possible free motion of the string, except
for one restriction: If the given point is an exact node
of some normal mode, then the corresponding Fourier component cannot be
represented in the velocity function. More specifically,
if the point in question lies at a fraction m/P of the string
length (where m and P are integers), the harmonics P, 2P, etc.,
must be absent from the velocity as observed at that point.
(One may ad that under any circumstances, the "zeroth
harmonie" must be absent from the velocity to
prevent any secular drift of the string at least so long as the
terminations are fixed.) With this understanding, we now ask what
the possible free string motions are for which the velocity
at location *L* alternates between two fixed values.

Let , as above, be the "duty cycle" of a proposed two-velocity motion, that is, the fraction of a period spend in one of the two states. [Again, to preclude a secular drift, the two-velocities must have opposite signs, and their magnitudes must be in the ratio ( I-)/).] We then see from the foregoing discussion that, if is not rational, any duty cycle can be constructed, but, for rational, this is not the case. We show in Appendix A that, if = m/P, where m and P are integers and the fraction is in its lowest terms, the possible values of are limited to integer multiples of 1/P. In particular, the shortest duty cycle of a free string vibration which is a two-velocity motion at = m/P is = I /P.

In physical cases, of course, the qualitative distinction between rational and irrational values of cannot be meaningful. Instead, we would expect the behavior of the string to differ between a regime of "small P" and one of "largeP," the comparison being with some "critical P" determined, perhaps, by the rounding of the Helmholtz kink due to string stiffness or bow width.

If harmonics of order P, 2P, etc., are removed, the
shape
of the force exerted by the string on the bridge ( which
can be thought of as the "signal" sent by the string
to the violin body) will be qualitatively modified: The displacement
very near the bridge will change from a "sawtooth"
to a "staircase," in which the uniform ramp of the sawtooth
is replaced by P horizontal "steps." ^{[10]-11} It is true,
of course, that if P is fairly high (as it would have to be for a bow at a normal playing position rather near the bridge), the quantitative
modification may not be very large. Nonetheless, there
is something unphysical about a phenomenon that seems to
exist when is rational but disappears abruptly when
it is irrational. As we shall see the existence of other
two-velocity motions will allow us to reformulate the phenomenon
so as to make its dependence upon more continuous.

With finite slopes, the question of stability requires a more detailed calculation, which does not become trivial no matter what the values of the slopes. Consider, for example, the (not particularly realistic) case in which both slopes are

from which one might be led to conclude that, if the
string spends a fraction of its period with a resistance
R_{g} attached, and the remaining fraction 1- with a resistance R^{g}, the net rate of exponential decay of the amplitude
of a perturbation corresponding to the nth normal mode will
be

In this expression, R_{a} and R_{g} are, respectively,
the sticking slope (which is positive) and the slipping slope (which
is negative); Z_{0} is the characteristic impedance
of the string;

is the (fractional) location of the bowing point;
and is the dutycycle.
(For a Helmholtz motion, and are, of course, equal.) This would lead us to formulate as a stability criterion for all modes.

In fact, as we shall see in more detail in Sec. VII,
this criterion is by no means sufficient. The fallacy is
that it is really necessary to define the "modes during sticking"
separately from the "modes during slipping" and to reexpand
the perturbation first in one set then in the other,
twice during each cycle. The coherent behavior of these successive
reexpansions can then cause instabilities whose overall
periodicity is nowhere near any of the normal modes, as
in the well-established subharmonic generation discussed by
Mc-lntyre el al.^{13} It is also possible as we shall see
in Sec. VII, to construct perturbations that do have the correct fundamen-
tal frequency, but that are so distributed over the string
as to be essentially absent from the bow location during the
sticking part of the cycle. We refer to them as "between-the-rain-
drops" motions, drawing on the metaphor of a person
remaining dry in the rain by "running between the
raindrops."
Under those circumstances, the growth due to the (nega-
tive) slipping resistance is not compensated by any
decay due to the (positive) sticking resistance, making the
perturbation considerably more unstable than the criterion
of Eq. (3) would suggest. Appendix B demonstrates how motions
of this type may be constructed.

Nonetheless, the qualitative idea expressed by Eq. (2)
that a relatively short duty cycle (that is, a short
slipping time) enhances the stability of a two-velocity motion
generally remains valid. Such a condition corresponds to
the statement made by Cremer, ^{14} that "the bow tries always
to maximize the sticking time," even though the explanation
given by him in terms of a simple friction-driven oscillator
seems to us to be insufficient.

The following specific consequence of such a condition is worth noting immediately. There is, in the common description of Helmholtz motion (or of any other two-velocity motion), an ambiguity concerning the duty cycle in that it is assumed that, of the two fractions and 1- (which, for Helmholtz motion, are the same as, and 1-), the shorter one always corresponds to slipping and the longer one to sticking. Kinematically, however, an "inverted" Helmholtz motion is equally possible. If, for example, the bow is placed at a point that lies at 10% of the string length, one can construct a Helmholtz motion corresponding to it which slips for 90% of the cycle and sticks for 10%, with the bow speed corresponding to the larger velocity. Our discussion makes it clear, however, that such an inverted motion (which, to our knowledge. has never been observed) would be very much more likely to run into stability problems.

Specifically. the clear theoretical proof that ideal
Helmholtz motion with perfect sticking is always unstable
leads us to ask which specific deviation from the model produces
the stability that is experimentally observed: Is it the
fact that the sticking is. in effect, not perfect (due perhaps
to torsion^{7-13} or other effects ), or the fact that the
motion deviates kinematically from Helmholtz motion? It should be added
that the presence of weak dissipation (which is surely
there in a real case) cannot be simply taken care of by slightly
raising the point Q_{a} relative to Q_{g} in Fig. 1. True,
that would provide a net energy influx from the bow, but the force
would no longer be constant in time, so that the Helmholtz
motion plus a fixed kink would no longer be a solution
of the string equations.

It is interesting, in this connection, to present some relevant data from our experiments with the "digital bow." In this setup, a real string is driven, not by a physical bow, but by an electronic system that substitutes for it by electrodyna- mically exerting a force that is determined by the string's instantaneous velocity at the imagined "bowing point." Specifically, the string's velocity is measured photoelectrically and the resulting signal digitized and sent to a computer. The computer, in turn, does a table lookup according to a previously programmed "frictional characteristic," then converts the corresponding "force" to an analog current that is sent through the string. By virtue of a permanent magnet mounted at the "bowing point," this results in a corresponding physical force exerted on the string. The process is repeated at a sampling rate up to 125 kHz.I In the present experiments, the string was 1.40 m long and had a fundamental frequency near 70 Hz; the magnetic field extended over a length of approximately 2.5 cm, which is to be thought of as the "width" of the equivalent bow; and the frictional characteristic was defined by the function.

The parameters in this function, taking account of all the amplifier gains as well as the sensitivities of the converters, had the (approximate) values

F_{0}=0.015N, _{0}=0.05m/s. (5)

The string itself (made of brass) had a linear density
of l.68 X l0^{-3} kg/m corresponding (for this length and
fundamental frequency) to a characteristic impedance of 0.329
kg/s.

Figure 2 presents data obtained in such a situation. The graph shows the velocity of the "bowed" point of the string as a function of time for three placements of that point, corresponding to =0.333 (dashed), =0.250(dotted),and , = 0.291 (solid). All approximate an alternation between "sticking" and "slipping," with the "sticking" velocity the same for all three cases and equal to the programmed bow speed. But whereas the observed duty cycle corresponds roughly to in the first two cases, it is clear thatthis is not so for the third (solid) curve: Even though , is intermediate between the other two, is smaller than either one.

Figure 3(a)-(c) corresponds, respectively, to the same
three cases as Fig. 2; this time, however, we show the
displacement of the string as a function of time, observed
at two locations: the bowing point itself (top graph), and
a point very near the bridge (bottom graph) . As we indicated
above, the latter can be considered proportional to the slope
of the string at the fixed endpoint and thus to the lateral
force exerted by the string on the bridge. In Fig. 3(a) and
(b) (corresponding to = 1/3 and 1/4), we see the characteristic "staircase" function near the bridge, corresponding
to the fact that certain harmonics must be missing, as discussed
above. In Fig. 3(c). on the other hand, there is (when
observed near the bridge) a marked enhancement of certain
harmonics, the effect of which is the narrowing of the
duty cycle at the bowing point, in agreement with what our
considerations of stability would suggest. This motion
is quite similar to the one described by Lawergren;'7 according
to McIntyre et al.,^{18} it is on occasion musically
useful as well.

FIG. 2. Velocity of the bowing point as a function of time obtained on the digital bow under three condition: = 0,333 (dashed), = 0.333 (doted), and = 0.293 (solid). The frequency is 70.16 Hzs and two complete periods are shown. The absolute phase is arbitrarily chosen, to superpose the "sliping" pulse of the three cue.

When the "bow" (that is, the magnet-photodetector assembly) is moved slowly from, say, = l/4 to a "more irrational" position like = 0.291, one observes the wave shape changing continuously from that of Fig. 3 (b) to that of 3 (c). In that way, the large quasisinusoidal admixture to the Helmholtz motion seen in the lower graph of 3 (c) joins continuously to the staircase characteristic of 3(b) [or 3(a)], making the latter intuitively more palatable.FIG. 3. Displacement of the slring as a function of lime, obtained on the digital bow under the conditions of Fig. (a) = 0.333. (b) 1 = 0.250, (c) = 293. In each case, the upper curve is taken at the bowing point, the lower one at a point near the bridge. The absolute phases and the vertical scales are arbitrary for each of the six curves.

Figure 4(a)-(c) shows computed results corresponding to the graphs in Fig. 3. For (a) we took m = l, P= 3, M = 1; for (b), m = l, P = 4, M = 1. For (c), was taken to be midway between the other two at 7/24, so that m = 7, P = 24; the choice M = 3 then produced the closest correspondence between experiment and simulation. In other words, Fig.4(c) is a two-velocity motion at the bowing point whose duty cycle is somewhat less than half of its Helmholtz value, that is, 3/24 compared to 7/24. Naturally, the absolute phase in these simulations is arbitrary but was chosen to resemble as much as possible the data of Fig. 3.

It should be noted that, since m = 1 for Fig. 4(a) and (b), no two-velociy motion of lower duty cycle is possible. However, in Fig. (c), is not as small as it can be. For comparison, Fig. 5(a) shows a pure Helmholtz motion for = 7/24, while 5 (b) illustrates the lowest possible duty cycle, corresponding to = 1/24 (that is, M = 1 ) for the same . Clearly, the observed motion, as simulated in Fig. 4(c), is narrower than the Helmholtz motion but not as narrow as is kinematically possible.FIG. 4. Simulted displacements at the bowing point (top)and at a point near the bridge (bottom) contructed as superpositions of Helmholtz motions, with conditions (including abslolute phase) chosen to correpond to the data of Fig. 3. (a) ==1/3; (b) ==1/4; (c) =7/24, =3/24.

That there must be a dynamical lower limit onl for a given is, of course, obvious, since, as P becomes larger, the purely kinematical limit gets smaller and smaller, becoming zero for irrational values of . In attempting to explain this dynamical lower limit, three factors come to mind: the stiffness of the string. the width of the bow, and the presence of energy losses. All three constitute violations of the assumptions that we have made in defining our model.

Although we are not able, at this moment, to treat these complications in detail. We found it heuristically interesting to calculate the energies corresponding to the two-velocity motions of given , the idea being that, when dissipation is introduced. a larger stored energy may also correspond to a larger rate of energy dissipation. The larger power requirement from the bow may then make the motion more difficult to maintain.

The energy tored in a simple Helmholtz motion is easily calculated at a sum of kinetic and potential energy densities integrated over the string. If the maximum lateral
displacement of the kink (which occurs when it passes the
center of the string) is A_{0}, the instantaneous string
displacement at time t is given by where p is the linear density and T = c^{2} is the static tension.

FIG. 5. Simulilated displacement at the bowing point (top) and at the point near the bridge (bottom), constructed as in Fig. 4:(a) for the pure Helholtz motion at = 7/24, (b) for the shortest possible duty cycle at the same

where c is the speed of transverse waves. The total energy then becomes

When more than one kink is present, the energies cannot, of course. be simply added, since the individual kinks are not orthogonal to each other. Nonetheless, the calculation for any given superposition is straightforward, especially when done by computer.

On the other hand, the two-velocity motions for various
but given , do not have the same lateral displacements; rather, they must (if the comparison is to be meaningful) be
normalized to the same value of bow speed _{b}. Since
the construction that determines this depends on the frictional
characteristic, as shown in Fig. 1, it cannot be given
for a general case. Nonetheless, an excellent approximation
is obtained by assuming, for this purpose, that the sticking
slope is infinite. In that case, the bow speed is related
to , the difference in velocities of the two-velocity motion, by

_{b}=

(8)

allowing the amplitude of the component Helmholtz motion to be determined.

Table I shows the computed results for various two-velocity motions with = 7/24 = 0.291, the case discussed above, normalized to the simple Helmholtz motion. It is seen that the energy does, indeed, rise steeply as is made smaller, making this a possible factor in putting a lower limit on duty cycles actually observed.

TABLE 1. Relative energies of two-velocity motions, normalized to a fixed bow speed, for = 7/24 and varying . The energy for = is arbitrarily taken as unity. The middle column gives the number of Helmholtz kinks required to construct each motion.

Number or kinks | Relative energy | |
---|---|---|

1/24 | 7 | 49.00 |

2/24 | 14 | 18.577 |

3/24 | 21 | 6.050 |

4/24 | 4 | 4.261 |

5/24 | 11 | 3.923 |

6/24 | 18 | 2.216 |

7/24 | 1 | 1.000 |

8/24 | 8 | 1.449 |

9/24 | 15 | 1.278 |

10/24 | 22 | 0.743 |

11/24 | 5 | 0.811 |

12/24 | 12 | 0.852 |

13/24 | 19 | 0.580 |

14/24 | 2 | 0.379 |

15/24 | 9 | 0.460 |

16/24 | 16 | 0.362 |

17/24 | 23 | 0.170 |

18/24 | 6 | 0.246 |

19/24 | 13 | 0.272 |

20/24 | 20 | 0.170 |

21/24 | 3 | 0.123 |

22/24 | 10 | 0.154 |

23/24 | 17 | 0.093 |

Figure 6 shows some computed results. Each of the four graphs give the displacement of the point nearest one end of the string, plotted at the initial time and then once per period (that is, every 46 time increments) for a total time of 100 periods; they differ, however, in the values assigned to the sticking and slipping resistances.

FIG. 6 Perturbations computed in th linear regime for a "string" of 23 points, with = 7/23, for various value of the sticking and slipping resistances. The displacement of the point nearest one end is plotted once per period, for a total interval of 100 periods:In Fig. 6(a) and (b), the slipping resistance is taken to be R

(a) R_{a}= 50, R_{g}= O;

(b) R_{a}= 0.5, R_{g}= O;

(c) R_{a}= 50, R_{g}= -0.05;

(d) R_{a}= 0.5, R_{g}= -0.05.

In Fig. 6 (b) the slipping resistance is kept at zero
but the sticking resistance is lowered to R_{a} = 0.5. The subharmonic
is now so strongly damped as to be essentially unrecognizable, yet the nonzero asymptotic value is not only still there but entirely unchanged in magnitude. as is to be epected from a perturbation that never sees the sticking part
of the bow cycle at all.

Finally, in Fig. 6(c) and (d), the computation is
repeated after setting the slipping resistance to the mildly
negative value R_{g} = -0.05; in each figure, R_{a} is the same as in the graph immediately to its left. It is clear that the
previously constant asymptotic perturbation now becomes a growing
one, since it is absent from the bow position when the
bow sticks but present when it slips. The massive damping
caused by R_{a} in Fig. 6(d) has no appreciable effect on this
growth rate, even though the amplification is quite small compared
to this damping, and the duration of the slipping cycle
is shorter.

It may be worth, in this connection, to mention another
phenomenon observed with the digital bow by Chayé^{20};
for the case in which is a ratio of small integers but the numerator is not unity. She found that, if = m/P, the duty
cycle of the motion at the bowing point tends always to be 1/P. For example, for = 1/7, 2/7, 3/7, 4/7, 5/7, or 6/7, is always 1/7. In other words, in this case it is, indeed, the shortest that is chosen, consistent with the idea that, due to stability
considerations, the motion which is preferred is the
one that
minimizes the time spent in the negative-resistance
region of
the characteristic. It is not clear to what degree this
narrow-
ing can be avoided on real violins by "correct"
bowing; comparison with the digital bow is complicated by its relatively
large width, as well as the fact that the magnitude
of its force
is limited by effects of string heating.

Our discussion of Sec. IV, along with the computer simulations of Sec. VII, indicates that the apparent experimental stability of Helmholtz motion remains a puzzle, in that
(unlike what has been previously believed) this stability
cannot be explained by changing the "effective" sticking
resistance from an infinite to a finite value. As far as
we can see, the truth must lie in one or more of the following three
possibilities: (a) The stability comes from some other source
of dissipation (such as losses at the supports); (b) the
stability comes from other deviations from the "perfect"
model, such
as bow width. string stiffness, or perhaps bow dynamics:
the latter would make it impossible to represent the frictional
characteristic a an instantaneous. time-independent
function: and (c) the Helmholtz motion is, in fact, *not* stable
but its instabilities themselves stabilize at fairly small
amplitudes so that casual experiments appear to suport the notion
that
Helmholtz motion is being executed. It should be noted
in
this connection that the simulations of Sec. Vll were
computed in a linear regime and thus throw no light on
what happens to the perturbations after they have begun to
grow.

We begin by noting that = m/P is a node of all harmonics whose order is a multiple of P, so that the Fourier spectrum of the velocity observed at that point must lack those harmonics. To impose this condition, we first calculate the amplitude of the nth harmonic of a square pulse of duty cycle ; apart from an unimportant normalizing factor, it is given by

ABy setting A_{n}() = [1-exp(2in)]/n.(9)

PThus the Pth harmonic will vanish if and only if has the form M/P, where M is an integer. We note also that the vanishing of the Pth harmonic automatically guarantees the vanishing of all its multiples. as required._{}= integer.(10)

If does have the above form, we know from the discussion of Sec. II that a corresponding two-velocity motion
must exist. It may, nonetheless, be useful to facilitate
its visualization by giving an explicit construction for it
as a superposition of simple Helmholtz motions as advocated by
Kubota. ^{21-22}

The construction is as follows. Consider the multiples of m modulo P. If one such is equal to zero, it means that a certain multiple of m is equal to a certain multiple of P; and, since m and P have (by assumption) no common factors, it follows that the number by which m is multiplied to produce zero (modulo P) must itself be a multiple of P. In other words, if m is multiplied successively by 0,1,....,P-1, all of which are less than P, no two results can he equal (modulo P). On the other hand, there are exactly P numbers in the range of all integers modulo P, so that each of them must appear once and only once in the set of posible results of multiplying m by some integer and reducing the result modulo P. Let M be all integer in the set 1,2,....,P-1 . and N the integer such that Nm mod P = M.

Consider now N simple Helmholtz motions, all of the same amplitude, coexisting on one string, and spaced apart by a fraction m/P of a period. In terms of a spatial distribution, it is convenient to viualize the string as an endless conveyor belt looped around a pair of pulleys located at the endpoints of the string, and moving with the speed of transverse waves on the string; half of the belt can then be thought of as carrying waves traveling to the right, the other half to the left. The N Helmholtz motions under consideration can then be represented as marks on the conveyor belt indicating the location of the kinks. Note that the total length of the belt is 2L, so that the spatial interval between successive kinks is 2(m/P)L.

Consider now the location (m/P)L along the string as measured from one end which, for convenience, we shall call the "bridge." Because the kinks are spaced apart by twice that distance, it follows that, whenever a kink passes this observation point moving toward the bridge, another will be passing it moving in the opposite direction. Since the velocity transition of a Helmholtz motion occurs whenever the kink passes the observation point, we see that in our present case each such transition will be canceled by an equal and opposite one. The two exceptions are when the first kink of the sequence passes in the direction toward the bridge, and when the last one passes in the other direction. Since there are N kinks, spaced apart by m/P of a period, the time interval between the first and the last, figured as a fraction of a period, is Nm/P. In order to reduce it to a propr fraction, we reduce the numerator modulo P; and since Nm mod P = M by construction, we see that the fraction of a period that elapes between the two transitions that do occur exactly M/P, as was required.

For the "multikink simulations" of Sec. VI, we used the above construction but modified it further by complementing each Helmholtz kink with Pnegative "subsidiary" kinks in order to produce the "staircase" behavior explained in Sec. III. This modification does not, of course, affect the motion observed at = m/P and is hence irrelevant for the theorem stated at the beginning of this Appendix. It is, however, important for describing the motion at other points and for the calculation of energies.

Let our conveyor belt, whose total (unfolded) length is 2L, have associated with it a displacement pattern that is periodic with period 2L; since is, by assumption, not rational, it is clear that somewhere there will be a junction point where this periodicity is broken. If we now imagine the belt mounted on the pulleys (of which one represents the "bridge") and put in motion with the correct wave velocity, the equations of the free string will be satisfied, in that the resulting displacement will contain a wave going to the right and another going to the left, each of which returns from its respective endpoint reflected with a change of sign (since the "top" of the conveyor belt going to the right turns into the "bottom" going to the left ) .

Consider now the "bow position" L. Since the two points of the belt passing it at a given moment are separated by a distance 2L (measured around the "bridge" pulley), and since the pattern is, by construction, periodic with that quantity as period, the displacements corresponding to the left-going and right-going waves will, at each moment, be equal and opposite, resulting in a total displacement at the bow position which vanishes identically. This condition will not hold, however, when the above-mentioned junction point is passing from the "bow position" to the "bridge" and back again; and since this distance is a fraction, of the total belt length, the time for which the displacement is not zero will, be a fraction of the period. Thus the existence of between-the-raindrops" motions is proved.

It is worth noting that motions of even shorter duration at the bow position can be constructed by choosing displacement patterns whose spatial period along the belt is an integer submultiple of 2L. In fact, it is easy to convince oneself (in analogy with the construction of short duty cycles as described in Appendix A) that, for the case when is irrational, motions exist which vanish at the bow position for all but an arbitrarily small fraction of the period.

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