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the sand assumed similar circular forms, concentric and alter-
nating with the water elevations.

90. On strewing a little lycopodium over the water for the
purpose of gaining information relative to what occurred at the
surface during the crispation, it moved about over the fluid in
every possible direction, whilst the crispations existed of the
utmost steadiness beneath. The same thing occurred with
pieces of cork on very large crispations (98). But when much
lycopodium was put on, so that the particles retained each
other in a steady position, then it formed lines* parallel to the
arrangement of the heaps, the powder being displaced from
the parts over the heaps, and taking up an arrangement per-
pendicularly over the sand beneath. As the lycopodium forms
float on the water they are easily disturbed, and in no respect
approach as to beauty and utility to the forms produced by
the sand ; but lycopodium may be used with smaller crispations

than sand.

* Wheatstone.

344 On the Forms and States of Fluids [1831.

91. The crispations are much influenced by various circum-
stances. They tend to commence at the place of greatest
vibration ; but if the quantity of fluid is too little there, and
more abundant elsewhere, they will often commence at the
latter place first. Their final arrangement is also much affected
by the form of the plate, or of the pool of water on which they
occur. When the plates or pools are rectangular, and all parts
vibrate with equal velocity, the lines of heaps are at angles of
45 to the edges. But when semicircular and other plates
were used, the arrangement, though quadrangular, was un-
steady, often breaking up and starting by pieces into different
and changing positions.

92. When mercury was used (77), the film formed on it after
a few moments had great power, according to the manner in
which it was puckered, of modifying the general arrangement
of new crispations.

93. When a circular plate, supported by cork feet attached
where a single nodal line would occur, was covered with water
and vibrated by a rod resting upon the middle, the crispations
extended from the middle towards the nodal line ; these were
sometimes arranged rectangularly, but had no steadiness of
position, and changed continually. At other times the heaps
appeared as if hexagonal, and were arranged hexagonally, but
these also shifted continually. These and many other experi-
ments (83) showed that the direction and nature of the vibra-
tion of the plate (i.e. of the lines of equal or varying vibrating
force) had a powerful influence over the regularity and final
arrangement of the crispations.

94. The beautiful appearance exhibited when the crispations
are produced in sunshine, or examined by a strong concen-
trated artificial light, has been already referred to (78, 79).
When the reflected image from any one heap is examined (for
which purpose ink (75) or mercury (77) is very convenient), it
will be found not to be stationary, as would happen if the heap
was permanent and at rest ; nor yet to form a vertical line as
would occur if the heap were permanent but travelled to and
fro with the vibrating plate ; but it moves so as to re-enter
upon its course, forming an endless figure, like those produced
by Dr. Young's piano-forte wires, or Wheatstone's kaleido-
phone, varying with the position of the light and the observer,

1831.] on Vibrating Elastic Surfaces. 345

but constant for any particular position and velocity of vibra-
tion. Upon placing the light and the eye in positions nearly
perpendicular to the general surface of the fluid, so as to avoid
the direct influence of the motion of vibration, still the lumi-
nous, linear, endless figure was produced, extending more or
less in different directions, according to the relation of the
light and eye to the crispated surface, and occasionally corre-
sponding in its extent one way to the width of the heap, i. e. to
the distance between the summit of one heap and its neigh-
bours, but never exceeding it. The figure produced by one heap
was accurately repeated by all the heaps when the vibrating
force of the plate was equal (70) and the arrangement regular.

95. The view which I had been led to anticipate of the
nature of the heaps, from the effects described in the former
paper, were, that each heap was a permanent elevation, like
the cones of lycopodium powder (53. 58), the fluid rising at the
centre, but descending down the inclined sides, the whole
system being influenced, regulated, and connected by the cohe-
sive force of the fluid. But these characters of the reflected
image, with others of the effects already described, led to the
conclusion, that notwithstanding the apparent permanency of
the crispated surface, especially when produced on a small
scale, as by the usual method, the heaps were not constant,
but were raised and destroyed with each vibration of the plate ;
and also that the heaps did not all exist at once, but (referring
to locality) formed two sets of equal number and
arrangement, fig. 23, never existing together,

but alternating with, and being resolved into

each other, and by their rapidity of recurrence

giving the appearance of simultaneous and even

permanent existence. Provided this view were

confirmed, it seemed as if it would be easy to

explain the production of the heaps, their regular arrangement,

&c., and to deduce their recurrence, dimensions, and many

other points relative to their condition.

96. On producing a water crispation, having four or five
heaps in a linear inch, placing a candle beneath, and a screen
of French tracing-paper above it, the phenomena were very
beautiful, and such as supported the view taken. By placing
the screen at different distances, it could be adapted to the

346 On the Forms and States of Fluids [1831.

focal length due to the curvature at different parts of the sur-
face of fluid, so that by observing the luminous figure produced
and its transitions as the screen was moved nearer or further,
the general form of the surface could be deduced. Each heap
with a certain distance of screen gave a star
of light 0, fig. 24, which twinkled, i. e. ap-
peared and disappeared alternately, as the
heap rose and fell. At the corners x equi-
distant from these, fainter starred lights ap-
peared ; and by putting the screen nearer to - -
or further from the surface, lines of light, in
two or even four directions, appeared inter-
secting the luminous centres and apparently permanent, whilst
circumstances remained unchanged. These effects could be
magnified to almost any scale (72).

97. When heaps of similar magnitude were produced, with
diluted ink on glass (75), and white paper or an illuminated
screen looked at through them, a chequered appearance was
observed. In one position, lines of a certain intensity separated
the heaps from each other, but the square places representing
the heaps looked generally lighter. In another position, when
but little reflected light came from the surface of the heaps,
their places could be perceived as dark, from the greater depth
of ink there. By care, another position could be found in which
the whole surface looked like an alternate arrangement of light
and dark chequers, fig. 25, not steady, but with F - 2 5

a quivering motion, which further attention could
trace as due to a rapid alternation in which the
light spaces became dark and the dark light,
simultaneously. When, instead of glass, a bright
tin plate was used under the diluted ink, the
chequered spaces and their alternations could
be seen still more beautifully.

98. It was in consequence of these effects that very large
arrangements were made (72), giving heaps that w T ere two inches
and a half wide each*; and now it was evident, by ordinary

* This estimate is given in accordance with the mode of estimating the former
and smaller heaps, as if the heaps were formed simultaneously ; but it is evident
that if only half the number exist at once, each heap will have twice the width
or four times the area of those which can be formed if all exist together.

1831.] on Vibrating Elastic Surfaces. 347

inspection, that the heaps were not stationary, but rose and
fell ; and also that there were two sets regularly and alternately
arranged, the one set rising as the other descended.

99. Sand gave no indications of arrangement with these large
heaps (86) ; but when some coarse sawdust was soaked, so as
to sink in water, and then distributed in the fluid, its motions
were beautifully illustrative of the whole philosophy of the
phenomena. It was immediately washed away from under the
rising and falling heaps, and collected in the places equidistant
between these spots, as the sand did in the former experiments
(86), and by its vibratory motion to and fro, it showed distinctly
how the water oscillated from one heap towards another, as the
heaps sunk and rose.

100. When milk (75) was used instead of water for these
large arrangements in a dark room, and a candle was placed
beneath, the appearances also were very beautiful, resembling
in character those described (97).

101. Each heap (identified by its locality) recurs or is re-
formed in two complete vibrations of the sustaining surface*;
but as there are two sets of heaps, a set occurs for each vibra-
tion. The maximum and minimum of height for the heaps
appears to be alternately, almost immediately after the sup-
porting plate has begun to descend in one complete vibration.

102. Many of these results are beautifully confirmed by the
appearances produced, when regular crispations have been
sustained for a short time with mercury, on which a certain
degree of film has been allowed to form (77). On examining
the film afterwards in one light, lines could be seen on it, coin-
ciding with the intervals of the heaps in one direction ; in an-
other light, lines coinciding with the other direction came into
sight, whilst the first disappeared ; and in a third light, both
sets of lines could be seen cutting out the square places where
the heaps had existed : in these spaces the film was minutely
wrinkled and bagged, as if it had there been distended ; at
the lines it was only a little wrinkled, giving the appearance of
texture ; and at the crossing of the lines themselves, it was

* A vibration is here considered as the motion of the plate, from the time
that it leaves its extreme position until it returns to it, and not the time of its
return to the intermediate position.

348 On the Forms and States of Fluids [1831.

quite free from mark, and fully distended. All these are na-
tural consequences, if the film be considered as a flexible but
inelastic envelope formed over the whole surface whilst the
heaps were rising and falling.

103. The mode of action by which these heaps are formed
is now very evident, and is analogous in some points to that by
which the currents and the involving heaps already described
are produced. The plate in rising tends to lift the overlying
fluid, and in falling to recede from it ; and the force which it
is competent to communicate to the fluid can, in consequence
of the physical qualities of the latter, be transferred from par-
ticle to particle in any direction. The heaps are at their maxi-
mum elevation just after the plate begins to recede from them ;
before it has completed its motion downwards, the pressure of
the atmosphere and that part of the force of the plate which
through cohesion is communicated to them, has acted, and by the
time the plate has begun to return, it meets them endowed with
momentum in the opposite direction, in consequence of which
they do not rise as a heap, but expand laterally, all the forces
in action combining to raise a similar set of heaps, at exactly
intermediate distances, which attain their maximum height just
after the plate again begins to recede ; these therefore undergo
a similar process of demolition, being resolved into exact dupli-
cates of the first heaps. Thus the two sets oscillate with each
vibration of the plate, and the action is sustained so long as
the plate moves with a certain degree of force ; much of that
force being occupied in sustaining this oscillation of the fluid
against the resistance offered by the cohesion of the fluid, the
air, the friction on the plate, and other causes. Fig. 26.

104. A natural reason now appears for the
quadrangular and right-angled arrangement
which is assumed, when the crispation is most
perfect. The hexagon, the square, and the
equilateral triangle are the only regular figures
that can fill an area perfectly. The square and
triangle are the only figures that can allow of
one half alternating symmetrically with the
other, in conformity with what takes place be-
tween the two reciprocating sets of heaps,
fig. 26 ; and of these two the boundary lines

1831.] on Vibrating Elastic Surfaces. 349

between squares are of shorter extent than those between
equilateral triangles of equal area. Tt is evident therefore that
one of these two will be finally assumed, and that that will be
the square arrangement ; because then the fluid will offer the
least resistance in its undulations to the motions of the plate,
or will pass most readily to those positions into which the forces
it receives from the plate conspire to impel it.

105. All the phenomena observed and described may, as it
appears to me, be now comprehended. The fluid may be con-
sidered as a pendulum vibrating to and fro under a given im-
pulse ; the various circumstances of specific gravity, cohesion,
friction, intensity of vibrating force, &c. determining the extent
of oscillation, or, what is the same thing, the number of heaps
in a given interval. When the number of vibrations in a given
time is increased, these heaps are more numerous, because the
oscillation, to be more rapid, must occur in a shorter space.
The necessity of a certain depth of fluid (73) is evident, and
also the reason why, by varying the depth (82), the lateral ex-
tent of the heaps is changed. The arrangement of the sand and
lycopodium, by the crispations, and the occurrence of the latter
at centres of vibration, and only upon surfaces vibrating nor-
mally, are all evident consequences. The permanency of the
lateral extension of the heaps, when the velocity of the vibra-
ting plate varies, is a very marked effect ; and it is probable
that the investigation of these phenomena may hereafter im-
portantly facilitate inquiries into the undulations of fluids,
their physical qualities, and the transmission of forces through

106. As to the origin or determination of crispations, no
difficulty can arise ; the smallest possible difference in almost
any circumstance, at any one part, would, whilst the plate is vi-
brating, cause an elevation or depression in the fluid there; the
smallest atom of dust falling on the surface, or the smallest eleva-
tion in the plate, or the smallest particle in the fluid of different
specific gravity to the liquid itself, might produce this first effect;
this would, by each vibration of the plate, be increased in
amount, and also by each vibration extended the breadth of a
heap, in at least four directions : so that in less than a second
a large surface would be affected, even under the improbable
supposition that only one point should at first be disturbed.

350 On the Forms and States of Fluids [ 1 83 i .

107. I have thought it unnecessary to dwell upon the explana-
tion of the circular linear heaps (83. 93. 110) produced on long
or circular plates by feeble vibration. They are explicable
upon the same principles, account being at the same time taken
of the arrangement and proportion of vibrating force in the
various parts of the plates.

108. The heaps which constitute crispation (as the word has
been used in this paper) are in form, quality, and motion of
their parts, the same with what are called stationary undula-
tions ; and if the mercury in a small circular basin be tapped
at the middle, stationary undulations, resembling the ring-like
heaps (83. 110), will be obtained; or if a rectangular frame be
made to beat at equal intervals of time on mercury or water,
heaps like those of the crispations, arranged quadrangularly at
angles of 45 to the frame, will be produced. These effects
are in fact the same with those described, but are produced by
a cause differing altogether. The first are the result of two
progressing and opposed undulations, the second of four : but
the heaps of crispations are produced by the power impressed
on the fluid by the vibrating plate ; are due to vibrations of
that fluid occurring in twice the time of the vibrations of the
plate ; and have no dependence on progressive undulations,
originating laterally, as many of the phenomena described prove.
Thus, when the edges were beveled (72. 110), or covered with
cloth, or wet sawdust, so that waves reaching the side should
be destroyed, or when the limits of the water or plates were
round (91) or irregular, still the heaps were produced, and
their arrangement square. When the round plate (93) was
used, regular crispations were still produced, though, as the
water extended over the nodal line, and was there perfectly
undisturbed, no progressing and opposed undulations could
originate to produce them. Vellum stretched over a ring, and
rendered concave by the pressure of the exciting rod, pro-
duced the same effect.

109. When a plate of tin, rendered very slightly concave,
was attached to a lath (69), so as to have equality of vibratory
motion in all its parts, and a little dilute alkali (which would
wet the surface) put into it, the crispations formed in the mid-
dle, but ceased towards the sides, where, though well-wetted,
there was not depth enough of water, and whence also no

1831.] on Vibrating Elastic Surfaces. 351

waves could be reflected to produce stationary undulations in
the ordinary manner.

110. When a similar arrangement was made with mercury
on a concave tin plate, the effects were still more beautiful and
convincing. The centre portion was covered with one regular
group of quadrangular crispations ; at some distance from
the centre, and where the mercury was less in depth, these
passed into concentric, ring-like heaps, of which there were a
great many ; and outside of these there was a part wet with
mercury, but with too little fluid to give either lines or heaps.
Here there could be no reflected waves ; or, if that were thought
possible, those waves could not have formed both the circular
rings and the square crispation. When this plate was vibrated,
the mercury spread in all directions up the side ; a natural
consequence of the production of powerful oscillations at the
middle, which would extend their force laterally, but quite
against their being due to the opposition and crossing of waves
originating at the sides.

111. A limited depth of fluid is by no means necessary to
produce crispations on the surface (73). A circular glass basin
about five inches in diameter and four inches deep was attached
to a lath (69), filled with water and vibrated, the exciting rod
being applied at the side (71). The surface of the water was
immediately covered with the most regular crispations, i. e.
heaps arranged quadrangularly. On taking out part of the
water and filling it up with oil, the oil assumed the same super-
ficies. On putting an inch in depth of mercury under the water,
the mercury became crispated. The experiment was finally
made with water fourteen inches in depth. Particles at a very
moderate depth in the water seemed to have no motion except
the general motion of the fluid, and the whole of the lower
part of the water may be considered as performing the part of
a solid mass upon which the superficial undulating portion
reposed. In fact, it matters not to the fluid, what is beneath,
provided it has sufficient cohesion, is uniform in relation to the
surface fluid, and can transmit the vibrations to it in an undis-
turbed manner*.

* I have seen the water in a pail placed in a barrow, and that on the head
of an upright cask in a brewer's van passing over stones, exhibit these

352 On the Forms and States of Fluids [1831.

112. The beautiful action thus produced at the limits of two
immiscible fluids, differing in density or some other circum-
stances, by which the denser was enabled most readily to
accommodate itself to rapid, regular and alternating displace-
ments of its support when that support was horizontal, sug-
gested an inquiry into the probable arrangement of the fluid
when the displacements were lateral or even superficial.

113. On arranging the long plate (67. 81) verti- Fig. 27.
cally, so that the lower extremity dipped about one-
third of an inch into water, fig. 27, and causing it to
vibrate by applying the rod at x , or by tapping the

plate with the finger, undulations of a peculiar cha-
racter were observed : those passing from the plate
towards the sides of the basin were scarcely visible
though the plate vibrated strongly, but in place of
such appeared others, in the production of which
the mechanical force of the vibrating plate exerted upon the
fluid was principally employed. These were apparently per-
manent elevations, at regular intervals, strongest at the plate,
projecting directly out from it over the surface of the water,
like the teeth of a coarse comb gradually diminishing in height,
and extending half or three-quarters of an inch in length.
These varied in commencing at the glass, or having intervening
ridges, or in height, or in length, or in number, or in breaking
up into violently agitated pimples and drops, &c. according as
the plate dipped more or less into the water, or vibrated more
or less violently, or subdivided whilst vibrating into parts, or
changed in other circumstances. But when the plate (sixteen
or seventeen inches long) dipped about one-sixth of an inch,
then four of these linear heaps occupied as nearly as possible
the same space as four heaps formed with the same plate in the
former way (83) and accompanied with the same sound.

114. By fixing a wooden lath (69) perpendicularly down-
wards in a vice, plates of any size or form could be attached
to its lower end and immersed more or less in water ; and by
varying the immersion of the plate, or the length of the lath,
or the place against which the exciting rod (71) was applied,
the vibrations could be varied in rapidity to any extent.

115. On using a piece of board at the extremity of the lath,
eight inches long and three inches deep, with pieces of tin

1831.] on Vibrating Elastic Surfaces. 353

plate four inches by five, fixed on at the Fig. 28.

ends in a perpendicular position to pre-
vent lateral disturbance at those parts,

very regular and beautiful ridges were ! 1 1 1 1 1 1 ! f ! 1 1 ! 1 1 j
obtained of any desired width, fig. 28.

These ridges, as before, formed only on the wood 5 and were
parallel to the direction of its vibration. They occurred on
each side of the vibrating plane with equal regularity, force
and magnitude, but seemed to have no connexion, for some-
times they corresponded in position, and at other times not ;
the one set shifting a little, without the others being displaced.

1 16. It could now be observed that the ridges on either side
the vibrating plane consisted of two alternating sets ; the one
set rising as the other fell. For each fro and to motion of the
plane, or one complete vibration, one of the sets appeared, so
that in two complete vibrations the cycle of changes was com-
plete. Pieces of cork and lycopodium powder showed that
there was no important current setting in the direction of the
ridges ; towards the heads of the ridges pieces of cork oscil-
lated from one ridge towards its neighbour, and back again.
The lycopodium sometimes seemed to move on the ridges from
the wood, and between them to it ; but the motion was irregu-
lar, and there was no general current outwards or inwards.
There was not so much disturbance as amongst the heaps (90).

117. A very simple arrangement exhibits these ripples beau-
tifully. If an oval or circular pan, fifteen or eighteen inches
in diameter, be filled with water, and a piece of lath (69) twelve
or fifteen inches long be held in it, edge upwards, so as to
bear against the sides of the pan as supporting points, and cut
the surface of the water, then on being vibrated horizontally
by the glass rod and wet finger, the phenomenon immediately
appears with ripples an inch or more in length. When the

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