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them with the turning qualities of the ships as found by actual experiment ;
and then the models might be handed over to the officers of the ships, and
they might practise encounters and manoeuvres until they knew not only
what they could do with their ships, but what it was best to do in order to
outmanoeuvre each other, and this without any cost or risk.
The behaviour of the models would be in all respects similar to that of the
ships, the only difference being that the manoeuvres would be on a smaller
scale ; and the scale of the manoeuvres would be the same as that of the
models, so that the step from the models to the large ships would be easy;
and familiarity with the working of the ships as well as the models under
ordinary circumstances would prepare the officers for using the ships in an
actual fight as they have been accustomed to use the models in their friendly
encounters. The scheme here proposed has its parallel in military schools.
26] ON THE INVESTIGATION OF THE STEERING QUALITIES OF SHIPS. 197
Although "autumn manoeuvres" and sham fights afford soldiers a much
better opportunity of preparing themselves for battle than anything at
present within reach of the sailors, still the war game appears to be
growing in favour, and this is nothing more than practising manoauvres in
miniature.
Independently of their value as a means of training naval officers, such
models would afford a means of studying naval tactics. From them might be
learnt the way in which a ship should strive to approach another of nearly
equal power and speed, so as to use her ram to the greatest advantage ; and
of this as yet but very little can be known ; and, except on models, it can only
be learnt from experiments on the ships.
Important as are the laws which have been verified by the Committee on
the steering of screw-steamers, it appears to me that the most important
lesson to be learnt from their investigation is, that there is nothing capricious
in the behaviour of these ships. To realize the value of this lesson the
investigation must be followed up ; and it appears that the best way to do
this would be by the aid of model launches on the plan thus roughly sketched
For continuation see p. 204.
ON THE RATE OF PROGRESSION OF GROUPS OF WAVES
AND THE RATE AT WHICH ENERGY IS TRANSMITTED
BY WAVES.
[From "Nature," Aug. 23, 1877.]
(Head before Section A of the British Association, 1877.)
WHEN several waves forming a discontinuous group travel over the
surface of deep water, the rate of progression of the group is always much
less than the rate at which the individual waves which compose the group
are propagated.
As the waves approach the front of the group they gradually dwindle
down and die out, while fresh waves are continually arising in the rear of the
others. This, which is a well-known phenomenon, presents itself to our notice
in various ways.
When a stone is thrown on to the surface of a pond, the series of rings
which it causes gradually expands so as. finally to embrace the entire surface
of the water ; but if careful notice be taken it is seen that the waves travel
outwards at a considerably greater rate than that at which the disturbance
spreads.
Or, when viewing a rough sea, if we endeavour to follow with the eye any
wave which is larger than its neighbours, we find, after following it in its
course for a short distance, that it has lost its extra size, while on looking
back we see that this has been acquired by the succeeding wave.
But perhaps the most striking manifestation of the phenomenon is in the
waves which spring from the bows of a rapid boat, and attend it on its course.
A wave from either bow extends backwards in a slanting direction for some
27] ON THE RATE OF PROGRESSION OF GROUPS OF WAVES, ETC. 199
distance and then disappears ; but immediately behind it has come into
existence another wave parallel to the first, beyond which it extends for some
distance when it also dies out, but not before it is followed by a third
which extends still farther, and so on, each wave overlapping the others
rather more than its predecessor. Although not obvious, very little con-
sideration serves to show that the stepped form of these columns of waves, is
a result of the continual dying out of the waves in the front of the group,
and the formation of fresh waves behind. For as each wave cuts slantwise
through the column formed by the group, one end is on the advancing side or
front of the group, and this is continually dying, while the other is in the rear,
and is always growing.
So far as I am aware, no general explanation of these phenomena has as
yet been given. It has been shown, and I believe first by Prof. Stokes, that
if two series of parallel waves of equal magnitude, but differing slightly in
length, move simultaneously in the same direction over the same water so
as to form a series of groups of waves separated by bands of interference,
that these groups will advance with half the velocity of the individual
waves. This is doubtless an example of the same phenomenon, and shows
that the theory of wave motion is capable of explaining the phenomena ;
but it appears to leave something to be desired, for instance, why should
the bands of interference only progress with half the velocity of propagation
in a deep sea, whereas in sound the corresponding bands of interference
which constitute the beats move at the same velocity as the waves ?
My object in this paper is to point out a fact in connection with wave
transmission which appears to have hitherto passed unnoticed, at all events in
connection with the phenomena described above, of which it affords a clear
and complete explanation. One of the several functions performed by waves
progressing through a medium, is the transmission of energy. Thus the
energy which we receive from the sun is brought to us in the waves of light
and heat ; so in the case of sound the work done by the arm of the drummer
is transmitted to our ears by the waves of sound. It is possible, however, to
have waves which travel through a medium without conveying energy ; such
are the waves caused by the wind on a field of corn. This kind of wave may
be well understood by suspending a series of small balls by threads, so that
the balls all hang in a row, and the threads are all of the same length. If we
then run the finger along, so as to set the balls oscillating in succession, the
motion will be such as to give the idea of a series of waves propagated from
one end to the other ; but in reality there is no propagation, each pendulum
swings independently of its neighbours, there is no communication of energy,
the waves being merely the result of the general arrangement of the
motion.
In this case there is no communication of energy, neither is there any
200 ON THE RATE OF PROGRESSION OF GROUPS OF WAVES [27
propagation of disturbance. Any one ball may be set swinging without in
the least disturbing the others ; and what is indicated here is a general law
that wherever a disturbance is transmitted through a medium by waves,
there must always be communication of energy. The rate at which energy is
transmitted in different media, or by different systems of waves, is very
different. This may be illustrated at once by experiment. If the balls just
described are all connected by an elastic thread, then they can no longer
swing independently. If one be set in motion, then, by virtue of the
connecting thread, it will communicate its motion to its neighbours until
they swing with it, so that now waves would be propagated through the balls.
The rate at which a ball would impart its motion, i.e. its energy, to its
neighbours, would clearly depend on the tension of the connecting thread. If
this was very slight compared with the weight of the balls, it would stretch,
and the ball might accomplish several swings before it had set its neighbours
in full motion, so that of the initial energy of disturbance a very small
portion is communicated at each swing. But if the tension of the thread be
great compared with the weight of the balls, one ball cannot be disturbed
without causing a similar disturbance in its neighbours, and then the whole
energy will be communicated. This is simply illustrated by laying a rope or
chain on the ground, and fastening down one end ; if then the loose end be
shaken up and down the wriggle caused will travel to the other end, leaving
the rope perfectly straight and quiet on the ground behind it, so that in this
case it is at once seen that the wave carries forward with it the whole energy
of the disturbance.
The straight cord and the pendulous balls represent media in which the
waves are at the opposite limits ; in one case none of the energy of dis-
turbance is transmitted, and in the other case the whole is transmitted.
Between these two limits we may have waves of infinite variety, in which any
degree of energy from all to nothing is transmitted. Now the waves of
sound belong to the class of the cord in which all the energy is transmitted ;
but what I want particularly to make clear, is that when the waves on water
are between the limits, they are analogous to the waves in the balls suspended
when connected by an elastic string. And I have so to show that according
to the accepted theory of wave motion the waves on deep water only carry
forward half the energy of disturbance.
In regular trochoidal waves the particles move in vertical circles with a
constant velocity, and are always subject to the same pressure. Of the energy
of disturbance half goes to give motion to the particles, and half to raise them
from their initial position to the mean height which they occupy during the
passage of the wave.
Now the mean horizontal positions of the particles remain unaltered by
27] AND THE RATE AT WHICH ENERGY IS TRANSMITTED BY WAVES. 201
the waves, hence, since their velocities are constant, none of their energy of
motion is transmitted ; nor since the pressure on each particle is constant,
can any energy be transmitted by pressure. The whole energy, therefore,
which remains to be transmitted, is the energy due to elevation, and that this
is transmitted is obvious, since the particles are moving forward when above
their mean position, and backwards when below it. This energy constitutes
half the energy of the disturbance, and this is, therefore, the amount trans-
mitted.
For a definite mathematical proof that
In waves on deep water the rate at which the energy is carried forward is
half the energy of disturbance per unit of length multiplied by the rate of
Let AO be the initial height occupied by a particle supposed to be of unit
weight, AJ the height of the centre of the circle in which it moves as the wave
passes, r the radius of the orbit, and the angle the radius vector makes with
the horizontal diameter, then the height of the particle above its initial
position is AJ A + r sin 6 ; adding to this height due to its velocity, we have
the whole energy of disturbance
= 2 (hi - A ) + r sin Q.
The velocity of the particle is
J2g(h 1 -h ),
and the horizontal component of this is
j h ) sin 6.
Therefore the rate at which energy is being transmitted by the particle is
{2 (A! - A ) + r sin 6} Jty (hi - A ) sin 6,
and the mean of this is
- 1 2 " {2 (Ai - h ) + r sin 0} J%g (A, - A ) sin 0.d0
ATT J Q
and if X be the length of the wave and n\ the rate of propagation
, Trr 2 , 2gr
/? a fi = } and -^- = 4-7rr' 2 .
A, A,
Therefore the mean rate at which energy is transmitted by this particle
= n\ (A! - ho),
or the rate of propagation multiplied by half the energy of disturbance.
[Q. E. D.
It now remains to come back to the speed of the groups of waves, and to
show that : if ike rate at which energy is transmitted is equal to the rate of
propagation multiplied by half the energy of disturbance, then the velocity of
a group of waves will be half that of the individual waves.
Let P 1} P 2 , P 3 , P 4 be points similarly situated in a series of waves which
gradually diminish in size and energy of disturbance from P 3 to P x , in which
direction they are moving. Let E be the energy of disturbance between 2\
and P 2 at time t, E + a the energy between P 2 and P 3 , E + 2a between P 3
and P 4 , and so on.
Then at the time t + n after the wave has moved through one wave-
length, it follows that the energy between PI and P 2 will be
E + E+a
and between P 2 and P 3 will be
E + a + E + 'la 3a
and again, after another interval n, the energies between P t and P 2) P 2 and
P 3 will be respectively
and - 3 - =E + 2a.
So that after the waves have advanced through two wave-lengths, the distri-
bution of the energy will have advanced one, or the speed of the groups is
half that of the waves. [Q. E. D.
Of course this reasoning applies equally to the waves on the suspended
balls, when connected by an elastic string, as to water ; and in this case the
conclusions may be verified, for, as on water, the groups of waves travel at a
slower rate than the waves. This experiment tends to throw light on the
manner in which the result is brought about. When a ball is disturbed, the
disturbance is partly communicated to the adjacent ball by the connecting
string, and part retained in the form of pendulous oscillation ; that part
which is propagated forward, is constantly reduced in imparting oscillations to
the successive balls, and soon dies out, while the motion retained by the
swinging pendulum, constantly gives rise to succeeding waves until it is all
absorbed. If the tightness of the cord be adjusted to the length of the
suspending threads, waves may be made to travel along in a manner closely
resembling the way in which they travel on water, the speed of the group
being half the speed of the individual waves.
Although the progression of a group has hitherto been spoken of as if
the form of the group was unaltered, this is by no means the case as
a rule.
In the mathematical investigation it was assumed that the motion of
the particles is circular ; this, however, cannot be the case when the succeed-
ing waves differ in size by a sensible quantity, and hence in this case the form
of the group cannot be permanent. And it may be further shown, that as a
small group proceeds, the number of waves which compose it will continually
increase, until the graduation becomes indefinitely small ; and this is exactly
what is observed, whether on water or on the strings.
So far as we have considered deep water, when the water is shallow
compared with the length of the waves, the results are modified, but in this
case the results as observed are strictly in accordance with the theory.
According to this, as waves enter shallow water, the motion of the
particles becomes elliptical, the eccentricity depending on the shallowness
of the water ; and it may be shown that under these circumstances, the rate
at which energy is transmitted is increased, until, when the elliptic paths
approach to straight lines the whole energy is transmitted, and consequently
it follows that the rate of the speed of the groups to the speed of the waves
will increase as the water becomes shallower, until they are sensibly the same.
In which case only the groups of waves are permanent, and Mr Scott Russell's
solitary wave is possible. Besides the explanation thus given of these various
phenomena, it appears that we have here a means of making some important
verifications of the assumptions on which the wave theory is based ; for the
relative speed of the groups, and the waves which compose them, affords a
criterion as to whether or not the particles move in circles.
ON THE EFFECT OF PROPELLERS ON THE STEERING OF
VESSELS.
[From the "British Association Report," 1877.]
Report of the Committee, consisting of JAMES R. NAPIER, F.R.S., Sir W.
THOMSON, F.R.S., W. FROUDE, F.R.S., J. T. BOTTOMLEY,' and OSBORNE
REYNOLDS, F.R.S. (Secretary), appointed to investigate the JSffect of
Propellers on the Steering of Vessels.
SINCE the meeting of the British Association held in Glasgow last year,
the Committee has been able to carry out some further experiments on
steering as affected by the reversing of the screw.
The largest vessel experimented upon last year was the barge No. 12, of
about 500 tons, and it appeared, on comparing the behaviour of this vessel
with the behaviour of those of smaller size, that the larger the ship the more
important would the effect of reversing the screw become. This view has
been completely borne out by the experiments of this year, made with one
vessel of 850 tons and another of 3594 tons.
In May last the 'Melrose,' a new vessel belonging to Messrs Donald
Currie & Co., was tried at the instance and under the superintendence of
Mr James R. Napier. The ' Melrose ' is 228 feet in length by 29 feet in
breadth, and 16 feet 3 inches in depth. She is 850 tons gross register ;
her propeller makes 90 revolutions per minute with the vessel going at
a speed of lOf knots.
The following is Mr Napier's report of the trials : " These experiments
were made on 3rd of May 1877, between Wemyss Bay and Rothsay. There
28] ON THE EFFECT OF PROPELLERS ON THE STEERING OF VESSELS. 205
was little or no wind ; the sea was glassy smooth. The draft of water was
9 feet 1 inch forward, and 12 feet 5 inches aft ; the diameter of the propeller
was 11 feet 6 inches, the pitch 14 feet 3 inches, it had 4 blades and was
right-handed. The maximum speed at the nautical mile was lOf knots ; but
the speed was about 10 knots when the trials were made.
"A trial was made with the rudder said to be amidships, and the ship's
head turned to starboard ; but it was found afterwards that the pointer on
the bridge had been misplaced, and, as it was difficult at the time to ascertain
the rudder's position, the result was uncertain.
"First mock collision trial. The vessel was steaming about 10 knots when
the telegraph bell warned the engineer to stand by his engines, and shortly
after the bell was rung for him to reverse at full speed (no intermediate
order to slow or stop being given) ; in 15 seconds after this order was given
the engines began to reverse, and in 2 minutes 15 seconds after the giving
of the order to reverse, the forward motion of the ship had entirely stopped.
"At the instant that the engineer below telegraphed to the captain on
deck that his engines were reversing, the captain gave the order 'Hard
aport,' which was quickly obeyed by the two men at the wheel. The vessel's
head almost immediately commenced turning to port, and when the ship's way
was stopped, or about 2 minutes after the order to port was given, the vessel's
head had turned 26 or 28 degrees to port.
" Second mock collision trial. Everything was done in the same manner
as in the first trial, except in this case the order was to starboard hard.
The vessel's way was lost in about the same time. The vessel's head com-
menced to turn to starboard almost immediately after the engines began to
reverse, and when the forward way was lost, her head had gone round 40 to
starboard.
" These results were so contrary to the expectation of some of the nautical
party on board, that they made a third mock collision trial (a second one with
the helm hard aport) ; but on this occasion the orders to reverse the engines
and to port the helm were given simultaneously. The result was similar to
the first trial, the head turning a long way to port ; but I was not on the
bridge to note the angle through which her head moved before head-way was
" Mr Currie, one of the owners of the ship, most of the nautical men and
visitors on board learned, I think, something regarding the steering of screw-
steamers, and a cause of some, if not of many, collisions which they did not
know before. The Captain of the ship, however, when asked before the
206 ON THE EFFECT OF PROPELLERS ON THE STEERING OF VESSELS. [28
trials what would be the result of the sudden reversal of the engines, with
the helm aport or starboard, stated the direction in which the ship's head
would turn as it actually happened."
The Committee wish to thank Mr Currie for allowing them the use of his
ship for the experiments.
It will be seen, from Mr Napier's report, that the ' Melrose ' behaved in
precisely the same way as did the vessels last year, except that the effect of
the reversed screw on the action of the rudder was even more apparent than
in the previous trials. This was obviously owing to the greater size of the
ship, and the consequently greater time taken by the reversed screw in
bringing her to rest, and the result led the Committee to conclude that with
still larger ships the result would be yet more pronounced.
This conclusion has been verified in a somewhat unexpected although in
a most satisfactory manner; for, after arriving at Plymouth, the Secretary
received the following account of trials made in the s.s. ' Hankow,' of London,
3594 tons, by Captain Symmington, the commander, in response to the
circular issued by the Committee last year, but otherwise at his own
instance.
Capt. Symmingtoris Report.
" S.s. ' Hankow,' of London,
8th March, 1877.
" Gross tonnage 3594 12 , net 2331 75 tons.
" Length 389 feet, breadth 42'1, depth 28'8.
" Some experiments were conducted this forenoon from 9.20 A.M. to
11.20 A.M., in lat. 8 50' S., long. 153 58' E., in order to determine how
the ship's head turned on reversing the engines suddenly when going full
speed ahead with the helm amidships, port, and starboard ; also the time
and diameter of the circles made when going slow and full speed ahead on
the port helm.
" Sea smooth or between No. 1 and 2 of the Beaufort scale ; ship drawing,
on leaving Sydney on the 28th ult., 26 feet forward and 24 feet 3 inches aft ;
to-day the probable draft will be 24 feet 8 inches forward and 23 feet 8 inches
aft, mean 24'2.
" First Experiment.
" Ship going ahead full speed, engines were suddenly reversed, helm put
hard aport ; immediately the engines started, time noted and bearing of ship's
head by standard (Admiralty compass) noted, and the bearing of the ship's
head also noted at every 15 seconds until the ship came to a dead stop.
Time. A.M.
Ship's Head by
Head tu
rned to
h. m. s.
N. 62 W.
15 84i
15 85$
15 79|
:8*
" Ship came to a dead stop in 3 min. 30 sec., and turned to port 26
in 2 min., then turned to starboard 8 in 1 min. 30 sec.
"Second Experiment.
" Ship going ahead full speed, say 10 knots. The engines were suddenly
reversed full speed astern, helm put hard astarboard ; bearing of ship's head
taken and time. At every 15 seconds the bearing of ship's head was also
noted until the ship came to a dead stop.
Head ti
2lJ
" Ship came to a dead stop in 3 min. 23 sec. Her head payed off to port
2 during the first 15 sec., and afterwards turned to starboard 39 before
coming to rest.
" Third Experiment.
" Ship going full speed ahead, say 10 knots, the engines were suddenly
reversed, full speed astern, the helm put amidships, and the bearing of the
ship's head noted by the standard azimuth compass (Admiralty) at every
15 seconds until the ship came to absolute rest. Wind and weather as before.
Going full speed ahead 10 knots, reversed full speed astern, helm amidships.
Head tt
irned to
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