Frederick William Lanchester

“ It must be borne in mind that an unsymmetrical design of sound generator may conceivably emit pressure waves containing momentum in one or more directions, and rarefaction waves in others, or perhaps the air displaced by the pressure waves emitted in one direction may be replaced by a steady flow in other directions. On the other hand, it is possible that by some highly refined method the true pressure of a continuous wave train may be detected and measured, and the theoretical result that it is due to the energy passed into the thermodynamic system may some day receive confirmation. ”

“ When wave motion exists in such a gas, heat is abstracted where the gas is rarefied and added where the gas is compressed, but more heat is added than subtracted; the difference represents the work done, according to well-known thermodynamic principles. We can therefore look upon the adiabatic wave as a Boyle's law wave in which heat has been added to one part and abstracted from another part, but in sum an addition of heat has been made to the contents of the enclosure, and the mean pressure increase can be calculated therefrom. ”

“ Therefore the time during which any pressure acts across the imaginary plane is inversely as that pressure, or constant for any small unit of mass. But is the momentum communicated across the imaginary plane by pressure per unit area, and we have shown the total units of mass in any wave is the same as in undisturbed air. Consequently in a plane wave in a fluid obeying Boyle's law the momentum communicated by the pressure of the wave is precisely that communicated by the undisturbed fluid. ”

“ Where we have a train of waves in a gas following the adiabatic law, it has been shown that there must be a pressure increase due to the energy that enters the thermodynamic system. Where the train is continuous, as in the Kundt's tube, no complication arises from this cause, but where we are dealing with a limited train, it is difficult to see in what manner this pressure can be confined to the region occupied by the wave train ”

“ If we suppose the enclosure rigid and the fluid elastic, the change of pressure due to the vortices on the boundary walls depends upon the law of elasticity, and is not a function of the magnitude or energy of the vortex system alone. ”

“ The influence of aerofoil weight (wing weight) is less important in the case of a small aerodrome or bird than of a large one. Consequently nature can endow a small bird relatively with an extent of wing surface not "commercially " possible in the case of a larger bird, so that the smaller bird can, in normal active flight, fly slower than a large one, but by adopting the leaping mode it can, in effect, divest itself of its superfluous surface, and can then rival the larger birds in velocity. ”

“ If the turbulence contain only one harmonic component, or if, which amounts to the same thing, the bird is only able to take advantage of the harmonic component in the line of flight, the available energy for a given maximum velocity will be only half that on the basis of circular motion ”

“ If the adiabatic wave is examined by the foregoing method an excess of mean pressure is found to exist, and without doubt, if the source of sound emits a continued succession of waves, momentum accompanies such waves as an ever-spreading field of excess mean pressure, but it is not clear that if the source ceases to emit, this pressure region will be confined to and move with the advancing waves; it appears more probable to the author that the air contained within the wave sphere shares in the excess pressure. ”

“ The modified Newtonian medium of our present hypothesis resembles in many ways the perfect gas of kinetic theory, but differs in one very important respect. The molecules of a perfect gas are not only in a state of motion, but are undergoing frequent encounters one with another. Whether these encounters are due to gross impact or to some kind of action at a distance is immaterial from the point of view of the present discussion. The particles or corpuscles of the hypothetical medium have no magnitude, consequently they do not encounter one another, and therefore the medium has no continuity. ”

“ It is evident that if a bird can, by altering its angle and altitude, so manipulate the wind coming within its grasp, that the portions that are moving in excess of the mean velocity have their velocity reduced, and those that are moving at less than the mean velocity are accelerated, the total energy of the wind will be reduced, and the energy thus taken from the wind may become available for the purposes of propulsion. ”

“ The leaping mode is, in fact, a means of adjustment, by means of which the conditions of least resistance can be approximated under considerable variations of velocity. If one of the larger birds, with its limited relative area, were to force its velocity up to the point at which leaping flight would pay, it would require an amount of energy per second far beyond its actual horse-power capacity. ”

“ Under these conditions the greater the energy of the vortex system set up in the fluid the lower will become the pressure in the internal part of the region, so that the plus and minus momentum of the equal and opposite flow taking place across any imaginary barrier plane is accounted for by the ordinary static pressure on the confines of the region, and does not give rise to any added pressure. ”

“ If we suppose now a resistance to be inserted in the electrical circuit, the current, and therefore the force F, will tend to fall off, but the applied force F1 continues, so that the conductor is set in motion in the magnetic field and is maintained in motion, the energy expended by the applied force F1 being accounted for as energy lost in the electrical circuit ”

“ Thus the present application of the principle of no momentum is in no way antagonistic to modern views and discovery as to the transference of momentum by light and other manifestations of electric radiation. ”

“ Larmor gives a theorem which purports to be a general proof of the transmission or communication of momentum by wave motion. Poynting [17] has given a condensed edition of this alleged proof, which may be quoted, as follows:—
Let us suppose that a train of waves is incident normally on a perfectly reflecting surface. Then, whether the reflecting surface is at rest, or is moving to or from the source, the perfect reflection requires that the disturbance at its surface shall be annulled by the superposition of the direct and reflected trains. The two trains must therefore have equal amplitudes. ”

“ The motion of the conductor under the influence of the force F1 corresponds in our analogy to the descent of an aerodone in its gliding path, the gliding angle being represented by the velocity of the conductor divided by the velocity of the magnetic flux. ”

“ The sailing boat, for example, offers a very promising field for the application of the peripteral principles of flight, and furnishes strong confirmation of the present theory. We may look upon the sailing boat, and especially the racing craft with its fin or deep keel, as an aerofoil combination in which the under-water and above-water reactions balance one another. ”

“ But a self-contained system consisting of a simple enclosure containing fluid of uniform mean density (regarding the individual waves of the train as small) cannot suffer change of momentum without infringing the third law of motion; consequently the wave train (if of the same mean density as the quiescent fluid) cannot carry momentum. ”

“ The compression regions are always endeavouring to move faster and the rarefaction regions slower than the mean velocity. From the present standpoint this is evidently due to the pressure increase becoming proportionately greater than the density increase (Fig. 161) , and vice versâ, thus destroying the necessary balance between the pressure reaction and the communication of momentum by which it is maintained. The more usual and equally correct point of view is to attribute the difference of velocity of different portions of the wave to the difference of temperature of its parts. ”

“ Thus in the case of a limited wave train, if it carry momentum, that momentum can be represented by some definite value of mv, and the remainder of the system with which the wave is associated must, relatively to the common mass centre, have an equal and opposite momentum at every instant of time. ”

“ The result of this supposition is evidently that the minimum angle at which the boat can shape its course relatively to the wind is the sum of the under and above-water gliding angles.
If the boat had no body (hull) , and the conditions of our supposition be complied with, this reasoning shows that the minimum angle of the course relatively to the apparent direction of the wind would be the sum of [30] the γ for water and the γ for air, which is probably a degree or so less than 20 degrees, or rather less than two "points." ”

“ Incidentally it may be remarked that it is probably for this reason that the soaring mode of flight, in which energy is captured from the wind, is principally employed by the larger birds, many of which are otherwise incapable of prolonged flight.
↑ Other methods of soaring are practised by many of the larger birds. In some cases soaring is accomplished by merely gliding on an up-current whose velocity is equal to or in excess of the late of fall for gliding in still air ”

“ In all probability the velocity of least resistance for a given bird in active flight is somewhere about 20 per cent. greater than for the gliding or soaring mode. ↑ Ceteris paribus, the horse-power of any animal or machine varies as the square of its linear dimension, whereas the weight varies as the cube. Thus the power per unit weight is greater for a small bird than a large one. ”

“ If the conductor be situated in a magnetic field, it will experience a force at right angles to the direction of the field just as has been shown to exist in the case of the peripteral system, so that again we find the analogy holds. ”

“ It is probable that the difference in the behaviour of air or any other gas, and the medium, will be least at very high and very low velocities; at intermediate velocities the present mode of treatment is unlikely to be of any utility. ”

“ Another form of soaring depends upon the proximity of masses of air having different velocities, as the live stream and "dead-water" region in the wake of an obstacle; the bird circles round and round, playing off the one mass of air against the other. ”

“ In this instance the species of bird was not identified. This proportion means that the reaction sustained by the wings when in action is approximately four times the weight of the bird, on which computation the flight velocity should be about twice that proper to the actual weight and wing area measurement. [26]
It is difficult to assess accurately the speed of flight of a bird under any circumstances, and most of all under the conditions now under discussion. Travelling at somewhat over thirty miles ”

“ Phys. Soc., l.c. ante. ↑ At this time Poynting evidently has no misgivings as to the soundness of Larmor's theorem, and therefore must still have supposed that a Boyle's law wave-train carries momentum, apart from the evidence already given. ↑ "Sensations of Tone," Appendix VII., B II. ↑ It has already been pointed out (§ 187) that the problem is in all likelihood modified by the conditions of active flight, so that the tabulated figures, which relate to the gliding mode, may require to be multiplied by some unknown coefficient. ”