THE WAVE THEORY OF SOUND CONSIDERED.
By HENRY. A. MOTT, Ph.D., LL.D.
Before presenting any of the numerous difficulties in the way of accepting the wave theory of sound as correct, it will be best to briefly represent its teachings, so that the reader will see that the writer is perfectly familiar with the same.
The wave theory of sound starts off with the assumption that the atmosphere is composed of molecules, and that these supposed molecules are free to vibrate when acted upon by a vibrating body. When a tuning fork, for example, is caused to vibrate, it is assumed that the supposed molecules in front of the advancing fork are crowded closely together, thus forming a condensation, and on the retreat of the fork are separated more widely apart, thus forming a rarefaction. On account of the crowding of the molecules together to form the condensation, the air is supposed to become more dense and of a higher temperature, while in the rarefaction the air is supposed to become less dense and of lower temperature; but the heat of the condensation is supposed to just satisfy the cold of the rarefaction, in consequence of which the average temperature of the air remains unchanged.
The supposed increase of temperature in the condensation is supposed to facilitate the transference of the sound pulse, in consequence of which, sound is able to travel at the rate of 1,095 feet a second at 0°C., which it would not do if there was no heat generated.
In other words, the supposed increase of temperature is supposed to add 1/6 to the velocity of sound.
If the tuning fork be a Koenig C3 fork, which makes 256 full vibrations in one second, then there will be 256 sound waves in one second of a length of 1095/256 or 4.23 feet, so that at the end of a second of time from the commencement of the vibration, the foremost wave would have reached a distance of 1,095 feet, at 0°C.
The motion of a sound wave must not, however, be confounded with the motion of the molecules which at any moment form the wave; for during its passage every molecule concerned in its transference makes only a small excursion to and fro, the length of the excursion being the amplitude of vibration, on which the intensity of the sound depends.
Taking the same tuning fork mentioned above, the molecule would take 1/256 of a second to make a full vibration, which is the length of time it takes for the pulse to travel the length of the sound wave.
For different intensities, the amplitude of vibration of the molecule is roughly 1/50 to 1/1000000 of an inch. That is to say, in the case of the same tuning fork, the molecules it causes to vibrate must either travel a distance of 1/56 or 1/1000000 of an inch forward and back in the 1/256 of a second or in one direction in the 1/512 of a second.
I might further state that the pitch of the sound depends on the number of vibrations and the intensity, as already indicated by the amplitude of stroke—the timbre or quality of the sound depending upon factors which will be clearly set forth as we advance.
Having now clearly and correctly represented the wave theory of sound, without touching the physiological effect perceived by means of the ear, we will proceed to consider it.
We must first consider the state in which the supposed molecules exist in the air, before making progress.
The present science teaches that the diameter of the supposed molecules of the air is about 1/250000000 of an inch (Tait); that the distance between the molecules is about 8/100000 of an inch; that the velocity of the molecules is about 1,512 feet a second at O°C., in its free path; that the number of molecules in a cubic inch at O°C. is 3,505,519,800,000,000,000 or 35 followed by 17 ciphers (35)17; and that the number of collisions per second that the molecules make is, according to Boltzmann, for hydrogen, 17,700,000,000, that is to say, a hydrogen molecule in one second has its course wholly changed over seventeen billion times. Assuming seventeen billion or million to be right for the supposed air molecules, we have a very interesting problem to consider.
The wave theory of sound requires, if we expect to hear sound by means of a C3 fork of 256 vibrations, that the molecules of the air composing the sound wave must not be interfered with in such a way as to prevent them from traveling a distance of at least 1/50 to 1/1000000 of an inch forward and back in the 1/256 of a second. The problem we have to explain is, how a molecule traveling at the rate of 1,512 feet a second through a mean path of 8/100000 of an inch, and colliding seventeen billion or million times a second, can, by the vibration of the C3 fork, be made to vibrate so as to have a pendulous motion for 1/256 of a second and vibrate through a distance of 1/50 to the 1/1000000 of an inch without being changed or mar its harmonic motion.
It is claimed that the range of sound lies between 16 vibrations and 30,000 (about); in such extreme cases the molecules would require 1/16 and 1/30000 of a second to perform the same journey.
It must not be forgotten that a mass moving through a given distance has the power of doing work, and the amount of energy it will exercise will depend on its velocity. Now, a molecule of oxygen or nitrogen, according to modern science, is a mass 1/250000000 of an inch in diameter, and an oxygen molecule has been calculated to weigh 0.0000000054044 ounce. Taking this weight traveling with a velocity of 1,512 feet a second through an average distance of 8/100000 of an inch, the battering power or momentum it would have can be shown to be in round numbers capable of moving 1/200000 of an ounce.
Now, when the C3 tuning fork has been vibrating for some time, but still sounding audibly, Prof. Carter determined that its amplitude of stroke was only the 1/17000 of an inch, or its velocity of motion was at the rate of 1/33 of an inch in one second, or one inch in 33 seconds (over half a minute), or less than one foot in one hour.
Assuming one prong to weigh two ounces, we have a two-ounce mass moving 1/17000 of an inch with a velocity of 1/33 of an inch in one second. The prong, then, has a momentum or can exercise an amount of energy equivalent to 1/200 of an ounce, or can overcome the momentum of 1,000 molecules.
It would be difficult to discover not only how a locust can expend sufficient energy to impart to molecules of the air, so as to set them in a forced vibration, and thus enable a pulse of the energy imparted to control the motion of the supposed molecules of the air for a mile in all directions, but also to estimate the amount of energy the locust must expend.
According to the wave theory, a condensation and rarefaction are necessary to constitute a sound wave. Surely, if a condensation is not produced, there can be no sound wave! We have then no need to consider anything but the condensation or compression of the supposed air molecules, which will shorten the discussion. The property of mobility of the air and fluidity of water are well known. In the case of water, which is almost incompressible, this property is well marked, and unquestionably would be very nearly the same if water were wholly incompressible. In the case of the air, it is conceded by Tyndall, Thomson, Daniell, Helmholtz, and others that any compression or condensation of the air must be well marked or defined to secure the transmission of a sound pulse. The reason for this is on account of this very property of mobility. Tyndall says: "The prong of the fork in its swift advancement condenses the air." Thomson says: "If I move my hand vehemently through the air, I produce a condensation." Helmholtz says: "The pendulum swings from right to left with a uniform motion. Near to either end of its path it moves slowly, and in the middle fast. Among sonorous bodies which move in the same way, only very much faster, we may mention tuning forks." Tyndall says again: "When a common pendulum oscillates, it tends to form a condensation in front and a rarefaction behind. But it is only a tendency; the motion is so slow, and the air so elastic, that it moves away in front before it is sensibly condensed, and fills the space behind before it can become sensibly dilated. Hence waves or pulses are not generated by the pendulum." And finally, Daniell says: "A vibrating body, before it can act as a sounding body, must produce alternate compressions and rarefactions in the air, and these must be well marked. If, however, the vibrating body be so small that at each oscillation the surrounding air has time to flow round it, there is at every oscillation a local rearrangement—a local flow and reflow of the air; but the air at a distance is almost wholly unaffected by this."
Now, as Prof. Carter has shown by experiment that a tuning fork while still sounding had only an amplitude of swing of 1/17000 of an inch, and only traveled an aggregate distance of 1/33 of an inch in one second, or one inch in 33 seconds, surely such a motion is neither "swift," "fast," nor "vehement," and is unquestionably much "slower" than the motion of a pendulum. We have only to consider one forward motion of the prong, and if that motion cannot condense the air, then no wave can be produced; for after a prong has advanced and stopped moving (no matter for how short a time), if it has not compressed the air, its return motion (on the same side) cannot do anything toward making a compression. If one such motion of 1/17000 of an inch in 1/512 of a second cannot compress the air, then the remaining motions cannot. There is unquestionably a "union limit" between mobility and compressibility, and unless this limit is passed, mobility holds sway and prevents condensation or compression of the air; but when this limit is passed by the exercise of sufficient energy, then compression of the air results. Just imagine the finger to be moved through the air at a velocity of one foot in one hour; is it possible that any scientist who considers the problem in connection with the mobility of the air, could risk his reputation by saying that the air would be compressed? Heretofore it was supposed that a præong of a tuning fork was traveling fast because it vibrated so many times in a second, never stopping to think that its velocity of motion was entirely dependent upon the distance it traveled. At the start the prong travels 1/20 of an inch, but in a short time, while still sounding, the distance is reduced to 1/17000 of an inch. While the first motion was quite fast, about 25 inches in a second, the last motion was only about 1/33 of an inch in the same time, and is consequently 825 times slower motion. The momentum of the prong, the amount of work it can do, is likewise proportionately reduced.
Some seem to imagine, without thinking, that the elasticity of the air can add additional energy. This is perfectly erroneous; for elasticity is a mere property, which permits a body to be compressed on the application of a force, and to be dilated by the exercise of the force stored up in it by the compression. No property of the air can impart any energy. If the momentum of a molecule or a series of molecules extending in all directions for a mile is to be overcome so as to control the character of the movements of the molecules, then sufficient external energy must be applied to accomplish the task: and when we think that one cubic inch of air contains 3,505,519,800,000,000,000 molecules, to say nothing about the number in a cubic mile, which a locust can transmit sound through, we are naturally compelled to stop and think whether the vibrations of supposed molecules have anything or can have anything to do with the transference of sound through the air.
If control was only had of the distance the vibrating molecule travels from its start to the end of its journey, then only the intensity of the sound would be under subjection; but if at every infinitesimal instant control was had of its amplitude of swing, then the character, timbre, or quality of the sound is under subjection. It is evident, then, that the blows normally given by one molecule to another in their supposed constant bombardment must not be sufficient to alter the character of vibration a molecule set in oscillation by a sounding body must maintain, to preserve the timbre or quality of the sound in process of transmission; for if any such alteration should take place, then, naturally, while the pitch, and perhaps intensity, might be transmitted, the quality of the sound would be destroyed.
Again, it is certain that no molecule can perform two sets of vibrations, two separate movements, at the same time, any more than it can be in two places at the same time.
When a band of music is playing, the molecule is supposed to make a complex vibration, a resultant motion of all acting influences, which the ear is supposed to analyze. It remains for the mathematician to show how a molecule influenced by twenty or more degrees of applied energy, and twenty or more required number of frequences of vibration at the same time, can establish a resultant motion which will transmit the required pitch, intensity, and timbre of each instrument.
When a molecule is acted on by various forces, a resultant motion is unquestionably produced, but this would only tend to send the molecule forward and back in one direction, and, in fact, a direction it might have taken in the first place if hit properly.
How any resultant can be established as regards the time necessary for the molecule to take so as to complete a full vibration for the note C11, which requires 1/16 of a second, and for other notes up to C''''', which only requires 1/4176 of a second, as when an orchestra is playing, is certainly beyond human comprehension, if it is not beyond the "transcendental mathematics" of the present day.
Unquestionably, the able mathematicians Lord Rayleigh, Stokes, or Maxwell, if the problem was submitted to them, would start directly to work, and deduce by so called "higher mathematics" the required motions the molecules would have to undergo to accomplish this marvelous task—the same as they have established the diameter of the supposed molecules, their velocity, distance apart, and number of bombardments, without any shadow of positive proof that any such things as molecules exist.
As S. Caunizzana has said: "Some of the followers of the modern school push their faith to the borders of fanaticism; they often speak on molecular subjects with as much dogmatic assurance as though they had actually realized the ingenious fiction of Laplace, and had constructed a microscope by which they could detect the molecule and count the number of its constituent atoms."
Speaking of the "modern manufacturers of mathematical hypotheses," Mattieu Williams says: "It matters not to them how 'wild and visionary,' how utterly gratuitous, any assumption may be, it is not unscientific provided it can be vested in formulæ and worked out mathematically.
"These transcendental mathematicians are struggling to carry philosophy back to the era of Duns Scotus, when the greatest triumph of learning was to sophisticate so profoundly an obvious absurdity that no ordinary intellect could refute it.... The close study of pure mathematics, by directing the mind to processes of calculation rather than to phenomena, induces that sublime indifference to facts which has characterized the purely mathematical intellect of all ages."
Tyndall, however, states in all frankness, and without the aid of mathematical considerations, that "when we try to visualize the motions of the air having one thousand separate tones, to present to the eye of the mind the battling of the pulses, direct and reverberated, the imagination retires baffled at the attempt;" and he might have added, the shallowness and fallacy of the wave theory of sound was made apparent. He, however, does express himself as follows: "Assuredly, no question of science ever stood so much in need of revision as this of the transmission of sound through the atmosphere. Slowly but surely we mastered the question, and the further we advance, the more plainly it appeared that our reputed knowledge regarding it was erroneous from beginning to end."
Until physicists are willing to admit that the physical forces of nature are objective things—actual entities, and not mere modes of motion—a full and clear comprehension of the phenomena of nature will never be revealed to them. The motion of all bodies, whether small or great, is due to the entitative force stored up in them, and the energy they exercise is in proportion to the stored-up force.
Tyndall says that "heat itself, its essence and quiddity, IS MOTION, AND NOTHING ELSE." Surely, no scientist who considers what motion is can admit such a fallacious statement, for motion is simply "position in space changing;" it is a phenomenon, the result of the application of entitative force to a body. It is no more an entity than shadow, which is likewise a phenomenon. Motion, per se, is nothing and can do nothing in physics. Matter and force are the two great entities of the universe—both being objective things. Sound, heat, light, electricity, etc., are different forms of manifestation of an all-pervading force element—substantial, yet not material.
[NATURE.]