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.