[From the American Journal of Science and Arts, s 7 J.] 
ON THE EXPERIMENTAL DETERMINATION OF 
THE RELATIVE INTENSITIES OF SOUND; 
AND ON THE MEASUREMENT OF THE POWERS OF VARIOUS 
SUBSTANCES TO REFLECT AND TO TRANSMIT SONO 
ROUS VIBRATIONS. 
By ALFRED M. MAYER, Ph.D., 
Professor of Physics in the Stevens Institute of TechnolojrY Hoboken, N. J 
(Road before the National Academy of Sciences, in Cambridge, Nov. 21, 1872.) (From the Am. Journal of Science and Arts, Vol. V, Jan. and Feb., 1873.] 
ON TIIE EXPERIMENTAL DETERMINATION OF 
THE RELATIVE INTENSITIES OF SOUND; 
ANI) ON THE MEASUREMENT OF THE POWERS OF VARIOUS 
SUBSTANCES TO REFLECT AND TO TRANSMIT SONO- 
ROUS VIBRATIONS. 
By ALFRED M. MAYER, Ph.I)., 
"rofessor of Physics in the Stevens Institute of TechnoloKy, Hoboken, N. J 
(Road before the National Academy of Sciences, in Cambridge, Nov. 21, 1872.) 
While the problems of the determination of the pitch of 
sounds and the explanation of timbre have received their com- 
plete elucidation at the hands of Mersenne, Young, De la Tour, 
Kdnig and Helmholtz, the problem of the accurate experimen- 
tal determination of the relative intensities of given sonorous 
vibrations has never been solved. 
The method I here present will, I hope, open the wav to the 
complete solution of this difficult and important problem; and 
I trust that the success I have met with will instigate others, 
more learned and patient, to attack with superior acumen a 
subject which must necessarily become of fundamental import- 
ance in the future progress of acoustic research. 
1. The determination of the relative intensities of sounds of the 
same pitch. 
If two sonorous impulses meet in traversing an elastic 
medium, and if at their place of meeting the molecules of the 
medium remain at rest, it is evident that at this place of quies- 
cence the two impulses must have opposite phases of vibration, 
and be of equal intensity. 
I have, in the following manner, experimentally applied this 
principle to the accurate determination of the relative intensi- 
ties of vibrations giving the same note, and propagated from 
their sources of origin in spherical waves. 
Clothe two contiguous rooms with a material which does not 
reflect sound, and place in each room one of the sounding 2 
bodies, and maintain these sounds of constant intensity ; or the 
two sources of sound may be placed in the open air, and sepa- 
rated from each other by a non-reflecting partition. Fix at a 
certain distance from each sounding body a resonator respond- 
ing to its note, and attach to each resonator the same length of 
firm gum-tubing; then lead these tubes to a forked pipe, so that 
the impulses from the two resonators meet at the confluence of 
the two branches of the forked tube, and connect the branch of 
the forked tube, in which the sounds meet, with one of Konig’s 
manometric capsules. Now sound continuously one of the 
bodies, and the manometric flame when viewed in a revolving 
mirror will present its well known serrated appearance. On 
sounding the second body impulses from it will meet those from 
the first body, and if the phases of vibration of the impulses on 
the manometric membrane are opposed and of equal intensity, 
the membrane will remain at rest and the flame will now appear 
in the mirror as a band of light with a rectilinear upper border. 
But although the intensities of the pulses can easily be ren- 
dered equal by altering the distance of one of the resonators 
from its sounding body, yet this change of position will alter 
the relation of the phases of the impulses reaching the membrane, 
so that, if by mere chance, we get them opposed in the first 
position of the resonator, they will no longer be so after its 
change of position. But on stopping the vibrations of one of 
the bodies and setting it in vibration at intervals we may finally 
succeed in causing the impulses on reaching the membrane to 
have opposite phases of vibration. Such a method, which relies 
only on chance, can be of little value on account of its uncer- 
tainty and the tediousness of its application. 
The above difficulty I have entirely removed by the follow- 
ing means. I cut a piece out of one of the tubes equal in 
length to a half-wave of the note we are experimenting on, and 
replace this piece of tubing with a glass tube of the same length, 
into which slides another glass tube also of half a wave in 
length. Now the experimentation becomes expeditious and 
certain. Sound both bodies continuously and place in a fixed 
position one of the resonators. Move the other to a certain dis- 
tance from its sounding body and then pull out the inner glass 
tube until exact opposition of phase of the impulses is brought 
on to the manometric membrane. This condition will be known 
when the serrations have dropped to their minimum of elevation. 
If the latter do not entirely disappear from the band of light in 
the mirror, we must place the movable resonator at another 
distance and readjust the sliding tube. A few trials will give in 
the mirror a band of light with a straight, unruffled top border; 
then we have opposed phases of vibrations at the confluence of 3 
the branches of the forked tube, and equally intense pulses 
are traversing the two tubes leading from the resonators. 
The distance of each resonator from its sounding body is now 
measured, and the inverse ratio of the squares of these 
distances will be the ratio of the intensities of the vibrations at 
the sources of the sounds, if the intensities of the impulses sent 
through a tube from a resonator varies directly with the inten- 
sities of the vibrations of the free air in the plane of the mouth 
of the resonator. 
It will be observed that the accuracy of the determinations 
by this experimental method depend on three conditions. 
First, that the vibrating effects of the same area of a spherical 
sonorous wave diminish in intensity as the reciprocals of the 
squares of the distances of this area form the point of origin of 
the wave. There is every dynamic reason to believe in the 
truth of this proposition. The second necessary condition is 
that the elongation of one of the resonator tubes over the other 
by half a wave-length of firm glass tubing does not diminish 
the intensity of the impulses which have traversed it. Numer- 
ous experiments, especially those of Biot and Regnault on the 
aqueduct pipes of Paris, show that this short connecting tube of 
glass cannot in any way affect the accuracy of the measures. 
The third condition is that the intensities of pulses sent through 
a tube from a resonator vary directly with the intensities of 
the vibrations of the free air in the plane of the mouth of the 
resonator. This is a very important consideration, and as I 
believe there is no entirely reliable discussion of the above rela- 
tion, the problem will have to be experimentally solved with 
the greatest care. If, however, the relations between the inten- 
sities of pulses inside the tube and those outside the mouth of 
the resonator shall be shown to be different (and I think they 
will be) from what we, for illustration, have here assumed, the 
process of the numerical reduction of the experiments will be 
only modified while the experimental method remains secure. 
Indeed, I cannot but consider that 1 have here, by applying the 
principle of interference, so fertile in results in optics, been the 
first to give an experimental method which will determine with 
precision the relative intensities of two sonorous vibrations pro- 
ducing the same note. 
Savart and many other experimenters have determined the 
relative intensities of two sounds by placing sand or other 
light particles on membranes and receding from the source of 
sound until no motions of the particles were visible. Also Drs. 
Renz and Wolf (Pogg. Ann., vol. clxxiv, p. 595) give the results 
of experiments on the determination with the ear of the inten- 
sity of the sounds of a ticking watch. More recently Dr. Heller 
(Pogg. Ann., vol. ccxvii, p. 566) lias made an elaborate research 4 
on the intensities of sounds; deducing mathematically his deter- 
minations from the observed amplitudes of vibration of a mem- 
brane ; and Mr. Bosanquet (L. E. and D. Phil. Mag., Nov., 1872) 
has just published a paper in which he proposes to measure the 
intensities of the sounds of pipes of different pitch by the deter- 
mination of the quantity of air which each pipe consumes in 
sounding. But all of these experimenters acknowledge the 
want of precision in their measures and the difficulties in the 
actual practice of their methods. 
When the resonators have such distances from their corres- 
ponding sounding bodies that the phases of the impulses on 
the membrane are opposed while their intensities are different, 
a residual action is given, and the intensity of this action on the 
membrane will depend on the relative intensities of the sources 
of sound and the relative distances at which the resonators 
are placed. It may here be interesting to consider the simplest 
case, that is, when the intensities of vibration at the two sources 
of origin of the sounds are the same, and the two resonators are 
placed at various distances from these points of origin, but 
always differ in their distances by one half wave-length. Let 
us call A one of the resonators, B the other. Let A be succes- 
sively placed at distances from its sounding body equal to 1, 2, 
3, &c., wave-lengths, and B successively at distances equal to 1£, 
2£, 3£, &c., wave-lengths. When the resonators are in the 
above positions we will suppose that the phases of vibration 
reaching the membrane are opposed. The following table gives 
the calculations made on the assumption that the intensities 
of the vibrations diminish as the reciprocals of the squares of 
their distances from the sounding bodies: 
A’s dist. in A. 
B’s dist. in 
Ratios of Intensities. 
Residual Kffects. 
1 
1 *5 
•444 
•556 
2 
2-5 
•640 
■360 
3 
3-5 
•734 
•266 
4 
4-5 
•7.-0 
•210 
5 
5-5 
•826 
•174 
6 
65 
•854 
•146 
7 
7*5 
"871 
•129 
8 
8-5 
•885 
•115 
9 
9-5 
•897 
•103 
10 
10-5 
*907 
•093 
11 
11-5 
•914 
•086 
12 
12-5 
•921 
•079 
13 
13-5 
•927 
•073 
24 
245 
*959 
•041 
25 
25-5 
•961 
•039 
We have projected these related numbers in the accompany- 
ing curve, whose abscissas represent the distances of A from 5 
the source of sound, and whose ordinates gives the ratios of in- 
tensities between A, taken at the distances on the axis of 
abscissas, and B at distances from its sounding body always 
one half wave-length greater than A's distance from its sound- 
ing body. The formula of the curve is 
a;2 
V 
If the curve be placed up side down, and referred to the 
corresponding numbers on the abscissas and ordinates (the 
latter being equal to unity minus the numbers at the corres- 
ponding points of the curve when in its first position), we have 
the graphical representation of'the variation of the resultant in- 
tensities, contained in the fourth column of the table. 
In the case of notes of different pitch, giving the same ampli- 
tude of swing to the aerial particles, the higher note will 
necessarily force the air to make its vibrations with a greater 
velocity, and the intensities will therefore not alone depend on 
the amplitudes of these vibrations but also on their velocities, 
and it has been deduced from established principles of dynamics 
that the apparent intensities of notes of different pitch will vary 
directly as the squares of the amplitudes, and inversely as the 
fourth power of the wave-length or periodic time. (See Mr. 
Bosanquet on the Relation between the Energy and Apparent 
Intensity of Sounds of different Pitch, L. E. & 1). Phil. Mag., 
Nov., 1872). Hence the determination of the relative intensities 
of notes of different pitch becomes very complicated, and the ex- 
perimental solution of the problem is encompassed with many 
difficulties. I however hope to be able, at some future day, to 
present some work in this direction when I have succeeded in 6 
obtaining results worthy of the appellation of* measures of 
precision. 
2. Measurement of the powers of various substances to transmit 
and to reflect sonorous vibrations.* 
After we have succeeded in obtaining a measure of the inten- 
sity of the vibrations of the air at a certain distance from the 
sounding body, we can measure the poewrs of various sub- 
stances to transmit, absorb and reflect sonorous vibrations. 
To accomplish this I place one of the sounding bodies in the 
focus of a parabolic reflector and bring the two resonators at 
such distances from their sounding bodies that the intensities 
of the pulses traversing their respective tubes are equal. We 
then place in front of, but not too near, the mouth of the reson- 
ater, in front of the reflector, the plane surface of the substance 
* In the Smithsonian Report for 1857 will be found an account of very inter- 
esting and valuable experiments, by Prof. Joseph Henry, bearing on “ Acoustics 
applied to Public Buildings.” In these investigations, Prof. Henry determined 
the relative powers of various substances to reflect, transmit and absorb sonorous 
vibrations by placing on the bodies the foot of a tuning-fork, and comparing the 
duration of its sound when thus placed with that given when the fork was sus- 
pended in the free air by a fine cambric thread. Thus suspended the fork vibrated 
during 252 seconds. Placed on a large, thin pine board, its vibrations lasted about 
10 seconds. In this case “ the shortness of duration was compensated for by the 
greater intensity of effect produced.” The fork having been placed successively on 
a marble slab, a solid brick wall, and on a wall of lath and plaster, its vibrations 
lasted respectively 115 seconds, 88 seconds, and 18 seconds. 
Placed on a cube of india-rubber, the sound emitted by the fork was scarcely 
greater than when it was suspended from the cambric thread, but its duration was 
only 40 seconds. Here Henry puts the question, what became of the impulses 
lost by the tuning-fork? They were neither transmitted through the india-rubber 
nor given off to the air in the form of sounds; but were probably expended in 
producing a change in the matter of the india-rubber, or were converted into 
heat, or both. Though the inquiry did not fall strictly within the line of this 
series of investigations, yet it was of so interesting a character in a physical point 
of view to determine whether heat was actually produced, that the following ex- 
periment was made. * * * The point of a compound wire formed of copper and 
iron was thrust into the substance of the rubber, while the other ends of the wire 
were connected with a delicate galvanometer. The needle was suffered to come 
to rest, the tuning-fork was then vibrated, and its impulses transmitted to the 
rubber. A very perceptible increase of temperature was the result. The needle 
moved through an arc of from one to two and a half degrees. The experiment 
was varied, and many times repeated; the motions of the needle were always in 
the same direction, namely, in that which was produced when the point of the 
compound wire was heated by momentary contact with the fingers.” We have 
pleasure in again calling attention to this most beautiful experiment of Prof. 
Henry, for he was, I believe, the first to obtain the production of heat on the 
absorption (so to speak) of sonorous vibrations; and although several experimen- 
ters have subsequently obtained the same results, not one of them gives Henry 
credit for antecedent work. In 1868 I published a full account of the above ex- 
periment in my Lecture, Notes on Physics, p. 79. Van Nostrand, N. Y. 
In the same paper Professor Henry obtained a few qualitative relations in the 
reflecting powers of various substances, by placing a watch between the centre 
and focus of a concave mirror; he then receded along the axis of the diverging 
sonorous beam, with a hearing trumpet. Paper and flannel were now stretched 
between the watch and the mirror, and the intensity of the sound was found to 
be diminished by the reflecting and absorbing powers of these substances. 7 
whose transmitting and reflecting powers we would determine. 
Serrations now appear in the flame, because part of the force of 
the pulses which previously sounded the resonator are now 
reflected from the interposed substance. The resonator which 
has not the reflecting surface in front of it is now gradually 
drawn away from its sounding body, and at each successive 
point of remove the pulses propagated through the two resona- 
tor tubes are brought to opposition of phase on reaching the 
membrane by means of the glass telescoping tube. Equality 
of impulses having been obtained, we measure the distance of 
the resonator, which has not the reflecting substance in front 
of it, from the origin of its sounding body, and this measure, 
together with the known previous distance of this resonator, 
when equality was attained before the interposition of the 
reflecting surface, gives the data for the computation of the 
intensity of the transmitted vibration. This number substracted 
from the measure of the intensity when the substance was not 
before the resonator, taken as unity, gives the reflecting power 
of the substance plus its absorbing power. 
It is very important, in such measures, to be sure that a plane 
wave surface is reflected from the mirror. This character of 
wave can be approximately obtained by placing the mouth of 
a closed organ-pipe at or very near the principle focus of the 
mirror and testing, by the method we have described above, 
the equality of intensity of the vibrating air in front of the 
mirror as we recede along its axis. We thus, by trial, at last 
succeed in obtaining a sufficiently plane wave-surface. Care 
must also be taken that the surface of the reflecting substance is 
so large that no inflected vibrations can act on the resonator. 
1 have made several measures of intensity and of transmit- 
ting and reflecting powers, but as the experiments were made 
in a room whose walls, ceiling and floor gave reflected sonorous 
waves, I will not present measures until 1 have arranged suit- 
able apartments for their accurate execution. 
November 13 th, 1872.