Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/133

of this is, that the segments of the plate ${\text{AOD}}$, ${\text{BOC}}$ always vibrate in the same direction, but oppositely to the segments ${\text{AOB}}$, ${\text{DOC}}$. Hence, when the pasteboard is in its place, there are two waves of same phase starting from the two former segments, and reaching the ear after equal distances of transmission through the air, are again in the same phase, and produce on the car a conjunct impression. But when the pasteboard is removed, then there is at the ear opposition of phase between the first and the second pair of waves, and consequently a minimum of sound.

Fig. 31.

97.A tubular piece of wood shaped as in fig. 31, and having a piece of thin membrane stretched over the opening at the top ${\text{C}}$, some dry sand being strewn over the membrane, is so placed over a circular or rectangular vibrating plate, that the ends ${\text{A}}$, ${\text{B}}$ lie over the segments of the plate, such as ${\text{AOD}}$, ${\text{COB}}$ in the previous fig., which are in the same state of motion. The sand at be set in violent movement. But if the same ${\text{A}}$, ${\text{B}}$, be placed over oppositely vibrating segments (such as ${\text{AOD}}$, ${\text{COD}}$), the sand will be scarcely, if at all, affected.

98.If a tuning-fork in vibration be turned round before the ear, four positions will be found in which it will be inaudible, owing to the mutual interference of the oppositely vibrating prongs of the fork. On interposing the hand between the ear and either prong of the fork when in one of those positions, the sound becomes audible, be cause then one of the two interfering waves is cut off from the ear. This experiment may be varied by holding the fork over a glass jar into which water is poured to such a depth that the air-column within reinforces the note of the fork when suitably placed and then turning the fork round.

99.Helmholtz's double syren (§ 51) is well calculated for the investigation of the laws of interference of sound. For this purpose a simple mechanism is found in the instrument, by means of which the fixed upper plate can be turned round and placed in any position relatively to the lower one. If, now, the apparatus be so set that the notes from the upper and lower chest are in unison, the upper fixed plate may be placed in four positions, such as to cause the air-current to be cut off in the one chest at the exact instant when it is freely passing through the other, and vice versa. The two waves, therefore, being in opposite phases, neutralise one another, and the result is a faint sound. On turning round the upper chest into any intermediate position, the intensity of the sound will increase up to a maximum, which occurs when the air in both chests is being admitted and cut off contemporaneously.

100.If two pipes, in exact unison, and furnished with flame manometers, are in communication with the same wind-chest, and the two flames be placed in the same vertical line, on introducing the current from the bellows, we shall find that the two lines of reflected images will be so related that each image in one lies between two images in the other. This shows that the air-vibrations in one pipe are always in an opposite phase to the other, or that condensation is taking place in the one when rarefaction occurs in the other. This arises from the current from the bellows passing alternately into the one and the other pipe. There will also be a remarkable collapse of the sound when both pipes communicate with the wind-chest com pared with that produced from one pipe alone.

101.If the two interfering waves are such as produce vibrations whose numbers per second are $n$, $n^{\prime }$ respectively, these being to each other in the ratio of two integers $m$, $m^{\prime }$ when expressed in its lowest terms, then the lengths of the waves $\lambda$, $\lambda ^{\prime }$ being inversely as $n$ to $n^{\prime }$, will be to each other as $m^{\prime }:m$, and consequently $m\lambda =m^{\prime }\lambda ^{\prime }$. Particles therefore of the air separated by this distance from each other will be in the same phase, that is, the length of the resultant wave will be $m\lambda$ or $m^{\prime }\lambda ^{\prime }$, and if ${\text{N}}$ denote the corresponding number of vibrations ${\text{N}}={\tfrac {n}{m}}$ or ${\tfrac {n^{\prime }}{m^{\prime }}}$.

Thus, for the fundamental and its octave ${\tfrac {n}{n^{\prime }}}={\tfrac {1}{2}}$ and therefore ${\text{N}}=n$ or; ${\tfrac {n}{2}}$; that is, the note of interference is of the same pitch as the fundamental.

For the fundamental and its major third, ${\tfrac {n}{n^{\prime }}}={\tfrac {4}{5}}$. Hence ${\text{N}}={\tfrac {n}{4}}$ or ${\tfrac {n^{\prime }}{5}}$, that is, the resulting sound is two octaves lower than the fundametal.

For the fundamental and its major sixth, ${\tfrac {n}{n^{\prime }}}={\tfrac {3}{5}}$; ${\text{N}}$ therefore $={\tfrac {n}{3}}$ or ${\tfrac {n^{\prime }}{5}}$, and the resulting sound is a twelfth below the lower of the two interfering notes.

If $m$ and $m^{\prime }$ differ by $1$, then ${\text{N}}=n-n^{\prime }$; for $m-m^{\prime }$ or $1={\tfrac {n}{\text{N}}}-{\tfrac {n^{1}}{\text{N}}}$. Hence, if the ratio of the vibrations of two interfering sounds is expressible in its lowest terms by numbers whose difference is unity, the resulting note has a number of vibrations simply equal to the difference of those of the interfering notes.

The results stated in this section may be tested on a harmonium. Thus, if the notes ${\text{B}}$, ${\text{C}}$, at the extreme right of the instrument be struck together, there will be heard an interference note four octaves lower in pitch than the above ${\text{C}}$, because the interval in question being a semi tone, is ${\tfrac {16}{15}}$, and, consequently, by last case, the interference note is lower than the ${\text{C}}$ by interval ${\tfrac {1}{16}}$.

Other notes may be heard resulting from the mutual interference of the overtones.

102.When two notes are not quite in tune, the resulting sound is found to alternate between a maximum and mini mum of loudness recurring periodically. To these periodical alternations has been given the name of Beats. Their origin is easily explicable. Suppose the two notes to correspond to $200$ and $203$ vibrations per second; at some instant of time, the air-particles, through which the waves are passing, will be similarly displaced by both, and consequently the joint effect will be a sound of some intensity. But, after this, the first or less rapidly vibrating note will fall behind the other, and cause a diminution in the joint displacements of the particles, till, after the lapse of ${\tfrac {1}{6}}$ of a second, it will have fallen behind the other by ${\tfrac {1}{2}}$ a vibration. At this moment, therefore, opposite displacements will be produced of the air-particles by the two notes, and the sound due to them will be at a minimum. This will be followed by an increase of intensity until the lapse of another sixth of a second, when the less rapidly vibrating note will have lost another half-vibration relatively to the other, or one vibration reckoning from the original period of time, and the two component vibrations will again conspire and reproduce a maximum effect. Thus, an interval of ${\tfrac {1}{3}}$ of a second elapses between two successive maxima or beats, and there are produced three beats per second. By similar reasoning it may be shown that the number of beats per second is always equal to the difference between the numbers of vibrations in the same time corresponding to the two interfering notes. The more, therefore, these are out of tune, the more rapidly will the beats follow each other.

Beats are also heard, though less distinctly, when other concords such as thirds, fifths, &c., are not perfectly in tune; thus, $200$ vibrations and $303$ vibrations per second, which form, in combination, an imperfect fifth, produce beats occurring at the rate of three per second.