MJ 



PIPE. 



PIPE. 



C34 



We shall first consider the pipe in a state of continued sonorous 

 vibration (no matter how produced), yielding the lowest note which it 

 will give : let it be a simple pipe open at both ends, and let it be sounding, 

 say the of the treble clef, which note requires 258 double vibrations 

 per second. If we now remember that the air at the two extremities 

 is in communication with the outer ah-, we see that no condensation or 

 rarefaction can take place at those extremities, or only very small ones 

 compared with those which take place in the interior of the tube. To 

 get approximately at the conditions of vibration, let us suppose that 

 no condensation or rarefaction takes place at the extremities. \Ve 

 then see [ACOUSTICS] that the state of the pipe, its two extremities 



interior being affected by the joint 

 both. Moreover, the distance between two uncondensed particles ia 

 always the whole length of the wave of condensation or that of rare- 

 faction, or a multiple of this length ; that is, the pipe must be either 

 the half-length of a double wave or a multiple of this half-length. 

 When the pipe sounds the lowest note, it must give the longest wave ; 

 that is, the length of the pipe must be that of the simple wave of con- 

 densation or rarefaction. Hence, the lowest note which a pipe can 

 yield, which is called its fundamental note, is that belonging to a double 

 wave of sound which is double of its length. Each double wave 

 answers to a complete or double vibration of a string. 



To compare this result with practice, let us suppose sound to 

 travel at the rate of 1125 feet per second (temperature t>2" Fahr.). The 

 note c having 258 double vibrations per second, this 1125 feet must 

 contain 258 double waves, or each double wave must be 4 36 feet. The 

 single wave then is 2'18 feet, or 2 feet 2 inches and '16 of an inch, 

 which is the theoretical length of the pipe. Now the organ-builders 

 say 2 feet [ORGAN, CONSTRUCTION OF], but this of course is a rough 

 description, since the French organ-builders also say 2 feet (according 

 to Biot), and the French foot is longer than the English. Further on 

 in the article referred to we see 2 feet 2 inches given as the length of 

 this o in an open pipe (the dulciana), and 1 foot 1 inch in a stopped 

 pipe (the stopped diapason), which, as we shall presently see, ought to 

 be half as long as an open pipe. The common flute, when everything 

 is stopped, gives this same c, and the length from the embouchure (or 

 mouth-hole) to the end of the instrument is a little more than 2 feet, 

 but certainly never 2 feet 2 inches. It must be remembered however 

 that this instrument is made up of the flute (so called) and the player, 

 whose lips, when they come over the embouchure, confine the air, and 

 are equivalent to a slight lengthening of the pipe. It is not the man- 

 ner of blowing which does this, but the approach of the lips, as may 

 bo tlni.-i shown. Take a common flute, and without holding it to the 

 lipa, atrike the uppermost hole with the linger ; a faint sound will be 

 heard. Now approach the lipa to the embouchure, but without blow- 

 ing, and then strike the same hole with the finger; another faint 

 sound will be heard, decidedly flatter than the former. It is well 

 known to those who play on this instrument (to those who play in tune 

 at least) that drawing the lips back, so as not so much to confine the 

 air contiguous to the embouchure, sharpens the tone, and what some 

 persons call humouring the instrument means continual alteration of 

 the position of the lips, so at to shorten or lengthen the pipe by turns, 

 according to the note to be sounded. It is also well known to players 

 that this humouring can be carried to a much greater extent with the 

 high notes than with the low notes ; but so little were the practical 

 musicians in connection with the theoretical in the time of Daniel 

 Bernoulli (who first gave the mathematical theory of this subject), 

 that this simple fact was only discovered by him from a new and some- 

 what complicated experiment. 



In the preceding theory all the parts of any section of the pipe per- 

 pendicular to ita axia are supposed to vibrate in the same manner. 

 This cannot be the case in the common flute or in the organ-pipe, 

 in which the cause of condensation is supplied at tfie side ; and in fact 

 all experiments in which the cause of undulation has been equally 

 applied over all the parU of a section perpendicular to the axis, have 

 agreed in the result that the time of vibration is wholly independent of 

 the diameter of the tube : while those in which the same was not 

 equally applied give the result that the greater the diameter the lower 

 is the tone. Moreover, when an orifice ia made in the side of a pipe, 

 aa in the flute, it ia not equivalent to the formation of a new pipe ter- 

 minating at that orifice, though the results are somewhat resembling. 

 Any note between the fundamental note and ita octave may be 

 obtained by an orifice of one size or another made at or near the 

 middle of a pipe. 



We have aeen that we may suppose the extremities of the open pipe 

 to contain between them 3, 8, ftc., half-wavea, which, the whole 

 pipe being one half-wave in length, will give the HAHMOSICS of the 

 fundamental note. This subject ia sufficiently treated in the article 

 cited. 



Various instruments yield different harmonica more or less readily ; 

 the general rule being that the more violent the agitation which pro- 

 duces the sound, the larger the number of half-wavea formed in the 

 tube, and the higher the harmonic : alao that a certain diameter, the 

 larger the greater the length of the tube, in necemary to the produc- 

 [' the fundamental note. Time, if nu organ-pipe be too small in 

 the bore, it will yield the octave of the fundamental note ; or if the 



latter, only with great attention to the voicing, or adjustment of the 

 orifice through which the wind enters. If the bore of a flute ba 

 too narrow (which we imagine to be the case ill modern iusta-uments), 

 the lower notes will be difficult to obtain. And the various harmonica 

 are produced with very different degress of facility; a circumstance of 

 which the theory can give no account. Thus, players on tha trumpet 

 find it exceedingly diincult to produce that tone which divides the 

 instrument into seven parts, or the flat seventh in the third octave 

 above the fundamental note ; while in the flute there ia no moderately 

 skilful player who cannot produce it. It is to be observed however 

 that all pipes of the trumpet class are of tapering diameter ; and 

 though they agree in all material points with the theory of cylindrical 

 and prismatic pipes, it is not remarkable, in the present state of the 

 mathematical analysis of this subject^that they should present circum- 

 stances difficult of explanation. 



It will be obvious, from the considerations in ACOUSTICS, that when 

 the extremities of the pipe contain between them n half-waves, thers 

 will be + 1 points (the orifices included) at which the velocities are 

 always greater than elsewhere, and no condensations or rarefactions ; 

 and n points (in the middle of the subdivisions), at which the condensa- 

 tions or rarefactions are always greater than elsewhere, and which are 

 always at rest or nearly so. These immoveable points are called nodes 

 of vibration ; and there is one of them in the middle of the tube only 

 when the number of half-waves in the pipe is odd. 



Let us consider the case of a pipe with one end closed. It is obvious 

 now that the open extremity is a point of no condensation, while the 

 closed extremity must be a node, or point of no velocity. Henoe the 

 tube must be the half of an odd number of simple waves in length, 

 twice the tube must be an odd number of simple waves, and four 

 times the tube an odd number of double wavea in length. Hence 

 the fundamental note belongs to a double wave of four times the 

 length of the tube ; so that the fundamental note of a pipe closed 

 at one end is an octave lower than that of the same pipe open at 

 both ends. It is the same thing to say that a pipe of half the length 

 of an open pipe, closed at one end, gives the same note as the open 

 pipe. This is the reason why the pipes of the stopped diapason 

 stop of an organ ore halves of the lengths of those of the open 

 diapason. 



Again, since the double length of the pipe is an odd number of 

 simple waves, the harmonics which the pipe can yield are not the com- 

 plete set yielded by the open pipe of double the length, but every 

 other one, beginning from the fundamental note. The number of 

 vibrations per second being 1, those of the harmonics producible by 

 the pipe cloaed at one end are 3, 5, 7, &o. We will leave the pipe 

 closed at both ends (a matter of no practical concern, since its sound 

 could not be heard) to the student ; the result he should arrive at by 

 the preceding considerations, ia that it ia in all respects analogous to 

 the vibrating CORD fixed at both ends. But he must not infer, by a 

 reversed analogy, that the vibrations of an elastic body fixed at on 

 end (aa tha spring of a tuning-fork) answer to those of a pipe closed at 

 one end, since their law is very different. 



It ia usual first to give the theory of a closed pipe, and then to 

 suppose the open pipe made of two closed pipes, with their closed enda 

 together, and their closing diaphragms removed. The opposition of 

 the vibrating movements will then keep the particles in the middle at 

 rest. This ia a sufficient explanation of those modes of vibration of 

 the open pipe In which there is a node in the middle. 



We now come to the explanation of the manner in which the 

 sonorous vibration of a pipe is maintained. If we suppose a vibrating 

 body placed at the orifice, it is found that if the vibrations of the 

 body be equal or nearly equal to those of the fundamental note of the 

 tube in the preceding theory, or one of its harmonics, the sound of the 

 vibrating body ia reinforced by the tube. A slight alteration of the 

 tube, though it may sharpen or flatten the note, does not by any means 

 produce auoh a difference aa would be caused by the same alteration, 

 if the sound were caused by the tube alone. We do not intend to go 

 into this subject ; the reader may find it discussed, both mathematically 

 and experimentally, in a paper by Mr. Hopkins, published iu tho 

 fifth volume of the ' Transactions of the Cambridge 1'hilosophical 

 Society." 



Wlifn the sound is caused by a current of air, as in the common 

 flute or simple organ pipe, a tolerably satisfactory explanation of the 

 phenomena haa been given in the case of the pipe closed at one end (to 

 which writers have confined themselves) ; but none whatever in that 

 of the pipe which is open at both ends. In the former case, as in a 

 reed of the Pan's pipe, a current of air is directed laterally over the 

 mouth of the pipe, with a slight obliquity of direction. A condensa- 

 tion is therefore produced in the tube, which travels to the closed end, 

 and is there reflected; so that by the time the condensation has 

 travelled over twice the length of the tube (down and back again), the 

 whole condensation, such aa it was when it began, is doubled. Henca 

 the air in the tube haa now become more powerful than the external 

 stream, and the condensed portion begins to be discharger!. This 

 continues until not only the whole of the condensation is discharged, 

 but also until all the velocity of the issuing particles has been destroyed; 

 and this ia not done until the effect of that velocity has produced a 

 rarefaction in the tube. The e0ect of the condensation is destroyed in 

 the same time as that in which it was prodnced ; and hence the com- 



