In the preceding theory all the parts of any section of the pipe per pendicular to its axis 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 pasta 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 en orifice is made In the side of a pipe, as in the flute, it is 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 Its octave may be obtained by an orifice of one size or another made at or near the middle of a pipe.
We have seen that we may suppose the extremities of the open pipe to contain between them 2, 3, &c., half-waves, which. the whole pipe being one half-wave in length, will give the FIARMONICS of the fundamental note. This subject is sufficiently treated In the article cited.
Various Instruments yield different harmonies 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 : also that a certain diameter, the larger the greater the length of the tube, is necessary to the produc tion of the fundamental note. Thus, if an 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 viicing, 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 in modern instruments), the lower notes will be difficult to obtain. And the various harmonica are produced with very different degrees of facility; a circumstance of which the theory oan give no account. Thus, players on the trumpet find it exceedingly difficult 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 is 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, there will be n + I 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. Renee 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 waves 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 ie the reason why the pipes of the stopped diapason stop of an organ are 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 closed at one end are 3, 5, 7, &a. 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, is that it is 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 one end (as the spring of a tuning-fork) answer to those of a pipe closed et one end, since their law is very different.