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ACOUSTICS Now from equation (3) in the investigation of the velocity of wave length X. The successive terms of (1) are called the harsound {supra), we see that monics of the first term. _ It follows from this that any periodic disturbance in air can be resolved into a definite series of simple harwhile putting V=1 monic disturbances of wave lengths equal to the original v— - dy -Z-. wave length and its successive submultiples, and each of dx Inserting these values and integrating the potential energy be- these would separately give the sensation of a pure tone. If the series were complete, we should have terms which tween # = 0 and x = , we have separately would correspond to the fundamental, its octave, its twelfth, its double octave, and so on. Now we can see that two notes of the same pitch, but of different But differentiating (1) above quality, or different form of displacement curve, will, when d JL^COs2* {x-Kt thus analysed, break up into series having the same hardx
monic wave lengths; but they may differ as regards the and as this has equal + and - values over a length x = the first members of the series present and their amplitudes and integral = 0. The second gives on integration epochs. We may regard quality, then, as determined by ir2 E a2 the members of the harmonic series present and their X ‘ amplitudes and epochs. It may, however, be stated here 2 Then, since E = pJJ , the potential energy in one wave length, that certain experiments of Helmholtz appear to show that cross section 1, the epoch of the harmonics has not much effect on the _ p7T2U2<22 quality. The total energy, kinetic + potential, in a length X of cross section Fourier’s theorem can also be usefully applied to the disturbance of a source of sound under certain conditions. The nature 1 is equally divided between the two kinds, and is of these conditions will be best realized by considering the case 2p7r2U2a2 of a stretched string. It is shown, in 0. A. § 55, how the vibrations X of a string may be deduced from stationary waves. Let us here If we regard this energy as travelling on with velocity U, the suppose that the string AB is displaced into the form AHB (Fig. quantity passing across one sq. cm. in one second is U/X times 8). Then let us imagine it to form half a wave length of the as great, or extended train ZGAHBKC, on an indefinitely extended stretched string, the values of y it equal distances from A (or from B) being 2/)7r2U3a2 2 equal and opposite. Then we may suppose the vibrations of the X string to be represented by the travelling of two trains in opposite This gives the measure of the intensity of the sound on the directions each with velocity supposition that loudness or intensity is to be measured by V t ension-^mass per unit length energy received per sq. cm. per second. From the values of the particle velocity and the pressure excess, it is easy to express the each half the height of the train represented in Fig. 8. For intensity in terms of these quantities. the superposition of these trains will give a stationary wave Any periodic curve may be resolved into sine or harmonic between A and B. Now we may resolve these trains bv Fourier’s theorem into harmonics of wave lengths X Fourier’s curves U Fourier’s theorem. 2’ 3’ etC'’ theorem. Suppose that any periodic sound disturbance, where X = 2AB and the condition as to the values of yean be consisting of plane waves, is being propagated shown to require that the harmonics shall all have nodes, coincid in the direction ABCD, Fig. 7. Let it be represented by ing with the nodes of the fundah mental curve. Since the velocity is the same for all disturbances they all Fig. 8. travel at the same speed, and the two trains will always remain of the same form. If then we resolve AHBKC into harmonics by Fourier’s theorem, a displacement curve AHBKC. Its periodicity implies we may follow the motion of the separate harmonics, and their that after a certain distance the displacement curve exactly superposition will give the form of the string at any instant. repeats itself. Let AC be the shortest distance after Further, the same harmonics with the same amplitude will which the repetition occurs, so that CLDME is merely always be present. AHBKC moved on a distance AC. Then AC = A is We see, then, that the conditions for the application the wave length or period of the curve. Let ABCD be of Fourier’s theorem are equivalent to saying that all drawn at such level that the areas above and below it are equal; then ABCD is the axis of the curve. Since disturbances will travel along the system with the same the curve represents a longitudinal disturbance in air it is velocity. In many vibrating systems this does not hold, always continuous, at a finite distance from the axis, and then Fourier’s theorem is no longer an appropriate resolution. But where it is appropriate, the disturbance and with only one ordinate for each abscissa. Fourier s theorem asserts that such a curve may be built up by sent out into the air contains the same harmonic series as the superposition, or addition of ordinates, of a series of sine the source. The question now arises whether the sensation produced curves of wave lengths X * * J ... if the amplitudes a, h, c by a periodic disturbance can be analysed in correspond. . . and the epochs e, fi g... are suitably adjusted, and the proof ence with this geometrical analysis. Using the term of the theorem gives rules for finding these quantities when the note for the sound produced by a periodic disturbance, original curve is known. We may therefore put there is no doubt that a well-trained ear can resolve a y = a sin ^{x-e) +6 sin ^{x-f) +c sin (a;-y) + etc. (1) note into pure tones of frequencies equal to those of the fundamental and its harmonics. If, for instance, a note is where the terms may be infinite in number, but always have struck and held down on a piano, a little practice enables wave lengths submultiples of the original or fundamental wave us to hear both the octave and the twelfth with the fundalength X. Only one such resolution of a given periodic curve is possible, and each of the constituents repeats itself not only in mental, especially if we have previously directed our its own wave length , but also evidently in the fundamental attention to these tones by sounding them. But the
