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FLUE-PIPES AND REEDS.                       107
The next simplest way in which the air in a stopped pipe can vibrate, must be that in which two nodes are formed, and these must necessarily occur as shown in B, fig. 56, where the end of the pipe, as we have seen, forms one node, the place of the other being represented by the vertical line. In order to understand the formation of a node at this point, let fig. 57 represent a stopped pipe, and let ab, be, cd, be each one-third of its length. Further, let it be supposed, that the pulses of condensation and rarefaction successively enter the open end, at inter­vals of time, each equal to that required for the pulse to travel from (a) to (c), that is, through two-thirds of the length of the tube. For the sake of simplicity, we will suppose, that the interval of time is one second, although of course it is really but a minute fraction of that period. First let a pulse of condensation C| enter the pipe. After the lapse of a second, that is at the beginning of the 2nd second, C| will be at (c) and the succeeding pulse of rare­faction E| will be just entering at (a). Neglecting E, for the present, let us see where C( will be at the beginning of the 3rd second; it will evidently have travelled through (cd) and back, and in fact will be at (c) again, but moving upwards. But by the supposition, another pulse of condensation C2 is now entering the tube at (a) and therefore moving downwards. These two equal pulses of condensation will meet at (b) and the air particles here being solicited by C( to move upwards, and by Ca to move downwards, will remain at rest. To return now to the pulse of rarefaction B(, which at the beginning of the 2nd second was entering the tube at (a): at the beginning of the 3id second it will be at (c), moving downwards: at the beginning of the 4th second it will be again at (c), but moving upwards. But, by the supposition, another pulse of rarefaction R2 is now entering at (a). These two equal pulses will meet at (b), and the air particles there being solicited by B( to move downwards, and by B2 to move upwards, will remain at rest. Thus the particles at (b) will be permanently at rest, that is, (b) will be a node. It will be noted, that it is per­fectly allowable to consider, as we have done, the pulses of con­densation and rarefaction separately; for we have already seen that two series of waves can cross, without permanently interfering with one another. It will be instructive, however, to consider the com­bined effects of the pulses of condensation C| and rarefaction R1, at the end of the 2nd second, or what is the same thing, at the