Page:Elementary Text-book of Physics (Anthony, 1897).djvu/185

§144] change of sign at the closed end, and returning, are reflected without change of sign at the open end. Any given wave $$a$$, therefore, starts up the tube the second time with its phase changed by half a period. The direct wave that starts up the tube at the same instant must be in the same phase as the reflected wave, and it therefore differs in phase half a period from the direct wave $$a$$. In other words, any wave returning to the mouthpiece must find the vibrations there opposite in phase to those which existed when it left. This is possible only when the vibrating body makes, during the time the wave is going up the tube and back, 1, 3, 5, or some odd number of half-vibrations. By constructing the curves representing the stationary wave resulting from the superposition of the two systems of vibrations, it will be seen that there is always a node at the closed end of the tube and an anti-node at the mouth. When there is 1 half-vibration while the wave travels up and back, the length of the tube is $$\tfrac{1}{4}$$ the wave length; when there are 3 half-vibrations in the same time, the length of the tube is $$\tfrac{3}{4}$$ the wave length, and there is a node at one third the length of the tube from the mouth.

If the tube be open at both ends, reflection without change of sign takes place in both cases, and the reflected wave starts up the tube the second time in the same phase as at first. The vibrations must therefore be so timed that 1,2, 3, 4, or some whole number of complete vibrations are performed while the wave travels up the tube and back. A construction of the curve representing the stationary wave in this case will show, for the smallest number of vibrations, a node in the middle of the tube and an anti-node at each end. The length of the tube is therefore $$\tfrac{1}{2}$$ the wave length for this rate of vibration. The vibration numbers of the several tones produced by an open tube are evidently in the ratio of the series of whole numbers 1, 2, 3, 4, etc., while for the closed tube only those tones can be produced of which the vibration numbers are in the ratio of the series of odd numbers 1, 3, 5, etc. It is evident also that the lowest tone of the closed tube is an octave lower than that of the open tube of the same length.