velocity. The pressure at an antinode is the same as that of the undisturbed air; hence, if a hole be made in the side of the pipe at an antinode, there will be no tendency for air to pass through the hole either way, and the state of things within the pipe will remain unaffected. If the pipe be cut across at an antinode, and one of the two portions removed, the vibrations in the remaining portion will go on as before.

Conversely, when waves travelling along a pipe arrive at an open end, a state of things is produced in the pipe which is the same as if a system of waves were entering the pipe at this end, and producing jointly with the incident waves an antinode at the open end. The reflected waves must therefore in this case be copies of the incident waves with disturbance of density reversed. A pulse of compression will yield a reflected pulse of rarefaction, and a pulse of rarefaction will yield a reflected pulse of compression.

107. Echo is a familiar example of the reflection of sonorous undulations.

We may mention, as illustrating both kinds of reflection, that there is a well at Kentish Town, belonging to the New River Company, where an eight-inch iron pipe descends from a little above the ground to some hundreds of feet, and the water stands in it at a depth of rather more than 200 feet from the top. Words spoken into the mouth of this tube are very distinctly echoed from the surface of the water, and if spoken loudly they are

echoed more than once. A word loudly shouted is repeated about seven times, becoming gradually feebler with each repetition. The explanation is that the sonorous waves are reflected backwards and forwards, between the surface of the water below and the open end of the tube above. To produce one echo, they must travel once down the tube and up again; to produce two echoes they must travel over twice this distance, and so on. When the reflected waves reach the open end of the tube they are reflected down, and then again reflected up from the surface of the water,

108. Resonance is another example of the reflection of sonorous undulations.

When a vibrating tuning-fork is held at the mouth of a tube of proper length, the sound is greatly intensified by the resonance of the tube. If the tube is open at the far end, this effect will be obtained when its length is about half the wave-length of the note of the fork; for every pulse originated by the fork is reflected from the far end with reversal of condensations and extensions (which we shall call, for shortness, reversal of form), and after travelling back to the near end is again reflected with a second reversal, which restores it to its original form. If the time occupied in this process (that is, the time of travelling over twice the length of the tube) is equal to the period of vibration of the fork, the next pulse from the fork will exactly concur with the reflected pulse, and their amplitudes will be added. As each

original pulse gives rise to a long series of reflections, a great number of amplitudes will be added together, if the length of the tube is such as to make the coincidence of period exact.

If the tube is closed at the far end, the pulses will have to travel four times over its length in order to be restored by two reversals to their original form. The tube will therefore respond if its length is one-fourth of the wave-length of the note emitted by the fork.

These are the shortest lengths that will suffice in the two cases. Resonance will also be obtained when the open tube is any multiple and the stopped tube any odd multiple of the shortest length, as will appear on tracing the successive reflections in each case.

109. Reflection such as we have here described takes place in organ pipes and wind instruments generally. From each end a reflected undulation is continually flowing through the pipe, and the combination of these two undulations travelling in opposite directions produces a stationary undulation, according to the principles of Chapter V. If the pipe is 'stopped,' there is a node at the stopped end; if it is open, there is an antinode at the open end; and in both cases there is an antinode at the end where the wind enters, which is always to a certain extent open.

The notes to which a pipe can respond are the same as those which it is fitted to yield. The lowest of these (which is the note that it is always made to yield in the

organ) is called its first or fundamental tone. The others are called its overtones. Their respective wave-lengths are most easily deduced from the following considerations.

1. At an open end there must always be an antinode, and at a stopped end a node.

2. The distance between a node and the nearest antinode is a quarter of a wave-length, and the distance between two consecutive nodes or two consecutive antinodes is therefore half a wave-length.

110. From these principles it follows that an open pipe must contain an even number, and a stopped pipe an odd number of quarter waves, so that if / denote the length of a pipe, and a the wave-length of one of its tones, we have, for an open pipe,

From these values it is easy to show that the values of λ are proportional to 1,,, &c. for an open pipe, and to I,,, &c. for a stopped pipe; whence it follows that the number of vibrations per second is proportional to 1, 2, 3, &c. for an open pipe, and to 1, 3, 5, &c. for a stopped pipe.

III. These statements would be exact if the air in the pipe vibrated in parallel plane layers, so that the motion

of all particles in the same cross section was the same, and was parallel to the length of the pipe. The actual wavelengths are rather greater, and the actual numbers of vibrations consequently rather less than the above calculations would make them. The pitch of the overtones is more affected by this correction than the pitch of the fundamental; so that, for example, the second tone of an open organ-pipe (especially if the pipe is wide in proportion to its length) has not quite double the number of

vibrations of the first.

112. The overtones of a musical string follow the same laws as those of an open organ-pipe.

When a pulse, consisting of a protuberance on one side of a string, runs along it, the particles of the string are drawn to this side as the protuberance reaches them, and return to their original position as it leaves them and passes on. On its arrival at one of the fixed ends of the string, it is unable to draw the fixed support to one side, and the additional resistance produces a rebound, throwing the protuberance over to the other side, and starting a reversed pulse, which travels along the string from this end to the other, where it is again reflected and reversed. The two portions of Fig. 42 will explain what is here One of them (it is immaterial which) shows the original, and the other the reflected pulse. Wherever we suppose the pulse to be at a given moment, it will have travelled over twice the length of the string before it comes back to its original position and circumstances.

meant.