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Volume 35, 1902
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Art. LIII. — On the Interpretation of Milne Earthquake Diagram.

[Read before the Philosophical Institute of Canterbury, 26th November, 1902.]

Plate L.

The question as to whether a horizontal pendulum seismograph acts as a clinograph, or whether its records must in part be ascribed to horizontal movement of the earth's surface, has received discussion by Milne,* Ōmori, and others, whose

[Footnote] * Nature, vol. lxv., p. 202, and B.A. Reports.

[Footnote] † Publications of the Earthquakes Investigation Committee, No. 5, Tokyo, 1900, p. 45 et seq.

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arguments I have not been able to peruse. But Professor Milne and Dr. Ōmori conclude that the tilts represented by the maximum displacement of the boom are too large to be admissible as tilts; and Ōmori discusses the accelerations of the earth-particles of four earthquakes which would result from the assumption that the maximum boom-movement was due to a series of waves of vertical displacement passing under the pillar of the instrument.

The following considerations do not appear to me to have been sufficiently realised:—

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The differential equation representing the motion of a body capable of free vibration of frequency n, but acted upon by a periodic force E cos pt, as d2u/dt2 + Kdu/dt + n2u = E cos pt*

where K is the constant of delay in the free vibration.

The solution of this is

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u = ✓ E/(n2p2)2 + K2p2 cos (pt − c)

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where tan C = pK/n2p2

This equation applies to seismograph of the Milne type and bodies also, as well as to other vibrating bodies.

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The result shows (1) that the vibrating body, in this case the boom of a seismograph, no longer vibrates in its natural period 2π/n, but takes the frequency of the disturbing force p; (2) that if friction be small compared with the difference of the squares of the frequencies, the resulting vibration has an amplitude E/n2p2; (3) the phenomenon of beats may occur between the forced vibration and the free period of the boom.

Considering the second conclusion first, it is evident that the maximum amplitude of swing of the boom gives no in-formation whatever of the amplitude of the disturbing cause, without also a knowledge of the periods of the free and forced vibrations. It does not follow, as appears to be supposed, that the maximum amplitude of swing of the boom is associated with the maximum amplitude of the disturbing cause. Of waves of equal amplitude but different wave-lengths those nearest in period to the free period will give the largest trace, and if p and n become equal the only thing which prevents the swinging of the boom from eventually becoming infinite is the term K2p2, which in this case must be considered, though it is very often small enough to be neg-

[Footnote] * Rayleigh's “Sound,” p. 38, 1st ed.

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lected in other cases. It is thus quite an erroneous proceeding to take the most marked phase of an earthquake diagram and to assume that this corresponds to the waves of largest amplitude in the earth; and it is also quite erroneous to derive the amplitude of the earth-wave by multiplying the amplitude of the trace by the conversion factor from millimeters to seconds of arc. Regard must be paid to the period of the earth-wave and of the free boom-vibration. In the tabulation of records of Milne seismographs no regard is at present paid to the period of the earth-wave, and hardly more to the period of free boom-vibration, and no information is supplied of the value of the constant K, which, though often negligible (i.e., when p and n are not very nearly equal), is occasionally of paramount importance. This constant can be deduced from the fact that the free vibration dies away according to an expression of the form

u = Ae — ½kt cos θ

Turning now to the result (1), that the forced vibration takes place in the period of the disturbing cause, this enables us to derive a value both of the periods and wave-lengths of the various sets of waves acting on the boom; and the record is thus capable of supplying the information absolutely essential for the correct valuation of the amplitude E of the disturbing cause, from which can be easily derived the total movements and accelerations of the earth-particles at any phase of the earthquake. But to do this the tape must be driven at a much higher rate than is at present the case. It is difficult to see the individual vibration, and it is impossible to estimate the time of vibration to within an accuracy of a second, as is certainly necessary for the correct determination of E, with the tape moving at the rate of only 1 mm. per minute.

The phenomenon of interference I have many examples of in earthquakes I have recorded; and what Professor Milne* describes as “earthquake echoes” may be explained in this way, combined very often with increasing or decreasing amplitude of the disturbing cause.

To give ocular demonstration of the truth of these theoretical conclusions, I decided to attempt to imitate a series of waves acting on the pillar, and of known period. For this purpose I had two boxes attached, one to the east and the other to the west of the pillar, and in these I placed sawdust. Depending from the roof-ceiling of the room, by a rope passing over two pulleys fixed vertically above the centres of the two boxes, were two chains. The length of the rope was such that the two chains just touched the two beds of sawdust together. By pulling these chains up and down at definite

[Footnote] * B.A. Reports, 1899, p. 288.

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rates a periodic tilting of the pillar, due to the loading of one side or the other by its proper chain and by more or less of it, was obtained bearing some resemblance to a sine curve, and of known period. Owing to an error of cutting the chains they were not of equal weight, one being 17½ Ib. and the other 21 Ib.,* but as in any case the resemblance to a sine curve was only a rough one, and as also the lighter chain acted upon the pillar at a somewhat greater area than the other, it was not considered worth while to alter it.

On removing one chain from its box and placing the other in its proper box the total boom-movement was 1·6 mm. I then proceeded to imitate in succession waves of 12 sec., 13 sec., 14 sec., 15 sec., 16 sec., 17 sec., 18 sec., 19 sec., and 20 sec. period, whilst the boom period throughout remained as nearly as I could determine at 16·5 sec. Before doing so I had increased the speed of the tape to 82 mm. per hour, so that the time-scale might be sufficiently open to enable me to count the individual vibrations of the boom. The various artificial seismograms thus obtained are appended. (Plate L.)

[Footnote] * I am indebted to the kindness of Messrs. E. Reece and Sons, of Christchurch, not only for lending me the chains, but for very courteously cutting lengths suitable for the purpose 1 had in view.

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In every case conclusion 1—viz., that the boom no longer vibrates in its natural free period, but adopts the period of the forcing cause—is exactly verified. The amplitude of the resulting swings is in every case larger than that of the static displacement of ½·6 mm., as with the period it theoretically should be, and as the synchronism becomes more perfect the boom-swing becomes larger; but I want to emphasize that, whatever has been the period of the forcing cause within the limits adopted, synchronism more or less complete is apparent. In no case, however, is the swing of the boom so great as it theoretically should be, supposing E to be 0·8 mm. This may be in part due to the imperfect imitation of a sine curve which resulted from the arrangement adopted, and there is no reason to conclude that because the tilt of the pillar due to the static alteration of a given disposal of weight is E, the displacement of the pillar due to a periodic alteration of this disposal will also be E.

Interference effects are seen in most of the artificial seismo-grams. Where they are absent the periods are so close that the free vibration has been damped down to comparative insignificance before opposition of place would occur. It appears to be probable, although the period of free vibration was determined as 16·5 sec., that its accurate value was more nearly 16·6 sec. It is difficult to determine the quantity accurately to 0–1 sec., and yet the result shows that it is important to do so.

As a result of this examination it appears—(1) that strict attention should be paid to accurately recording the period of free vibration; (2) that the tape should be driven at such a speed as will enable the period of forced vibration to be determined; (3) that the value of the constant K should be recorded.

We might then hope, by determining the amplitude of the earth-movement of any particular waves of given wave-length at different stations, to ascertain the law which governed the decrease of intensity with distance, and to determine how the velocity of the waves varied with their length.

As I am unacquainted with the practical details of Dr. Ōmori's instrument I have refrained from discussing the acceleration of the earth-particles given by him for certain earthquakes;* but if the considerations above set forth apply to his instrument, and if also they have been omitted in arriving at the results he gives, then by taking them into account the values of the accelerations would be so much reduced that it is improbable we should have felt the earthquakes he discusses.

[Footnote] * Publication of the Earthquakes Investigation Committee, No. 5, Tokyo, 1901, p. 45 et seq.