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Art. LX.—*Notes on the Comparison of some Elements of Earthquake Motion as observed in New Zealand, with their Theoretic Values*.

[*Read before the Philosophical Institute of Canterbury, 2nd November, 1898*.]

The complete mathematical discussion of earthquake movement-implies almost the whole range of the theory of elastic solids, and this involves twenty-one constants, with equations connecting them and their functions. A universal solution is impossible, but some very general solutions of particular cases have been obtained, and these can be applied in part to earthquake motion. This is particularly true in regard to moderate earthquakes, or to the motion at some distance from the origin.

One very valuable law, for instance, discovered and proved by Lord Kelvin, is that the most general form of small strain may be resolved into two independent small strains, one of which contributes dilatation and distortion without rotation,

and the other distortion and rotation without dilatation.^{*} That is to say, we may discuss the normal and transverse vibrations separately with some hope of getting an approximation to the actual facts of earthquake motion. If we consider the normal vibrations of an earth-particle at some distance from the origin of disturbance we may take the mode of motion to be wholly irrotational, and it has been shown ^{†} that the displacement potential *φ* then satisfies the equation

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Ω^{2}∇^{2} *φ* + *i*^{2} *φ* = *o* (i.)

Ω being independent of *i*, and, as will appear presently, equal to the velocity of propagation; and *i*^{2} a function of the initial impulse and of the elasticity moduli and the density of the system of particles for the series of vibrations in question.

It is also true that the form of this equation (and hence also the form of its solution) remains unchanged whatever be the value of *i* for any particular series (provided that the series be homogeneous and. isotropic). Now, in the case under consideration we may choose the axis of *x* in the line joining the origin and the particle considered, and neglect, all vibrations except those parallel to the axis of *x*—-in other words, if *u, v, w* be displacements parallel to the. co-ordinate axes, we may put

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*u* = *d* *φ*/*dx*, *v* = *o*, *w* = *o*.

So that *φ* is a function of *x* only, and equation (i) reduces to

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Ω^{2} *d*^{2} *φ*/*dx*^{2} + *i*^{2} *φ* = *o* (ii.)

every solution of which is of the form

*φ*_{1} = A sin.*ix*/Ω + B cos. *ix*/Ω

for any particular *i* series.

The corresponding partial solution for *φ* may be written in the form

C_{i} Sin. *i*/Ω (*x*—Ω*t*—S_{i}) + D_{i} sin. *i*/Ω(*x* + Ω*t* − T_{i})

where C_{i} D_{i}, S_{i}, T_{i}, *i*, Ω are unaltered as long as the same series of waves—that is, due to the same impulse at the origin—propagated through the same isotropic solid, are being considered.

The displacement potential has the same value when we put *t* = *o*, *t* = 2π/*i*, *t* = 4π/*i*, &c., *x* being constant—*i.e.*, while we

[Footnote] * Ibbetson, “Elastic Solids,” p. 286.

[Footnote] † Ibbetson, *i, c.*, p. 288.

are considering the same particle—that is, the waves have a period 2π/*i* seconds.

Again, if *t* be constant, the potential will have the same value when *x* = *x*_{1}, *x* = *x*_{1} + 2πΩ/*i*, *x*_{1} + 4πΩ/*i*, &c., or the same phase will occur at intervals from the origin of 2πΩ/*i*, which is therefore the wave-length.

Hence the velocity of propagation = distance between the. equipotential surfaces divided by the period = Ω, which is independent of *i*, and therefore the same everywhere in an isotropic elastic solid. Ω will be the same for all the waves proceeding from the same origin by the same path, provided they travel through an isotropic solid. Hence there will be no confusion between the different phases of the normal waves of an earthquake, or what in seismology we term the “maxima,” as at different places they will succeed one another at equal intervals, unless their paths have been through strata differing greatly from one another in character for some considerable distance: mere differences in surface strata would not appreciably affect the question.

The value of Ω is given by Ω^{2} = *k* + 4/3*n*/*d g*, where *k* = elasticity of volume, *n* = rigidity modulus, in the usual gravitation units; and *d* = density in units of mass per unit-volume.

Major Dutton has shown that the velocity of propagation is constant within the limits of errors of observation,^{*} and I have always made this assumption in calculating the elements of New Zealand earthquakes. It is here shown theoretically to depend on the hypothesis that the waves have travelled for the greater part of the distance through what may practically be regarded as a homogeneous solid. Hence we may infer that their path has been for the most part through the deeper strata, and that the origins are deep. All these inferences are borne out by the investigations of the best-observed shocks in New Zealand, by their self-consistency, and by the comparatively great depth—often twenty to twenty-five miles—which must be assigned to the origins in the cases where the data have been sufficient to determine them. It seems likely that we may, especially with the new instruments, have the means of determining the period of vibration, and (less truly) the amplitude; these, with the transit velocity, would enable us to draw conclusions as to the structure of the underlying rocks (from the value of Ω), and as to the character or amount of the impulse at the origin of disturbance. I have shown elsewhere that the transit-velocity of the normal waves in the

[Footnote] * “Charleston Earthquake.”

Wanganui earthquake of the 8th December, 1897 was about eighty-five miles per minute, or, say, 225,000 cm. per second. The value, according to Gray and Milne (calculated from experiments on Japanese rocks), for normal waves, in granite is 395,000cm. per second.^{*}

Approaching the question of the motion of an earth-particle from another point of view, and regarding it as inmost cases approximately simple harmonic vibration, we have 2μ*a* = VT, where *a* = amplitude, V = maximum velocity of particle, T = period in seconds, and the maximum acceleration = intensity = V^{2}/*a*.

Now, from cases in which the number of vibrations per second have been noted, I should judge that in the case of the most violent New Zealand earthquakes these have been about three per second. If we take T = ⅓ in the case of the Wanganui earthquake, and the intensity to be between viii. and ix. on the Rossi-Forel scale, or, say, 900 mm. per second according to Dr. Holden's equivalents, then we have V^{2}/*a* = 900, and 1/3 V = 2 φ*a*, ∴ *a* = 2.5 mm. nearly, Or the displacement of any earth-particle was about 1/10 in.

This is about one-third of, the maximum amount of horizontal displacement in the severe Japanese earthquake of the 15th January, 1887, according to the calculations of Professor Sekiya, based upon seismograph tracings. Our estimate would not, therefore, seem altogether improbable—possibly rather too high.

In the absence of accurate data, further speculation would be unprofitable; but I trust enough has been said to show that future investigation and. more accurate observation of earthquakes is likely to lead to interesting results.

[Footnote] * Quart. Jour. Geol. So., vol. 39, p. 139.