Or, neglecting y (u), and restoring t and x,

be

When we are considering the state of a solitary wave at a great distance from the origin, v or (at — x), which (see Article 31) is limited within the value B, may considered very small with regard to a; and in the factors of the small terms, for at +x or 2x + (at - x) we may put 2x; and we have

x;

+ 12 ($' (at — x)}°

+ 3* {4′ (at — x)}". 4′′ (at — x).

This degree of approximation will suffice for our

present purposes.

33. Progressive Change in the Character of the

Wave.

It is seen here that the law of displacement of the particles undergoes change as the wave travels on. The original function receives the addition of new functions, which are affected with multipliers depending on the distance of the disturbed point from the origin of the disturbance. Supposing that we assume the terms at which we have arrived in the last Article to suffice for our information of what happens when the wave has travelled through a considerable but not an enormous distance, we may interpret their effect thus:

which is 0 when v' = 0, is + from v = 0 to v'

2

to v B, and is 0 when

O when v

==

is from v

=

2

B

which is maximum + when v' = 0, is + till v' =

24

when

B

it vanishes, is- till v' =

when it is maximum -, is

2

- till '=when it vanishes, and is + till v'= B when

3B
4

it is maximum +.

Therefore the formula of Article 32 will give the following as shewing the nature of the principal alterations in the values of X;

X=$(v);

X=4(v) + two positive terms; one

depending on D, the other on D

X = $ (v) + a positive term depending on D2 - a very small positive term depending on D3.

when v' = B, X=$(∞).

The last very small term of the formula, which vanishes at these five critical points, has between them values which are successively +--+.

Remarking that the changes in the value of v and v' have the same sign as those of t, so that the smallest

value of v' corresponds to the beginning of the displacement of a particle, it will be seen that, in the state of the far-advanced wave,

The first part of the forward displacement is more rapid than in the primitive wave.

The latter part also is more forward, or the retreat is slower, than in the primitive wave. These peculiarities increase with increase of D.

In these points, the motion of a wave of air is closely analogous to that of a wave of water when its vertical movement is large and it runs for a considerable distance over a shallow bottom. See the Encyclopædia Metropolitana, Article, Tides and Waves, Section IV., Subsection 3.

34. Conjectured Change of Character of Wave when it has travelled very far.

It is difficult to say what will be the form of the wave when at +x is very large. It would be necessary to carry on the steps of the successive substitution to an indefinite extent, or rather, to find a function which would represent the infinite series thus produced. It appears not improbable that at length the continuity of the atmospheric particles may be destroyed, and that something may take place analogous to the bore of a tidal river or the surf of a sea, in which the form and properties of a wave are ultimately lost. (This idea is also suggested by Mr Earnshaw.)

SECTION IV.

INVESTIGATION OF THE MOTION OF A WAVE OF AIR

THROUGH THE ATMOSPHERE CONSIDERED AS OF

TWO OR THREE DIMENSIONS.

34*. Outline of the method to be employed; and cautions requiring attention in regard to the order of terms to be rejected.

The method of forming the equations of motion will be precisely the same, in principle, as that in the instance of air in a tube, Article 21. A symbolical displacement of particles, the most general which the circumstances permit, will be assumed; the symbolical density, and elastic force, and differences of elastic force in different directions, will be found; and these will be compared with the symbolical expressions for changes of velocity which they produce in different directions.

But care is peculiarly necessary, in consequence of the obscurity of the process treating of small terms of a higher order than those which we wish to preserve. In Article 21, we could perceive exactly the form and value of the terms which we rejected: here we can only draw inferences from general reasoning. These inferences, however, will enable us to judge with certainty whether a small quantity before us is of the