dX

dv

do and

dX

dt

would at that point be infinite, and our in

vestigation in Article 21 would be entirely inapplicable. Secondly, there can be no numerical disconti

ďaX d dx

nuity in '(v). For, in forming, or

a (noa. dx), by the limit of noa.

d

dX

ηθα.

ηθα

dv

dv

8. ' (v)

a dis

δυ

continuity or sudden interruption in the value of ' (v)

dX

would in like manner render infinite at that point;

dt

that is to say, there would be infinite force, infinite condensation of air, &c., all which is opposed to the ideas under which the investigation of Article 21 has been carried on. If, however, (v) and p' (v) are free from discontinuity, then the effect of a numerical discontinuity in "(v) would be that at special values of v the magnitude of the forces, condensations, &c., would change suddenly; but there does not appear to be any physical impossibility in this. If † (v), p′ (v), p′′ (v), are free from numerical discontinuity, then there is no sudden change even in the magnitude of the force or condensation. We conclude, however, that it is sufficient that the two first terms, & (v) and p' (v), be free from numerical discontinuity.

29. Forms proper for the Functions representing Continuous Series of similar Waves.

If we propose to represent a series of waves in which all the successive waves are exactly similar, the most

general formula which we can adopt for (v) is a function of sin v and cos v; and if we reject fractional and negative powers of sin v and cos v (which is necessary in order that (v), ☀′ (v), p′′ (v), &c. may never be infi(v) may never be ambiguous), that is, if we adopt only integral powers of sin v and cos v, we can always (see Article 74) give the following form to the function,

nite, and that

A ̧. sin (Bv + C1) + A ̧. sin (2Bv + C2) + &c.

which satisfies the conditions of Article 28.

The phænomena of music will usually be referred to this series and, in most instances, to the first term alone.

30. Introduction of the terms 'length of wave,' 'period of wave,' 'frequency of wave;' relation between their values and that of the velocity of waves;' remarks on the amplitude of vibration,' and its independence of the other quantities.

The displacement X of a particle being represented, in a continuous series of waves, by the expression

X = A1. sin (Bv + C1) + A ̧ . sin (2Bv + C2) + &c. upon making this maximum with respect to v, and positive (or negative, only confining ourselves to one sign), we find a definite value V for v, defined by numerical values (in terms of 41, A„, &c., C1, C2, &c.)

of sin BV and cos BV. These correspond to only one value of BV less than 27, but they correspond also to BV ± 2π, BV ± 4π, &c. And upon substituting in the expression for X, they all give the same value, which is every where the maximum. Thus we find that the maximum recurs, and the general character of the wave recurs, when Bu is increased or diminished by 2mm, or when v is increased or di

Now v=noa.t-x. If then we confine our attention to the wave at a certain instant of time, that is, if we regard t as constant; and if we survey the long series of waves, that is, if we contemplate the displacements and motions at that certain instant, of different particles; we find that when x is increased or

2π

diminished by an integer multiple of (and at no

B

other places) we come to particles in the same state of disturbance as that which was first considered.

2π

plain therefore that the 'length of a wave' is B

It is

But if we fix our attention on a certain particle, that is, if we regard x as constant; and if we examine its state of disturbance at different times, that is, if we consider different values of t; we find that the same state of disturbance recurs when noa.t is in2π

creased or diminished by an integer multiple of

B'

or when t is increased or diminished by an integer

the interval in time between the passage of two successive waves at the same point, and is therefore the 'period of wave.'

The frequency of wave,' or the number of waves

that occur in the unit of time, is evidently

ηθα. Β

2π

The 'velocity of wave,' Article 24, is nea. Therefore, comparing the expressions above, we find,

length of wave period of wave x velocity of wave;

=

or length of wave =

velocity of wave frequency of wave'

Now in these expressions, the factors 4, and A, &c. do not occur. If the formula for X is restricted to the first term A, sin (Bv + C1), A, does not enter at all into the determination of V; if there are other terms, the quotients 4, &c. enter, but not the absolute values

of A, A,, &c. Thus the theorems just found are independent of A, A,, &c. But the 'amplitude of vibration,' or maximum range in the amount of X, does depend on the absolute values of A,, 4,, &c. ; when the formula for X contains only one term, the amplitude is 24, Thus it appears that the amplitude (on one hand) and the length, period, frequency, and velocity of wave (on the other hand) are perfectly independent.

44

31. The Solitary Wave, and Functions proper to represent it; and Interpretation of their effect.

It is important to examine the case of the Solitary Wave: a wave which has been created by a single disturbance that occupied a limited time and was followed by absolute quiescence. Its algebraical conditions will be the following:

Till v has a certain value A, (v) = 0.

บ

When v has a value included between A and A+B, (v) is to have a real value.

When the value of v exceeds A+ B, $ (v) = 0.

The function (v) must be such that in no part there be numerical discontinuity in the values of p (v) and of p′ (v).

When v=A, and also when v=A+B; & (v) and '(v), depending on that form of the function which applies from v = A to v = A + B, must=0; inasmuch as the values of (v) and p' (v), before v=A and after v = A + B, are = 0; and numerical discontinuity is to be avoided.

=

Functions can be found, in infinite variety of form, which satisfy these conditions. For instance,

$ (v), from A to A + B‚ = C . (v − A)3. (A + B − v)3,