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Propagation of Sound in Air of one dimension.

b. Let us then conceive, that, in general, the section or stratum of the molecules a abb, whose distance from the initial place A of the section A is represented by x, shall, after the lape of any time t, have been transported into the situation ɑ ɑ ßß, at a distance Aα = y from the same fixed point A. Let x'x", &c. from the fixed

be the distances of the next consecutive sections point A, in their state of rest, and y', y', &c. their distances after the lapse of the same time t. Then will

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be the thicknesses (supposed infinitely small) of these strata, or the spaces occupied by them (taking the area of the section for unity) in their quiescent state, and

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dy', y'"' — y"
= dy", &c.
Now as these strata were in

the same in their state of motion.

contact at the origin of the motion, and are held together by the pressure of the surrounding fluid, they will remain in contact, and advance and recede along the pipe as one mass, only the space they will occupy at different points of their motion will be variable, according to the degree of condensation or dilatation they may have undergone in virtue of their motion itself. If, for instance, at any moment the hinder of them dy be in the act of urging forward the next dy', it will be condensed; if retreating, rarefied in comparison with the state of the preceding one dy".

c. Now any stratum of molecules d y', interjacent between two others d y and d y', can only undergo a change in its velocity when urged by some force, and the only force which can urge it is the difference of pressures it may experience on its two faces by the difference (if any) of the elasticities of the adjacent strata d y' and dy. If we can estimate this, the laws of Dynamics will enable us to express the consequent change of motion.

To this end, then, let the elasticity of the air in its quiescent state be represented by E, which is a given quantity, and is measured by the weight of a column of mercury sustained by it, or by the length of a homogeneous column of air of the same density, whose weight shall suffice to keep it so compressed, or be

Expression of the acting Forces.

equal to that of the column of mercury in the barometer. Then, since the elasticity of air is inversely as the space it occupies, (cæteris paribus),

dy: dx

elasticity of the air when occupying d x = E its elasticity when occupying d

dx

y = E

dy

Similarly the elasticities of the air occupying d y' and d y' are

d x1 dy'

represented by E. and E

d x" dy"

Hence the plane separat

ing the strata d y and dy' will be pressed forward by the elasticity E

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the differentials being taken on the supposition that d x is constant,

or that d x, d x d x", &c. are originally equal.

Now if we denote by H the length of a homogeneous column of air necessary to counterbalance the elasticity of the quiescent air, and by D its density, we have

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dx'. D d x. D the weight of the stratum d x',

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Equation of the motion of Sound in a Pipe.

d. Now the distance of the mass thus urged from the fixed point A, at the expiration of the time t, is y'. Hence the velocity of the particle d y' or, which comes to the same, of the particle dy in contact with it is the increase of velocity during the instant

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dy
d t

; and we have by Dynamics

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gravity being the unit of accelerating force, and t being expressed in mean solar seconds; and all linear quantities such as H, x, y being expressed in feet.

e. This is, in fact, an equation of 'partial differentials, y being at once a function both of x the original distance of the stratum dx from the origin of the motion, and of t the time elapsed. In its present form, simple as it appears, it is altogether intractable and incapable of integration. In fact, it embraces a class of dynamical problems of very great complexity; for it is evident that, since no hypothesis has been made in any way limiting the extent of the excursions of the original or subsequent strata from their points of quiescence, this equation must contain the general expression of all possible motions of elastic fluids in narrow pipes, whether great, as when urged by pistons or driven by bellows, or small, as are the tremors which cause sound. In the theory of sound we suppose the agitations of each molecule so minute as not to move it sensibly from its point of rest. Experience confirms this. Sounds transmitted through a smoky or dusty atmosphere cause no visible motion in the smoke or floating dust, unless the source of sound be so near as to produce a wind, which, however, is always insensible at

Equation of the Motion of Sound in a Pipe.

a very moderate distance. If we introduce this condition, the equation (I.) admits of integration; for the whole amount of motion of each molecule being extremely minute, their differences for consecutive molecules, or the amount of the rarefactions and condensations undergone must be much more so. Hence the value of which expresses the ratio of the condensation of the stratum dy in motion and in rest, may be regarded as equal to unity, and the equation becomes simply,

dy

dx

d2 y d t2

2 H.
=
g

d2 y d x2

(II.)

which is the equation of sound regarded as propagated in one dimension, that of length, only; or, as prevented from spreading laterally by a pipe.

f. The complete integral of this equation is

y = F(x+√2g Ht) + f (x

√2gHt)

(III.)

where F and ƒ denote arbitrary functions of the quantities within the parenthesis, and which must be determined by a consideration of the initial state of the fluid, or by the nature of the motion originally communicated to its molecules.

g. Let us, then, suppose that, at the commencement of the motion, we have impressed on each section of the fluid, along its whole extent, any arbitrary velocities and condensations, by any means whatever, so as to comprehend in our investigation all possible varieties of initial motion, whether expressible by regular analytical functions, or depending on no regular law whatever.

It is manifest that these conditions will be expressed by assuming arbitrary functions of x, such as ø (x) and y (x) for the initial valdy dy

ues of the two partial differentials and whereof the former d t dx'

represents in all cases the velocity (v) of a particle which would be at the distance x from the origin of coördinates in the state of equilibrium, and the latter the linear extent (e) of that particle compared with its original extent, to which its density and elasticity

State of any Molecule at any Instant.

are reciprocally proportional. Now differentiating (III.) we get for the general values of v and e,

v=

dy

--

d = √ √ 2 g H ( F (x + √2 g H t) — f'(x —√/2g Ht)),(IV.)

d t

dy

e= = F(x+2gHt)+f'(x-2gHt); (V.)

d x

consequently their initial values, making t = 0, will be

+ (x) = √/25 H (F (x) — f (x)),

g

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1

2/2g H √28 H 4 (x) + p (x)) d x,
S (v

1

ψ

ƒ (x) = 2 √2zHS (√ 25 H y (x) — ¢ (x) ) d x ;
Night (+

and thus the forms of the functions F and ƒ become known when those of and y are given.

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h. The question of the propagation of sound, however, does not require us to concern ourselves with these functions, as a knowledge of the actual velocity and density of any molecule at any instant is sufficient for our purpose. Substituting then in (IV.) and (V.) the values of F and f' given in (VI.), we get

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V=

2

g H ( y (x + √ 2 g H 1 ) — 4 (x — √2 g H 1))

+ į ( p (x + 2 √ 2 g H t ) + 4 ( x − √2 g H t)),

(VII.)

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