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a function of the normal x. cos x+y.cos ẞ+z. cosy.

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= $(at
(at-x.cos a-y. cosẞ-z.cos y)

+ (at +x. cos a+y.cos B+%. cos y),

= (at — normal) + ↓ (at + normal).

.

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w=x.cos a+y.cos B+z. cosy.

This shews that the disturbance of every kind is the same through that plane; and the factors cos a, cos ß, cos y, in the expressions for X, Y, Z, shew that the movements of the particles in that plane are perpendicular to the plane. The first term exhibits a wave moving, so as to increase the normal, with velocity a; and the second exhibits a wave diminishing the normal with the same velocity.

40. Combination of two plane waves from two

sources.

Suppose that F consists of two terms G and H, of which G depends on

w=x cos a + y cos ẞ + z cos y,

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which, in consequence of our assumptions as to the difference of form of the two terms, will require the separate equations,

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Adopting the first wave only in each solution, and

taking the same form of function,

F= G+H=$(at -x cos ay cos B - ≈ cos y)

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Therefore, by Article 38,

1

Density = D {1 — — $′ (at — x cos a — y cos 8 — z cos y)

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一部

— 1 p′ (at − x cos a − y cos ẞ +z cos y)

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·cos y. ' (at — x cos a - y cos 8-z cos y)

+cosy.' (at-x cos a-y cos ẞ+zcos y);

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;

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It appears therefore that, if the plane ry were a material boundary of the air, the motions of the particles of air would not be altered; since it would permit

motion parallel to the plane, and there is no motion perpendicular to the plane. But the density of the air in contact is variable; and therefore, if the barrier completely cuts off communication with tranquil air on the other side, it must be a rigid barrier.

41. Theory of the simple echo.

The number of solutions which the equations of Article 36 admit is infinite; embracing not only different forms of function, as in Article 23, but also functions corresponding to waves passing in any different directions in space of three dimensions, to converging waves, to diverging waves, &c. And any one or any combination of these functions, as need requires, may be considered as admissible in quality of the 'undetermined functions' to which attention is called in the author's Partial Differential Equations.

If now we wish to ascertain the law of motion of a plane wave, we proceed as in Article 39, and we obtain a solution which shews that the wave preserves its plane character, moving with a certain velocity. And this solution is sufficient, as long as we introduce no condition limiting the space occupied by the air, &c.

But suppose that we introduce this condition, "The air is bounded by an immoveable barrier, constituting

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