TRUMPET

TRUMPET, Articulate, comprehends both the speaking and the hearing trumpet. The former appears to have been known to the ancient Greeks and natives of Peru as early as 1595, but does not seem to have been an object of scientific investigation until the middle of the seventeenth century, when it occupied the attention of Athenasius Kircher, and subsequently in 1670, when Sir Samuel Morland proposed as a question to the Royal Society of London, "What is the best form for a speaking-trumpet?"

The subject is an exceedingly difficult one; but the most important fact for our purpose is this: An echo from a small plane surface in the midst of an open field is not heard, unless we stand in such a position that the angle of reflected sound may be equal to that of incidence. According to the theory of undulations, this small surface should become the centre of a new undulation, which should spread in all directions; but it is a fact, that if we go a very small distance on either side of the line of deflection we shall hear nothing. Whatever, then, may be the nature of the elastic undulations, sounds are reflected from a small plane in the same manner as light; and we may avail ourselves of this fact, though we cannot explain it in a satisfactory manner. The following view of the subject is derived from A Dissertation on Acoustic Instruments, by Mr Lambert of Berlin, in the Berlin Memoirs for 1763.

Let the trumpet be a cone $BCA$, $CN$ the axis, $DK$ a line perpendicular to the axis at the mouth-piece, $DEA$ the path of a reflected sound in the plane of the axis. Let $\backslash alpha$ be the angle of the cone; $\backslash theta\_1,\; \backslash theta\_2,\; \backslash dots,\; \backslash theta\_n$ the angles of incidence at each of $n$ reflections. Then, considering the angles of incidence and reflection to be equal, we have $\backslash theta\_n\; =\; \backslash theta\_\{n-1\}\; -\; \backslash alpha$, $\backslash theta\_\{n-1\}\; =\; \backslash theta\_\{n-2\}\; -\; \backslash alpha$, &c. $\backslash theta\_2\; =\; \backslash theta\_1\; -\; \backslash alpha$; therefore by addition, $\backslash theta\_n\; =\; \backslash theta\_1\; -\; n\backslash alpha$; whence we see that as the length of the cone, and therefore as $n$ increases, the last angle of reflection dimi-

nishes, and at some point, therefore, the direction of the reflected ray will be approximately parallel to the axis of the trumpet. Let $A$ be this point; draw $AB$ perpendicular to the axis $CN$. Let the trumpet be assumed of such dimensions that $AB$ is the diameter of the larger end, $KD$ that of the mouth-piece, and $TKt$ the plane section of a sphere, whose centre is $C$, and diameter $Tt\; =\; AB$. Draw the tangents $TB,\; tA$; these are parallel to $CN$. Then, following up the analogy of sound to light, we may assume that the effect upon an ear situated within the prolonged cylindrical surface $AtTB$ will be the same as if lines of sound proceeded from every point of the surface of a sounding sphere $TKDt$; and also that the sound is magnified in the proportion of the surface $TKDt$ to that of $KOD$. These considerations enable us to determine the dimensions and magnifying power of the trumpet. There are two conditions to be fulfilled—(1.) The diameter of the mouth-piece must be a fixed quantity, generally one inch and a half; (2.) The diameter of the larger end must be equal to that of the fictitious sonorous sphere.

Let $m$ = diameter of the mouth-piece, $y$ that of the larger end, $x$ the required length of the trumpet. Then, Dimensions of trumpet.

In practice it is usual to find the diameter of the larger end of the trumpet from the second of these equations, corresponding to a proposed length of trumpet. Let $M$ represent the magnifying power of the trumpet; then, as before observed, $M\; :\; 1\; =\; \backslash text\{surface\; \}\; TKDt\; :\; \backslash text\{surface\; \}\; KOD\; =$ Magnifying power.

$\backslash frac\{\backslash alpha\}\{4\}$ being a very small angle in practice, we may, without

sensible error, write $\backslash frac\{m\}\{y\}$ for $\backslash sin\; \backslash frac\{\backslash alpha\}\{4\}$; in which case we have

The extending power of the trumpet is the distance at which a person would hear the trumpet, compared with power. Extending power.

that at which he would hear the unassisted voice. Let $D,\; d$ represent these distances, $E$ the extending power of the trumpet, so that $E\; =\; \backslash frac\{D\}\{d\}$. Now, if we suppose that the

audibility of sounds varies inversely as the squares of the distances; and also that the magnifying power of the trumpet causes a voice to be heard as well at distance $D$ as it would, if unassisted, be heard at distance $d$, we

Trumpet, Articulate. The following table gives examples of a variety of trumpets, $x$ being expressed in feet, $y$ in inches, $m=1.5$ inches in all cases:—

Parabolic trumpet. Amongst other forms of trumpets which have been suggested, we may mention the parabolic and the hyperbolic. The former of these would be very advantageous if the mouth-piece were a point coinciding with the focus, for in that case all the reflected sounds would be parallel to the axis. But every point of an open mouth must be considered a centre of sound, and none of it must be kept out of the trumpet. Hence it will be found that the conical trumpet will disperse the reflected sounds less than the parabolic, and still less than the hyperbolic.

The elastic matter of the trumpet is thrown into tremors by the undulations from the mouth-piece. The agitations arising from these tremors tend greatly to hurt the distinctness of articulation; hence it is found advantageous to check all tremors of the trumpet by encasing it in woollen list.

Hearing trumpet. By diminishing the aperture of the smaller end, the same instrument may be employed as a hearing trumpet. In this case the investigation is much simplified by the consideration that all the lines of sound may be supposed to enter the large end of the tubes in directions parallel to the

Dimensions of hearing trumpet. Now, employing the same figure as in the preceding investigation, let $x$ = required length of the hearing trumpet, $z$ that of the part cut off, $x\text{'}$, $x\text{'}\text{'}$, $x\text{'}\text{'}\text{'}$, &c., the distances from C of each successive point of reflection. Then, if we trace the progress of a line of sound from A to D, we perceive

that the first angle of incidence and reflection is $\backslash frac\{\backslash alpha\}\{2\}$; and that each successive angle is greater than that which precedes it by the angle $\backslash alpha$. The greatest possible value of the angle of incidence is $90^\backslash circ$, and we may assume this to

be the angle at the smaller end DK. Now $\backslash frac\{x+z\}\{x\text{'}\}\; =\; \backslash frac\{\backslash sin\; \backslash frac\{3\}\{2\}\backslash alpha\}\{\backslash sin\; \backslash frac\{\backslash alpha\}\{2\}\}$,

$\backslash frac\{x\text{'}\}\{x\text{'}\text{'}\}\; =\; \backslash frac\{\backslash sin\; \backslash frac\{5\}\{2\}\backslash alpha\}\{\backslash sin\; \backslash frac\{3\}\{2\}\backslash alpha\}$, $\backslash therefore\; x+z\; =\; x\text{'}\text{'}$, $\backslash frac\{\backslash sin\; \backslash frac\{5\}\{2\}\backslash alpha\}\{\backslash sin\; \backslash frac\{\backslash alpha\}\{2\}\}$. Similarly, we should

find $x+z\; =\; x\text{'}\text{'}\text{'}\; =\; \backslash frac\{\backslash sin\; \backslash frac\{7\}\{2\}\backslash alpha\}\{\backslash sin\; \backslash frac\{5\}\{2\}\backslash alpha\}$; and so on till we arrived at

$x+z\; =\; z\; \backslash frac\{\backslash sin\; 90\}\{\backslash sin\; \backslash frac\{\backslash alpha\}\{2\}\}\; =\; \backslash frac\{z\}\{\backslash sin\; \backslash frac\{\backslash alpha\}\{2\}\}$.

Approximating power. Let A represent the approximating power of the trumpet; that is, the proportion in which a voice is brought apparently nearer. Then, if we consider sounds to be constituted in the ratio of the areas of the larger and smaller ends, that is, in the ratio of $(x+z)^2$ to $z^2$, we shall obtain

$A\; =\; \backslash frac\{x+z\}\{z\}\; =\; \backslash frac\{1\}\{\backslash sin\; \backslash frac\{\backslash alpha\}\{2\}\}$. But taking the usual diameter of the

smaller end to be $\backslash frac\{1\}\{6\}$ of an inch, we have $\backslash sin\; \backslash frac\{\backslash alpha\}\{2\}\; =\; \backslash frac\{1\}\{6z\}\; =\; \backslash frac\{A-1\}\{6z\}$;

hence $A\; =\; \backslash frac\{6z\}\{A-1\}\; \backslash therefore\; x\; =\; \backslash frac\{A(A-1)\}\{6\}$.

The above affords a simple rule for the construction of a hearing trumpet. Thus, suppose it be required to approximate the sound 12 times, making it 144 times stronger than the natural voice at the same distance; then in this case $A=12$, and we have for the required length $\backslash frac\{12\; \backslash times\; 11\}\{6\}\; =\; 22$ inches.

As in the case of speaking trumpets, all reverberation of the instrument should be avoided, by making it thick and of the least elastic materials, and by covering it externally with cloth.

The paraboloid is the best form for the hearing trumpet, because all the reflected sounds will pass through the focus. The necessary conditions for a trumpet of this form are, that it be cut off at the smaller end through the focus; that the parameter be $\backslash frac\{1\}\{2\}$ of an inch, and the focus $\backslash frac\{1\}\{2\}$ of an inch from the vertex,—the length being determined by the proposed approximating power, as before. (G. N. S.)