1()Q SPEAKING TRUMPET. 



IV. 



pet 



A Mathematical Theory of the Speaking Trumpet. By John 

 Gotjgh, Efq. 



To Mr. NICHOLSON. 

 SIR, 



Theory of the 1 HE fpeaking trumpet is a conical tube, which receives the 

 fpeakingtrum. human voice from the mouth, and encreales the range of it. 

 The ftructure of thefe inftruments, and the materials of which 

 they commonly confift, teach us to refer this Angular faculty to 

 two caufes. In the firft place, the aeriel pulfes undergo cer- 

 tain modifications in the cavity of the trumpet ; the nature and 

 effecls of which I am going to inveftigate. Secondly, the 

 metallic fiiell of the inftrument augments the power of the 

 voice, by its aptitude to conduct thofe impulfes; which are 

 imprefled upon it, partly by the pulfes of the included air, and 

 partly by the direct action of the Larynx, tranfmitted through 

 the medium of the face to the mouth-piece of the inftrument. 

 Before I enter upon the inveftigaticn of the effects refulting 

 from the firft cauie, it will be neceflary to premife ^he follow- 

 ing lemma : 



Article I . Let O R S r (Fig. IV. Plate 7) be an evanefcent 

 fphere of air, which is agitated by two or more pulfes, be- 

 ginning and ending together, and radiating from the points P/>, 

 &c. the vibatory motion of O R S r, in any direction what- 

 ever is equal to the joint fum of the forces, which the pulfes 

 would imprefs upon it feparately. Join PO, Op; and draw 

 the lines R O r, SOs perpendicular to P O, Oj^ ; now as 

 thefe lines are evanefcent, every point of each of them is 

 equally prefled at any inftant by its refpective pulfe. But the 

 force imparted to R O ;• by P, agitates the fphere ORSr 

 equally in all directions, Principia, 422; the fame holds true 

 of the force of p upon SOs. Now thefe forces cannot co- 

 alefce or be compounded fo as to act in a fingle direction, — 

 Manch. Mem. V. 5. p. 660; confequently they act with their 

 joint powers in all directions. 



Art. 2. This being premifed, let the fruftum A B C D (Fig, 

 3.) reprefent a trumpet, the (hell of which is a non-conductor 

 of found. After a pulfe of the voice has patted through the 



aperture 



