172 BELL SYSTEM TECHNICAL JOURNAL 



waves. It might seem as if we could measure their speed by picking 

 out one crest, as A of Fig. 3, and checking off with a stop-watch the 

 moments when it passes two fixed markers placed a known distance 

 apart. Not so; for we cannot see or hear or in any way perceive the 

 individual crests. The wave train produces a perfectly uniform tone 

 in the ear which it strikes. If two listeners are stationed at different 

 points along the path of the sound, neither can recognize the moment 

 at which any particular crest glided by. All they can recognize, all 

 they can compare, is the moment of passage of a perhirbation of the 

 wave train ; a sudden beginning, a sudden ending, a transient swelling 

 of the sound. Most measurements of the speed of sound, in fact, are 

 measures of the speeds of something violent — the crack of a pistol or an 

 electric spark, the roar of an explosion — something very unlike a uni- 

 form train of sine-waves.^ 



Now a sine-wave with a perturbation is in effect a sum of two or 

 more sine-waves each of endless extent and constant amplitude, but 

 having different wave-lengths and different amplitudes. This state- 

 ment is the content of Fourier's principle from which the method of 

 Fourier analysis is derived. One might represent even the sudden and 

 violent pulsation of air due to an explosion, or the electrical spasm due 

 to an outburst of static, by a summation of properly-chosen endless 

 monochromatic sine-wave trains. I take however the simplest con- 

 ceivable case: the wave train composed of only two sine-waves of dif- 

 ferent wave-lengths. 



The reader will probably recall that when the difference between the 

 wave-lengths is only a small fraction of either, this composite wave 

 train resembles a sine-wave with regular fluctuations of amplitude — 

 that is to say, with "beats" (Fig. 3). The maximum or centre of a 

 beat occurs where a crest of one sine-wave coincides with a crest of the 

 other — the minimum between beats, where crest falls together with 

 trough. Denote the two wave-lengths by X and X -|- AX. One 

 sees by inspection that a wave-length is the same fraction of the dis- 

 tance D between two consecutive beat -maxima, as the discrepancy AX 

 is of the wave-length -.^ 



D/\ = X/AX. (17) 



Of course this statement is exactly true only in the limit of vanishingly 

 small AX. We shall always stay close to this limit, though some of the 

 following statements would be valid even otherwise. 



^ I except so-called measurements of the velocity of sound which are really measures 

 of frequency and wave-length in stationary' wave-patterns, these being then multi- 

 plied together. 



^ The principle of the vernier. 



