High Order Beats

Beats of Fundamental Tone and its Overtone

Abstract

When two tones, whose frequencies are ω and nω +ε (n = 2, 3, 4, ……, ω>>| ε |), are sounded simultaneously, and the beats are heard, they are named false beats or high order beats. This paper explains this phenomenon mathematically. Judging from the ordinary beats phenomena, high order beats are defined as the envelopes of the combined oscillation (with their frequencies ω and nω+ε), and also as those whose period is determined by ε. Then this paper shows,
(1) high order beats is expressed generally by the following equations;
y_e= sin ωt + sin (nω+ ε) t
∂y_e/∂ω=0 (ω : parameter)
and the period of those envelopes is certainly determined by ε, and the value of y_e are able to be calculated.
(2) graphically the high order beats are obtained and it is proved that the period is exactly 2π/ε.
(3) the period of the same maximum value of y_e (which is equal to the period of the high order beats) is analytically 2π/ε, independent of n.