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*To*: AUDITORY@xxxxxxxxxxxxxxx*Subject*: Re: two sine tones simultaneously within one critical band*From*: Reinhart Frosch <reinifrosch@xxxxxxxxxx>*Date*: Sat, 8 Oct 2005 21:08:23 +0200*Comments*: To: Bob Masta <audio@DAQARTA.COM>*Delivery-date*: Sat Oct 8 15:43:53 2005*In-reply-to*: <4347AE48.20627.35132A@localhost>*Reply-to*: Reinhart Frosch <reinifrosch@xxxxxxxxxx>*Sender*: AUDITORY Research in Auditory Perception <AUDITORY@xxxxxxxxxxxxxxx>

On October 8, 2005, Bob Masta wrote: >There is no question that the sum of two sines can be expressed >as the product of two cosines of sum-and-difference frequencies, >as in: > >sin(A) + sin(B) = 2*cos(.5(A-B))*sin(.5(A+B)) > >But this most definitely does NOT demonstrate the >production of real acoustic beat frequencies. [...] I also believe that no real beat frequencies are produced. However, beats ARE produced. In order to explain, I would like to present a corrected version of my formulae (thanks to Jim Beaucham, Fred Herzfeld, and Piotr Majdak): Sound pressure of first sine-tone: p_1(t) = p_0 * sin(99 * 2pi * t); sound pressure of second sine-tone: p_2(t) = p_0 * sin(101 * 2pi * t). [ * = multiplication sign; t = time in seconds.] Total sound pressure: p(t) = p_1(t) + p_2(t) = 2p_0 * cos(2pi * t) * sin(100 * 2pi * t). According to that last formula, the amplitude of the 100-hertz sine function is modulated by the cosine function. Whenever the argument of the cosine function is an integral multiple of pi, the cosine function amounts to +1 or to -1, so that the amplitude of the sine function, and thus the sound level of the signal, is maximal. These level maxima occur at times t = 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, ... seconds. There are two level maxima per second, i.e., two beats per second. It is easy to draw a graph of p(t) according to the last equation or, equivalently, of the sum p_1(t) + p_2(t) according to the two previous equations, e.g. by means of EXCEL. A Fourier analysis of the signal yields a first line at 99 hertz and a second line at 101 hertz. Reinhart Frosch. Reinhart Frosch, Dr. phil. nat., Sommerhaldenstr. 5B, CH-5200 Brugg. Phone: 0041 56 441 77 72. Mobile: 0041 79 754 30 32. E-mail: reinifrosch@xxxxxxxxxx

**References**:**Re: two sine tones simultaneously within one critical band***From:*Bob Masta

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