# Re: two sine tones simultaneously within one critical band

```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

```