Analytic derivation of the (k/j)-order modulation envelopes in the sum of two mistuned (co)sine waves

The beating effects which may occur in the sum of two mistuned cosine (or sine) functions whose frequencies f1, f2 satisfy f2 ≈ (k/j)f1 (where k/j is a reduced integer ratio) are known as “beats of mistuned consonances”. They have already been investigated in the 19th century as for their beat frequencies; but interestingly, nothing has been said on the mathematical shapes and properties of these beats. In the present contribution we derive analytically the equations of the modulating envelope curves which accurately outline these beats. Although this may seem to be a straightforward problem, the modulating envelope curves in cases other than k = j = 1 turn out to be surprisingly complex. Denoting by n the value max(k, j), we show that for any k and j we have 2n modulating envelope curves, and that the equations of these curves are the 2n solutions of a polynomial equation of order 2n in y, whose coefficients are periodic functions of x. As expected, the simple case of k = j = 1 reduces into the classical cosine sum-to-product identity, where the modulating envelopes are cosinusoidal and have the frequency (f1 – f2)/2. Due to the complexity of the curve equations in cases other than k = j = 1, a simplified approximation using 2n cosinusoidal curves is also provided, which can be used when the exact analytic solutions are not really required. Further properties of the beating effects in question are also discussed and illustrated. Similar results hold for the modulation envelopes of mistuned sine waves, too. Applications may concern various fields, including acoustics, optics, etc.