Beats — Question 5

Question 5

Pairwise beat measurements can leave a common sign ambiguity

Tones A and B beat at $5$ and $8\,\mathrm{Hz}$ respectively against a $400\,\mathrm{Hz}$ reference. A and B together beat at $3\,\mathrm{Hz}$. All frequencies are stable.

1. Find every pair $(f_A,f_B)$ consistent with these observations.

2. After A is raised by exactly $1\,\mathrm{Hz}$, its beat rate against the reference is $4\,\mathrm{Hz}$. Select the original pair and predict the new A–B beat rate.

Question 5 - Solution

Pairwise beat measurements can leave a common sign ambiguity

(a) Method and calculation

The first observations give $f_A=395$ or $405$ and $f_B=392$ or $408\,\mathrm{Hz}$. Their differences are $3$ when they lie on the same side and $13$ on opposite sides. Thus the allowed pairs are$$(f_A,f_B)=(395,392)\ \text{or}\ (405,408)\,\mathrm{Hz}.$$

(b) Interpretation and further calculation

Raising A gives $396$ or $406\,\mathrm{Hz}$, whose reference beat rates are $4$ and $6$. Hence the original pair is $(395,392)\,\mathrm{Hz}$. The new intertone beat is$$f_{b,AB}=|396-392|=4\,\mathrm{Hz}.$$Pairwise absolute differences determine relative positions but may retain a reflection ambiguity about the reference.

Physical check

The controlled perturbation breaks the common sign ambiguity without needing to identify either original tone directly.