Beats Knowledge-Based Study

Study notes

1. What Are Acoustic Beats?

   When two sound waves with identical amplitudes but slightly different frequencies travel through the same medium in the same direction, they undergo superposition. Instead of hearing two distinct, steady tones, a listener detects a single tone whose loudness rises and falls periodically. This periodic variation in perceived intensity is called beats.

   Unlike standard two-source interference patterns in space—where stable alternating regions of destructive and constructive interference exist at fixed locations—beats are an example of temporal interference. At a fixed observation point, the phase relationship between the two overlapping waves constantly shifts over time.

   • In phase (Constructive Interference): The wave crests align, resulting in maximum amplitude and maximum sound intensity (loudness).
   • Out of phase (Destructive Interference): A crest aligns with a trough, causing cancellation and minimum amplitude (momentary quietness).

   Deriving Beat Frequency

   Consider two sound waves of equal amplitude $A$ detected at a single point in space:

   $$y_1(t) = A \sin(2\pi f_1 t)$$

   $$y_2(t) = A \sin(2\pi f_2 t)$$

   Applying the trigonometric identity for the sum of two sines, $\sin\alpha + \sin\beta = 2 \cos\left(\frac{\alpha - \beta}{2}\right)\sin\left(\frac{\alpha + \beta}{2}\right)$, the combined displacement is:

   $$y(t) = y_1(t) + y_2(t) = \left[2A \cos\left(2\pi \frac{f_1 - f_2}{2} t\right)\right] \sin\left(2\pi \frac{f_1 + f_2}{2} t\right)$$

   Key Quantities

   • Perceived Pitch (Carrier Frequency): The sound wave oscillates at the average frequency, $f_{\text{avg}} = \frac{f_1 + f_2}{2}$.
   • Modulation Envelope: The term inside the brackets represents a slowly varying amplitude envelope with frequency $\frac{|f_1 - f_2|}{2}$.
   • Beat Frequency ($f_{\text{beat}}$): Because sound intensity is proportional to the square of the amplitude, the listener hears a volume maximum every time the cosine equals either $+1$ or $-1$. Thus, two beats occur per cycle of the cosine function:

   $$f_{\text{beat}} = |f_1 - f_2|$$

   The time interval between successive loud pulses is the beat period, given by $T_{\text{beat}} = \frac{1}{f_{\text{beat}}}$.

   Tuning and Ambiguity in Beat Analysis

   Beats provide a sensitive acoustic method for tuning musical instruments. When a musician sounds an out-of-tune string simultaneously with a reference tuning fork, beats are heard. The musician adjusts the string tension until the beat frequency decreases to zero, indicating that the two frequencies match exactly.

   The Frequency Ambiguity Problem

   If a reference fork of frequency $f_{\text{ref}} = 440\text{ Hz}$ generates a beat frequency of $4\text{ Hz}$ when sounded with an unknown source, the unknown frequency $f_x$ can be either:

   $$f_x = 440\text{ Hz} + 4\text{ Hz} = 444\text{ Hz} \quad \text{or} \quad f_x = 440\text{ Hz} - 4\text{ Hz} = 436\text{ Hz}$$

   To resolve this ambiguity, a known change must be made to the system:

   • If the unknown source is a string and its tension is slightly increased (raising its frequency), an increase in beat frequency confirms that $f_x$ was already higher ($444\text{ Hz}$).
   • If increasing the tension causes the beat frequency to decrease, $f_x$ was lower ($436\text{ Hz}$) and is now approaching the reference frequency.

   Common Misconceptions

   • Auditory Limit: The human ear can distinguish individual beats up to roughly $15\text{–}20\text{ Hz}$. Above this limit, beats blur into a rapid roughness or a faint third difference tone.
   • Pitch vs. Loudness: Beats do not represent a changing frequency or pitch over time; the perceived pitch remains constant at $f_{\text{avg}}$, while only the amplitude (volume) pulsates.