Beats and Musical Sound

A note can grow and fade without either source changing

Sound two nearly matched tuning forks together. Instead of a steady loudness, you hear repeated increases and decreases: beats. Each fork can vibrate steadily, but the two displacements at your ear do not keep a constant relative phase because their frequencies differ slightly. Sometimes they reinforce; later they oppose. The relative phase then wraps through another full cycle and the loudness pattern repeats. The fast oscillation is heard as a note, while the slow variation is heard as waxing and waning loudness.

The frequencies should be close enough that the slow envelope can be distinguished from the rapid vibration. Comparable amplitudes make the effect easy to hear. Exactly equal amplitudes can cancel fully at minima; unequal amplitudes leave a nonzero minimum. Widely separated sound frequencies need not be heard as clear slow beats. The mathematical sum remains valid, but the particular slow-modulation interpretation has a limited regime.

Separate the fast oscillation from its slow envelope

Fix the listening location so we need not carry x through the calculation. Choose equal amplitude a and a convenient common phase: s₁ = a cos(ω₁t) and s₂ = a cos(ω₂t). Let ω₁ > ω₂ for now. Superposition gives s = a[cos(ω₁t) + cos(ω₂t)]. Using cos U + cos V = 2 cos[(U − V)/2] cos[(U + V)/2] separates the slow difference from the fast average.

If |ω₁ − ω₂| is much smaller than either frequency, the first cosine changes slowly while the second completes many cycles. The rapid oscillation has frequency fₐ = (f₁ + f₂)/2. Its slowly varying physical amplitude is 2a|cos[(ω₁ − ω₂)t/2]|. The absolute value matters: a negative prefactor reverses the rapid oscillation’s phase, but does not produce a negative loudness or amplitude.

Why beat frequency is the full difference

Loud maxima occur when the envelope cosine equals either +1 or −1. Therefore the amplitude magnitude repeats when its argument changes by π, not only when the signed cosine completes a full 2π cycle. Write |ω₁ − ω₂|Tᵦ/2 = π. Rearranging gives Tᵦ = 2π/|ω₁ − ω₂| = 1/|f₁ − f₂|. There are two loudness maxima during one full period of the signed envelope cosine. This removes the common but incorrect factor of two.

For a linear acoustic wave, mean intensity is related to the square of the oscillation amplitude. Averaging over the rapid vibration while the envelope changes slowly gives intensity proportional to cos²[(ω₁ − ω₂)t/2]. Its repetition rate is again |f₁ − f₂|. Beats are a modulation of intensity, not a series of occasions on which the source starts and stops. If the frequencies become exactly equal, the relative phase stops drifting and the reinforcement or cancellation is stationary rather than beating.

Tuning uses both the beat rate and the direction of change

A beat rate alone gives two possibilities. If one string has frequency 427 Hz and the beat rate is 5 Hz, the other could be 422 Hz or 432 Hz. To decide, perturb a known parameter. For a fixed length and linear density, a string’s natural frequencies are proportional to √F. Raising tension raises frequency. Observe whether that makes the frequency gap grow or shrink, using a sufficiently small adjustment so it does not cross the reference frequency without being noticed.

Pitch, harmonic mixture and timbre

An ideal normal mode produces one sinusoidal frequency, but a played instrument generally excites a mixture of modes. Two instruments can share the same fundamental, and hence the same nominal pitch, while differing in the amplitudes and phases of their overtones and in the way the sound begins and fades. These differences contribute to timbre, the quality that lets us distinguish a violin from a sitar. The point of bowing or plucking, the instrument body and its coupling to air all affect the mixture. One frequency number alone does not specify the complete time waveform.

The connection extends beyond strings and air columns. Resonant stone pillars can produce distinct notes when tapped because their elasticity, density, geometry and mechanical support determine their vibrations. The musical-pillar example in the Nellaiappar temple in Tamil Nadu illustrates this physical idea; related examples occur at Hampi and other southern Indian sites. The source groups pillars by their musical role: Shruti for notes or swaras, Gana Thoongal for tunes associated with ragas, and Laya Thoongal for rhythmic response. A stone does not need to be a conventional manufactured instrument to have resonant modes.

Sound can also carry information about obstacles

A bat emits ultrasound and analyses returning echoes. A simple travel-time estimate gives obstacle distance d = vΔt/2 because the sound must go out and back. Differences in arrival direction and timing at the ears help locate the obstacle. The strengths and frequency patterns of returned waves depend on how the target scatters sound; wavelength relative to object size affects the available detail. Thus echoes can carry clues to direction, size and nature as well as distance. This is information transmitted by the wave pattern, not a flow of air from the obstacle to the animal.

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