Modal frequency density scales with dimensionality from constant to f-squared

Evidence

The density of modes increases rapidly as a function of frequency, and is proportional to the square of the frequency. Note that the density of modal frequencies df is approximately constant for one-dimensional acoustic systems, such as strings (Figure 2.13) and tubes (Figure 2.12), directly proportional to the frequency (df ∝f) for two-dimensional acoustic systems, such as a membrane (Figure 2.15), and proportional to the square of the frequency (df ∝f 2) for three-dimensional acoustic systems, such as rooms. — Ville Pulkki, Matti Karjalainen, Communication Acoustics- An…, 294

Categorization

Category: Factual Claims

• DEFINE: Clarifies the scaling law relating modal density to the dimensionality of the acoustic system.

Related

• Modal density scales with spatial dimension, from flat in 1D to squared in 3D. (similarity 0.95)
• Two- and three-dimensional systems show rapidly rising modal density (similarity 0.82)
• Three-dimensional rooms yield denser mode distributions at higher frequencies (similarity 0.82)
• Room mode density rising smoothly with frequency signals good acoustics (similarity 0.80)
• Loudness scales with frequency squared, requiring large amplitude at low frequencies (similarity 0.80)
• Loudness is not reducible to sound pressure level or signal power (similarity 0.78)

Broader Connections

• Loudness scales with frequency squared, requiring large amplitude at low frequencies #complicates (0.75) — Eric J. Heller, Why You Hear What You Hear- An…, 667