Whether echoes constructively add in Fourier analysis depends on wavelength relative to echo timing.

Evidence

At low frequency, f1, the cosine in the Fourier analysis of the signal is long wavelength, and the signal multiplied by the cosine adds up all the echoes with the same sign (first inset, figure 12.3), giving a strong proximity resonant enhancement to the power spectrum. At higher frequency, f2, there is some cancellation, and a much weaker power spectrum (second inset). At still higher frequency, f3, many of the peaks again line up with the positive parts of the cosine, resulting in a local maximum in the power. The frequencies f1, f2, and f3 are indicated on figure 12.4. — Eric J. Heller, Why You Hear What You Hear- An…, 1137

Categorization

Category: Causal Relationships

Related

• Fourier analysis of echoes yields alternating strong and weak spectral bands (similarity 0.90)
• A returning echo creates power-spectrum peaks spaced by its inverse time (similarity 0.85)
• Frequency measured in Hertz is dimensionally inverse seconds (similarity 0.84)
• Loudness-time functions reveal echoes and reverberation better than level-time ones (similarity 0.82)
• Echoes rearrange spectral energy without changing its total (similarity 0.82)
• Power fluctuations persist because echo arrival times unevenly populate Fourier components (similarity 0.80)
• Resonance arises when echo round-trip time equals an integer number of wave periods. (similarity 0.80)
• The last echo time sets the minimum width of spectral peaks. (similarity 0.80)
• Time-delayed sound self-addition imposes spectral notches via autocorrelation (similarity 0.79)
• Echo interference creates sinusoidal power fluctuations that average to zero (similarity 0.79)
• Any echo-producing system inevitably exhibits frequency power fluctuations (similarity 0.78)

Broader Connections

• Repetition pitch equals the reciprocal of the echo delay time #expands (0.85) — Eric J. Heller, Why You Hear What You Hear- An…, 2146
• Delays too short for echoes still audibly alter timbre through spectral combs #qualifies (0.80) — Eric J. Heller, Why You Hear What You Hear- An…, 2551
• Autocorrelation discriminates echoes from genuine distinct sounds. #expands (0.75) — Eric J. Heller, Why You Hear What You Hear- An…, 2532
• The Fourier transform pair converts signals between time and frequency domains #expands (0.70) — David Howard, Jamie Angus, coustics and Psychoacoustics, 2746