Half-Wave Resonance

acoustics resonance formants
Last updated: 2025-01-19

Half-Wave Resonance

For a tube that is closed (or open) at both ends, the resonance characteristics differ from the quarter-wave resonator. Understanding half-wave resonance helps explain phenomena such as the acoustic effects of complete lip closure and the difference between open and closed organ pipes.

Resonance Frequencies for Half-Wave Tubes

For a tube with identical boundary conditions at both ends, the resonance frequencies are:

Fₙ = n(c/2L)

where n is any positive integer (1, 2, 3, 4…). The derivation of this formula follows along the same lines as the quarter-wave derivation, with the exception that the reflection coefficients are identical at both ends (either near +1 or near -1).

Comparison with Quarter-Wave Resonance

Note that the first formant frequency for a half-wave tube is c/2L, which is twice as high as for a quarter-wave tube (for the same tube length). This important difference is best remembered by drawing an analogy with musical instruments.

Organ Pipe Analogy

Figure 6.5: Photograph of several ranks of organ pipes viewed from top. Closed pipes have lower resonance frequencies than open pipes.

In organ pipe design:

  • Open pipes (open at top): Produce the highest notes for a given length
  • Closed pipes (closed at top): Produce the lowest notes for the same length
  • Partial closure: Shifts half-wave resonance frequencies downward toward quarter-wave frequencies

For a given pipe length, the closed pipes produce lower tones. This principle directly applies to vocal tract acoustics.

Application to the Vocal Tract

Lip Rounding and Closure

In the vocal tract, lip rounding can be thought of as a partial closure of the open end. This has the acoustic effect of:

  1. Shifting formant frequencies downward - Moving from half-wave toward quarter-wave resonances
  2. Darkening the vowel quality - Lower formants create a darker timbre
  3. Increasing acoustic length - The tract acts acoustically longer than its physical length

Complete Lip Closure

If the lips were to close completely, as in the voiced consonant [b]:

  • The quarter-wave resonator would change to a half-wave resonator
  • F₁ would drop from 500 Hz to 0 Hz (pressure flat across the tube)
  • F₂ would drop from 1,500 Hz to 1,000 Hz (pressure maximum at center, minima at both ends)
  • All formant frequencies would shift downward by 500 Hz
  • The lowest formant would effectively drop out (no radiation at 0 Hz)
  • F₂ of the quarter-wave resonator would become F₁ of the new half-wave resonator

Formant Frequency Relationship

For tubes of the same length:

Quarter-wave (one end closed):

F₁ = c/4L = 500 Hz (for L = 17.5 cm)
F₂ = 3c/4L = 1,500 Hz
F₃ = 5c/4L = 2,500 Hz

Half-wave (both ends same):

F₁ = c/2L = 1,000 Hz (for L = 17.5 cm)
F₂ = 2c/2L = 2,000 Hz
F₃ = 3c/2L = 3,000 Hz

The half-wave tube has formants at all integer multiples of the fundamental, while the quarter-wave tube has only odd multiples.

Practical Implications

Lip Spreading vs. Lip Rounding

  • Lip spreading: Creates a flared opening (like a trumpet bell), reducing pressure before the lips, decreasing acoustic length, and raising formant frequencies
  • Lip rounding: Partially covers the opening, increasing pressure behind the lips, increasing acoustic length, and lowering formant frequencies

Jaw Lowering

Jaw lowering has a similar effect to lip spreading, particularly with regard to F₁. The increased mouth opening:

  • Reduces acoustic length
  • Raises formant frequencies
  • Is especially effective for raising F₁ (Sundberg, 1977)

This is particularly important in female singing at high pitches, where sopranos tend to align F₁ with F₀ to maximize output power.

Summary

Half-wave resonators have identical boundary conditions at both ends and resonate at all integer multiples of c/2L, compared to quarter-wave resonators which resonate at only odd multiples of c/4L. For the same length, half-wave resonators have higher formant frequencies. In the vocal tract, lip rounding shifts resonances from half-wave toward quarter-wave characteristics, lowering formant frequencies and darkening vowel quality.

Key Takeaways

  • ✅ Half-wave tubes have formant frequencies Fₙ = n(c/2L) for all positive integers n
  • ✅ First formant of half-wave tube is twice the frequency of a quarter-wave tube of equal length
  • ✅ Closed organ pipes produce lower tones than open pipes of the same length
  • ✅ Lip rounding shifts formants downward by increasing acoustic length
  • ✅ Complete lip closure would convert the vocal tract from quarter-wave to half-wave resonance
  • ✅ Lip spreading and jaw lowering raise formant frequencies by decreasing acoustic length

Further Reading

  1. Fant, G. (1960). Acoustic theory of speech production. The Hague: Mouton.
  2. Sundberg, J. (1977). The acoustics of the singing voice. Scientific American, 236, 82-91.
  3. Titze, I. R. (2000). Principles of voice production (2nd ed.). National Center for Voice and Speech.
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