MECH7029 Chap.7 Standing Waves, Junctions and Duct Modes
Standing Waves, Junctions and Duct Modes
Match fields at boundaries
A duct field is a superposition of waves, so reflection creates pressure and particle-velocity patterns with different node positions. At an area change, acoustic pressure and volume flow must be matched.
The plane-wave model is conditional: transverse modes become propagative only after their axial wavenumber becomes real, and open ends require an effective rather than purely geometric termination.
Reflection phase fixes nodes
A rigid termination imposes zero normal velocity, giving same-sign pressure reflection and a pressure antinode at the wall.
An ideal opening approaches pressure release, so reflected pressure reverses sign and the pressure node lies near the end. Real radiation moves that node outside the physical opening through end correction.
Pressure and velocity standing patterns are displaced from one another, and large local pressure does not automatically imply downstream power.
Junctions conserve pressure and total flow
At an ideal sudden area change, pressure remains continuous and volume flow is matched. Particle velocity alone is not continuous because cross-sectional area changes.
The backward wave carries an opposite velocity sign for a given pressure amplitude, so arrows should be drawn before equations are written. Reflection and transmission limits then become intuitive: a very large expansion approaches an open end, while a vanishing downstream area approaches a rigid closure.
Higher modes add transverse structure selected by wall boundary conditions.
The total wavenumber is divided between axial and transverse components. Below cut-on, the axial part is imaginary and the mode decays; above cut-on, it becomes real and transports structure downstream. Rectangular and circular ducts use different eigenvalues, so geometry must be named before a cut-on formula is chosen.
What this chapter covers
- 01
Travelling and reflected waves
- 02
Pressure and velocity nodes
- 03
Area-junction continuity
- 04
Reflection and transmission coefficients
- 05
Higher-mode cut-on
- 06
Open-end correction
Worked example · free
Decide whether a plane-wave model is admissible
- 3Substitute sound speed and radius with consistent units.
- 2Compute the cut-on frequency.
- 3Compare the operating frequency and qualify the model.
Key terms
- Standing Wave
- A spatial interference pattern formed by oppositely travelling waves.
- Volume Flow
- Particle velocity integrated over duct cross-sectional area.
- Reflection Coefficient
- The complex ratio of reflected to incident wave amplitude.
- Cut-on Frequency
- The threshold above which a duct mode can propagate axially.
- End Correction
- An added effective length accounting for air motion beyond an opening.
Standing Waves, Junctions and Duct Modes FAQ
Why are pressure and velocity nodes not at the same place?
For a reflected plane wave, the pressure and particle-velocity reflection signs differ. Their interference patterns are therefore shifted by a quarter wavelength, so a pressure maximum coincides with a velocity minimum for ideal total reflection.
What is continuous across an ideal area junction?
Acoustic pressure and total volume flow are matched. Particle velocity alone generally changes because the cross-sectional area changes, which is why using velocity continuity produces the wrong reflection coefficient.
What changes above the first cut-on frequency?
Transverse field structure can carry energy downstream. A single plane-wave amplitude no longer represents the full field, and junctions or controls can couple energy among propagating modes.
Exam move
Divide revision into boundaries, junctions and modes. For boundaries, sketch incident and reflected pressure arrows at rigid and open ends, then draw pressure and velocity nodes separately. Add the appropriate end correction and predict how it shifts resonance before calculating. For junctions, label every pressure amplitude with direction.
Write pressure continuity on one line and volume-flow continuity on the next, letting propagation direction determine the velocity sign. Test the solution in both large-expansion and closed-end limits, and check power with cross-sectional area included. For modes, derive the transverse condition from the wall boundary rather than beginning with a memorised cut-on formula.
Draw the wavenumber triangle and explain how the axial component changes from imaginary to real. Compute the first threshold for both a rectangular and a circular example, keeping their eigenvalues distinct. Place an operating frequency below, near and above cut-on and describe evanescence, slow decay or propagation.
End every one-dimensional control calculation with a mode-validity sentence and a note about local evanescent structure near a discontinuity. Maintain an error ledger for reflection sign, velocity direction, pressure versus volume flow, end correction, geometry choice and cut-on comparison. Repair the sketch first whenever the algebra gives the wrong rigid-wall or open-end limit.
As a final transfer exercise, place a sudden junction near an open termination and identify which conditions apply at each location; this prevents two valid boundary rules from being imposed on the same cross-section.
