Add pressure phase before squaring
Two same-frequency pressure signals can reinforce or cancel locally. Let the first peak pressure amplitude be 1 Pa and the second be b Pa. Their relative phase is φ=2πΔr/λ when their source phases match and the only phase difference comes from a path-length difference Δr in a uniform medium. Local amplitudes already include any unequal spreading or attenuation.
Learning outcome: Predict values at path differences 0,.25,.5,1 wavelengths for b=.5 and 1. Retain pressure amplitude and its square separately. Explain why halving a pressure amplitude reduces its mean-square by four, and why cancellation at one point is not a whole-field energy balance.
Starting recap
Separate local oscillation from propagation, amplitude from its square, and a calibration fit from a fresh prediction. Use declared phase, medium frame and boundary conditions.
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At b=.5 and Δr/λ=.5, the resultant peak pressure is .5 Pa and the normalized mean-square is .25. At the same amplitude ratio with zero phase difference, these become 1.5 Pa and 2.25. At b=1 and half a wavelength, the ideal local cancellation is exactly zero.
Open the experimentPredict
With amplitude ratio .5 and a half-wavelength path difference, what is the normalized local mean-square pressure?
Experiment
Predict values at path differences 0,.25,.5,1 wavelengths for b=.5 and 1. Retain pressure amplitude and its square separately. Explain why halving a pressure amplitude reduces its mean-square by four, and why cancellation at one point is not a whole-field energy balance.
Add pressure phase before squaring
Worked teaching model · synthetic values · Two coherent same-frequency local pressure signals have phase difference 2pi DeltaPath/lambda and first peak amplitude 1 Pa. Fixed amplitude ratio includes propagation losses. Report normalized local mean-square pressure, not a direction-independent net energy flux in a standing wave.
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Local pressure interference versus path phase
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: same amplitudes and coherent frequency (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Local pressure interference versus path phase
same amplitudes and coherent frequency
| path difference / wavelength | normalized mean-square pressure |
|---|---|
| 0 | 2.25 |
| 0.025 | 2.23769 |
| 0.05 | 2.20106 |
| 0.075 | 2.14101 |
| 0.1 | 2.05902 |
| 0.125 | 1.95711 |
| 0.15 | 1.83779 |
| 0.175 | 1.70399 |
| 0.2 | 1.55902 |
| 0.225 | 1.40643 |
| 0.25 | 1.25 |
| 0.275 | 1.09357 |
| 0.3 | 0.940983 |
| 0.325 | 0.79601 |
| 0.35 | 0.662215 |
| 0.375 | 0.542893 |
| 0.4 | 0.440983 |
| 0.425 | 0.358993 |
| 0.45 | 0.298943 |
| 0.475 | 0.262312 |
| 0.5 | 0.25 |
| 0.525 | 0.262312 |
| 0.55 | 0.298943 |
| 0.575 | 0.358993 |
| 0.6 | 0.440983 |
| 0.625 | 0.542893 |
| 0.65 | 0.662215 |
| 0.675 | 0.79601 |
| 0.7 | 0.940983 |
| 0.725 | 1.09357 |
| 0.75 | 1.25 |
| 0.775 | 1.40643 |
| 0.8 | 1.55902 |
| 0.825 | 1.70399 |
| 0.85 | 1.83779 |
| 0.875 | 1.95711 |
| 0.9 | 2.05902 |
| 0.925 | 2.14101 |
| 0.95 | 2.20106 |
| 0.975 | 2.23769 |
| 1 | 2.25 |
| 1.025 | 2.23769 |
| 1.05 | 2.20106 |
| 1.075 | 2.14101 |
| 1.1 | 2.05902 |
| 1.125 | 1.95711 |
| 1.15 | 1.83779 |
| 1.175 | 1.70399 |
| 1.2 | 1.55902 |
| 1.225 | 1.40643 |
| 1.25 | 1.25 |
| 1.275 | 1.09357 |
| 1.3 | 0.940983 |
| 1.325 | 0.79601 |
| 1.35 | 0.662215 |
| 1.375 | 0.542893 |
| 1.4 | 0.440983 |
| 1.425 | 0.358993 |
| 1.45 | 0.298943 |
| 1.475 | 0.262312 |
| 1.5 | 0.25 |
| 1.525 | 0.262312 |
| 1.55 | 0.298943 |
| 1.575 | 0.358993 |
| 1.6 | 0.440983 |
| 1.625 | 0.542893 |
| 1.65 | 0.662215 |
| 1.675 | 0.79601 |
| 1.7 | 0.940983 |
| 1.725 | 1.09357 |
| 1.75 | 1.25 |
| 1.775 | 1.40643 |
| 1.8 | 1.55902 |
| 1.825 | 1.70399 |
| 1.85 | 1.83779 |
| 1.875 | 1.95711 |
| 1.9 | 2.05902 |
| 1.925 | 2.14101 |
| 1.95 | 2.20106 |
| 1.975 | 2.23769 |
| 2 | 2.25 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Relative phase | 3.14159 | rad |
| Resultant peak pressure amplitude | 0.5 | Pa |
| Normalized mean-square pressure | 0.25 | ratio to first signal |
| Minimum possible normalized mean-square | 0.25 | ratio |
| Maximum possible normalized mean-square | 2.25 | ratio |
A half-cycle phase difference gives complete cancellation only when the local amplitudes match.
Add the two pressure phasors, then square their resultant magnitude: 1+b²+2b cos(phi). The equivalent nonnegative form avoids a negative round-off at exact cancellation. Path phase assumes equal source phase and a uniform propagation wavelength. In a multidirectional or standing-wave field this local pressure statistic alone does not determine net vector energy flow; do not label it unconditionally as transmitted acoustic power.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A half-cycle phase difference gives complete cancellation only when the local amplitudes match.
Add the two pressure phasors, then square their resultant magnitude: 1+b²+2b cos(phi). The equivalent nonnegative form avoids a negative round-off at exact cancellation. Path phase assumes equal source phase and a uniform propagation wavelength. In a multidirectional or standing-wave field this local pressure statistic alone does not determine net vector energy flow; do not label it unconditionally as transmitted acoustic power.
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