Longitudinal Sound: Pressure and Displacement
For a small-amplitude plane longitudinal wave, excess pressure is minus the bulk modulus times the spatial derivative of particle displacement, making pressure amplitude Bk times displacement amplitude and placing pressure and displacement ninety degrees out of phase.
Why this shows up in the exam
Pressure-displacement phase questions · Sound-amplitude conversions · Open- and closed-end boundary interpretation
Learn the idea
Pressure and particle displacement in a travelling sound wave are quarter-cycle shifted in space. Where neighbouring particles crowd together pressure rises; maximum displacement itself does not mark maximum compression, because compression depends on the displacement gradient.
🧠 Memory hook: Pressure follows the slope of displacement, not displacement itself.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- Delta p = -B partial(s)/partial(x) — pressure from longitudinal strain
- p_max = rho v omega s_max — pressure-amplitude and displacement-amplitude relation
How to approach it
- 1Write the displacement wave
- 2Differentiate with respect to position
- 3Apply the correct pressure or displacement boundary condition
Common slip-ups that cost marks
- •Putting pressure and displacement maxima at the same position
- •Using transverse-string formulas for air pressure
- •Confusing particle speed with sound speed
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
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