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Theory

Linear acoustics refresher

Pressure, particle velocity, impedance, intensity and decibels, the vocabulary every later step uses.

Week 1 of 4; the paper is due at the end of week 4

Goals

  • Write the linear wave equation for sound pressure and say where each term comes from.
  • Convert between peak pressure, RMS pressure, intensity and dB SPL without looking it up.
  • Estimate how much a 60 kHz carrier is absorbed over a few metres of room air.

Why this matters

Every number in this project is a sound pressure level, an intensity or an absorption rate. The parametric array also lives at the edge of where linear acoustics stops working, so you need to know exactly what the linear theory says before you can see where it fails.

The physics

For small disturbances in a still fluid, mass conservation, Euler’s equation and an equation of state combine into the wave equation for the acoustic pressure pp:

∇2p−1c02∂2p∂t2=0,c02=(∂P∂ρ)s.\nabla^2 p - \frac{1}{c_0^2}\frac{\partial^2 p}{\partial t^2} = 0, \qquad c_0^2 = \left(\frac{\partial P}{\partial \rho}\right)_s .

For a plane wave the pressure and particle velocity are in phase, p=ρ0c0up = \rho_0 c_0 u, so the characteristic impedance of air is Z0=ρ0c0≈413 Pa s/mZ_0 = \rho_0 c_0 \approx 413\ \mathrm{Pa\,s/m} at 20 °C. The time-averaged intensity and the sound pressure level are

I=prms2ρ0c0,Lp=20log⁡10prms20 μPa dB.I = \frac{p_\mathrm{rms}^2}{\rho_0 c_0}, \qquad L_p = 20 \log_{10}\frac{p_\mathrm{rms}}{20\ \mu\mathrm{Pa}}\ \text{dB}.

Real air also absorbs sound. A wave p∝e−αxcos⁡(ωt−kx)p \propto e^{-\alpha x}\cos(\omega t - kx) loses 8.686 α8.686\,\alpha dB per metre. At audio frequencies α\alpha is tiny, but at the 60 kHz carrier used in the simulation it is large enough to matter over the length of a lab, and most of it comes from the vibrational relaxation of oxygen and nitrogen, not from viscosity.

Hints

Hint 1Start from three equations, not one

The wave equation is not a starting point. Write the linearized continuity equation, the linearized Euler equation and p=c02ρ′p = c_0^2 \rho', then eliminate the velocity. Doing it once yourself makes the nonlinear version in a later step much easier to read.

Hint 2Check your dB arithmetic with one known number

What RMS pressure is 94 dB SPL? If you get 1 Pa you have the reference right. Then work out the peak pressure of a 140 dB tone. You will need that number for the shock-distance estimate in the KZK steps.

Hint 3Humidity changes the answer a lot

Compute α\alpha at 60 kHz for 20 % and 60 % relative humidity using ISO 9613-1. (kzk_sim.py now has the formula built in as iso9613_alpha(), so you can check your hand calculation.) How many metres does it take to lose 10 dB in each case? Keep this calculation; it reappears when you set up the simulation and when you explain your measured beam.

What past teams did

2026–27 team

This is the first team on the project, so there are no notes from earlier teams yet. When you finish this step, add what you wish you had known on day one.