Foundations — Acoustic loads — SPL, octave band analysis

Before you can read the parent note, you must own every symbol it throws at you. This page builds each one from nothing — a plain meaning, a picture, and the reason the topic needs it. Read top to bottom; each brick rests on the one below it. This is the foundation for Acoustic Loads — SPL, Octave Band Analysis.

1. Pressure — the thing that is actually oscillating

Picture a drum skin. Air molecules are constantly bumping into it from both sides. The steady background pushing is atmospheric pressure — about $10^5$ Pa (100,000 Pa). That never changes fast, so it does no shaking.

Sound is a tiny wobble on top of that steady value. When a sound wave passes, the pressure at one spot rises a little, then falls a little, then rises again — over and over. We give that wobble its own symbol.

Look at the figure: the flat dashed line is steady atmosphere; the wavy coral curve is the sound pressure riding on top. Why the topic needs this: everything — loudness, force on a panel, damage — comes from the size of that coral wobble, not the flat line underneath.

2. RMS — turning a wobble into one honest number

The wobble spends as much time positive as negative, so its plain average is zero — useless for measuring loudness. We need a number that says "how big is the swing" without cancelling itself out.

In the figure, the coral curve is $p(t)$; the lavender curve is $p(t{)}^2$ (always above zero); the mint dashed line is its average; and the butter line marks $p_{\text{rms}}$ — the square root of that average.

Where does the $\sqrt{2}$ come from? Take a clean wave $p(t)=p_{\text{peak}}\sin (t)$. To get the "mean of the squares" we average $p_{\text{peak}}^2{\sin }^2(t)$ over one full wobble. The only quantity we need is the average of ${\sin }^2(t)$. Look at the lavender curve in the figure: ${\sin }^2$ bounces between $0$ and $1$, and — because ${\sin }^2+{\cos }^2=1$ and both look identical just shifted — it spends exactly half its time above and half below, so its average is $\frac{1}{2}$. Doing the honest integral confirms it: $\text{mean of }{\sin }^2=\frac{1}{2\pi }{\int }_0^{2\pi }{\sin }^2(t)dt=\frac{1}{2}.$ So the mean of the squares is $p_{\text{peak}}^2\times \frac{1}{2}$, and the root of that is:

Why the topic needs it: the loudness formula, the intensity formula, and every force calculation use $p_{\text{rms}}$ — it is the single honest measure of "how big the squeeze."

3. Logarithms — squashing a giant range into small numbers

Human ears (and rocket fairings) handle pressures from a whisper ($20\times 10^{-6}$ Pa) to a jet ($200$ Pa). That is a ten-million-fold range. Writing "0.00002 Pa to 200 Pa" is clumsy. We want a scale where each step means "×10 bigger."

The figure shows the same pressure axis two ways: on top, a linear ruler where the whisper is crushed invisibly against zero; below, a log ruler where whisper, speech, and jet spread out into readable, evenly-spaced marks.

4. The decibel (dB) and SPL — the loudness number

Now we assemble the symbols. Loudness is a log of the ratio of your pressure to a reference whisper.

Reading the number: every +20 dB means ×10 more pressure; every +6 dB means ×2 pressure (because $20{\log }_{10}2\approx 6$). Why the topic needs it: fairing specs are always quoted in dB, so you must fluently convert both directions.

5. Intensity, density, speed of sound, impedance — the energy chain

The parent note derives where $p_{\text{ref}}=20\mu$Pa comes from, using four more symbols. Meet them.

Chaining these gives the key bridge the parent uses: $I=\frac{p_{\text{rms}}^2}{\rho c}$

6. Combining sounds — why intensities add, not decibels

Two rocket engines, 145 dB each, roaring together. You cannot add the dB numbers. Here is the machinery, using every symbol above. First, a shorthand: from now on we write $L$ for any single SPL value (a "level," in dB) — so $L_1$, $L_2$ are the levels of source 1 and source 2, and $L_{\text{total}}$ is the level of the two combined. $L$ is nothing new; it is just SPL wearing a shorter name.

Since doubling intensity is $+10{\log }_{10}(2)\approx +3$ dB, two equal uncorrelated sources = +3 dB, never +145. This same "un-log, add energy, re-log" recipe is exactly how octave bands are summed into one overall level.

When many bands or sources combine, the same recipe gives the OASPL (Overall Sound Pressure Level) — the single dB number that summarises the whole sound after all its parts are energy-added: $L_{\text{OASPL}}=10{\log }_{10}\left({\sum }_{i}10^{L_i/10}\right)\text{ dB}.$ This is the quantity the prerequisite map below points to, and the parent note computes it for real octave-band data.

7. Frequency and octave bands — sorting the wobble by speed

A real launch sound is not one clean wave — it is a jumble of slow and fast wobbles mixed together. We sort them by frequency.

The band runs from $f_{\text{lower}}=f_c/\sqrt{2}$ up to $f_{\text{upper}}=f_c\sqrt{2}$, so $f_{\text{upper}}/f_{\text{lower}}=2$ — exactly one octave.

Why the topic needs it: octave bands convert one messy sound into a short table of numbers (one SPL per band) that engineers compare against the panel's natural frequencies — and that connects straight to 3.6.11 Random Vibration — PSD, Miles' Equation.

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer aloud.