Foundations — Speed of sound — a = √(γRT) — derivation

Before you can derive that formula, you need to be fluent in every letter and squiggle inside it. This page builds each one from absolutely nothing, in the order that lets each idea lean on the one before it. Nothing here is assumed — if the parent note used it, we build it here. (We will not use the symbols $a$, $\gamma$, $R$ or $T$ in any calculation until each has its own section below.)

0 — The star of the show: $a$, the speed of sound

Let's name the quantity the whole topic is about, before anything else.

The picture: clap your hands. A thin shell of slightly-squeezed air races outward and reaches a friend's ear a moment later. The speed of that shell is $a$. Everything below exists to compute this one number.

1 — What a "gas" actually is (the picture behind everything)

Forget equations for a second. A gas is a huge crowd of tiny balls (molecules) flying around in all directions, bouncing off each other and off the walls. There is mostly empty space between them.

Three everyday things we can measure about this crowd become our three main symbols:

• How many balls are packed into a box → density.
• How hard they hammer the walls → pressure.
• How fast they zip around → temperature.

Everything in the topic is a relationship between these three. Let's meet each one.

2 — Density $\rho$ (how heavy the gas is)

The picture: in figure s01, count the balls inside one box. More balls in the same box = higher $\rho$. Air at sea level has $\rho \approx 1.2kg/{\mathrm{m}}^3$.

Why the topic needs it: $\rho$ is the inertia half of the tug-of-war. Heavy gas is sluggish — hard to get moving — so a push travels through it more slowly. When you see $\rho$ in the denominator later ($a^2=\gamma p/\rho$), that is inertia slowing sound down.

3 — Pressure $p$ (how hard the gas pushes)

The picture: each ball that hits a wall gives it a tiny kick. Millions of kicks per second, added up and spread over the wall's area, is pressure. Sea-level air pushes at $p\approx 101,000\mathrm{Pa}$.

Why the topic needs it: $p$ is the stiffness half. When you squeeze a gas, its molecules crowd together and push back harder — pressure rises. That "push back" is exactly what lets one layer shove the next, passing the sound wave along. Stiffer (bigger push-back per squeeze) → faster sound.

4 — Temperature $T$ (how fast the molecules zip)

The picture: in figure s02, cold gas (left) has short arrows — slow, lazy molecules. Hot gas (right) has long arrows — fast, frantic molecules. Same crowd, more energy.

Why the topic needs it: the punchline of the whole chapter is that $a$ depends only on $T$. Fast molecules pass the "push" along fast — sound literally rides on the thermal jiggling. Hot gas = zippy molecules = faster sound.

5 — The ideal gas law: tying $p$, $\rho$, $T$ together

These three are not independent — squeeze the gas or heat it and they move together. The rule that links them is the ideal gas law.

Rearranged, this gives the single most important combination in the topic: $\frac{p}{\rho }=RT$

Why this matters for sound: later we find $a^2=\gamma \frac{p}{\rho }$. Because $p/\rho =RT$, the $p$ and $\rho$ collapse into just $RT$. That is why pressure and density individually drop out and only temperature survives. Learn this line — it is the hinge of the whole derivation. See Ideal gas law and specific gas constant.

6 — The specific gas constant $R$ (and why it's not $8.314$)

The picture: think of $R$ as a gas's personality. Light molecules (small $M_{\text{mol}}$) → big $R$ → given the same $T$, they move faster and carry sound faster. That is why helium ($M_{\text{mol}}=4$ g/mol, huge $R$) has sound near $1000$ m/s. See Ideal gas law and specific gas constant.

7 — The heat-capacity ratio $\gamma$ (the "no time for heat" correction)

Here is the subtle one — the symbol that makes this topic different from a plain gas-law problem.

When you compress a gas you can do it two ways:

• Slowly — heat leaks out, temperature stays constant. This is isothermal.
• Fast — heat has no time to escape, so the trapped energy also heats the gas. This is adiabatic.

A sound wave squeezes gas thousands of times a second — far too fast for heat to leak. So it is adiabatic, and the extra self-heating makes the gas push back even harder than the slow case. That extra stiffness is captured by one number, $\gamma$. See Adiabatic vs isothermal processes.

The picture: $\gamma$ is a "stiffness bonus." Newton forgot it and got $\approx 280$ m/s (isothermal). Laplace put it back and got $340$ m/s, matching reality. The $\sqrt{\gamma }$ is exactly that $1.4$ correction. See Isentropic relations p ∝ ρ^γ.

8 — Reading the squiggles: $d$, $\partial$, and the derivative

The derivation talks about tiny changes and slopes, so you need this notation.

The picture: in figure s03 the curve is $p$ against $\rho$. Pick a point, zoom in until the curve looks straight, and the steepness of that little line is $dp/d\rho$. A steep slope means "a small squeeze causes a big pressure jump" — a stiff gas. That is why $a^2=dp/d\rho$: the slope literally is the stiffness.

9 — Entropy $s$, "isentropic", and the constant $C$

For an isentropic ideal gas, pressure and density are locked together by the rule $p=C{\rho }^{\gamma }$.

Why the topic needs it: the subscript $s$ in $(\partial p/\partial \rho {)}_s$ is what forces us onto the adiabatic curve $p=C{\rho }^{\gamma }$ instead of the isothermal one. That single letter is where $\gamma$ sneaks into the answer. See Isentropic relations p ∝ ρ^γ.

10 — Mach number $M$ (why we bother at all)

The picture: $M=1$ means moving at the speed of sound; $M=0.5$ is half that. Below $M\approx 0.3$ the air behaves as if incompressible and you can ignore all of this. Above it, density changes matter — which is the whole reason we need $a$ in the first place. See Mach number and flow regimes and Compressibility and why M > 0.3 matters.

How the foundations feed the topic

Equipment checklist

Cover the right side and test yourself — if any answer is fuzzy, re-read that section before the derivation.