Speed of a longitudinal wave (speed of sound)

Clap your hands in a large empty hall and the sound comes back to you, delayed, off the far wall. Shout across a valley and the echo lags by seconds. Sound takes time to travel — and how much time depends entirely on what it is travelling through. A push transmitted hand to hand down a line of people is slow; the same push transmitted through a steel rod is almost instant. Why? Sound is a longitudinal wave: a chain of compressions and rarefactions, each layer of the medium shoving the next one forward and springing back, exactly like the coils of a slinky pushed from one end. Two properties of the medium decide how fast that shove-and-spring-back relay runs. First, how stiffly the medium resists being squeezed — a stiffer medium snaps back faster and passes the disturbance on quicker. Second, how heavy (dense) each layer is — a heavier layer is more sluggish to get moving, slowing the relay down. Stiffness speeds the wave up; inertia slows it down. That single trade-off, stiffness over density inside a square root, is the entire physics of this topic. Isaac Newton worked out this formula for gases nearly three and a half centuries ago — and got the speed of sound in air wrong by about 16%. It took Pierre-Simon Laplace, more than a century later, to spot the missing physics: sound compresses and expands air far too fast for heat to leak away, so the correct "stiffness" is not the ordinary (isothermal) one Newton assumed but a hotter, stiffer adiabatic one. That one-word correction — replacing an isothermal process with an adiabatic one — turned a 16% error into a match good to about 2%. This article builds the general longitudinal-wave-speed formula, walks through the Newton-Laplace correction for sound in air, and uses both to compare wave speeds across steel, water and air, and to solve the same kind of $v = f\lambda$ problems you will see in your board and competitive exams.

Key takeaways

A longitudinal wave moves as a sequence of compressions and rarefactions along the direction of travel, like the coils of a slinky pushed end to end — sound is the everyday example.
The general speed of a longitudinal wave in any medium is $v = \sqrt{B/\rho}$ , where $B$ is the bulk modulus (resistance to compression) and $\rho$ is the density.
This has exactly the same "restoring effect over inertia, inside a square root" shape as $v = \sqrt{T/\mu}$ for a stretched string.
Newton's formula used the isothermal bulk modulus $B = P$ and predicted the speed of sound in air too low by about 16%.
Laplace's correction: sound compressions happen too fast for heat exchange, so the process is adiabatic, giving $B = \gamma P$ and the corrected formula $v = \sqrt{\gamma P/\rho}$ , which matches experiment (about $332$ - $343$ m/s near room temperature) to within about 2%.
Solids have a far larger bulk modulus than gases, so sound travels much faster through steel (about $4530$ m/s) than through air (about $340$ m/s), even though solids are also denser.
Key idea: the speed of a longitudinal wave is v = sqrt(B/rho), and for sound in a gas Laplace's fix, replacing the isothermal bulk modulus P with the adiabatic bulk modulus gamma P, turns Newton's too-slow estimate into the value we actually measure.

The core insight

The speed of any longitudinal wave is $v = \sqrt{B/\rho}$ (elastic stiffness over density), and for sound in a gas the correct stiffness is the adiabatic bulk modulus $B = \gamma P$ , not Newton's isothermal $B = P$ , because compressions happen too fast for heat to equalise.

Where it fits in the syllabus

This topic sits inside the CBSE Class 11 Waves chapter (Unit: Oscillations and Waves), directly after the speed of a transverse wave on a stretched string. The prescribed scope is the general longitudinal-wave-speed formula $v = \sqrt{B/\rho}$ and a qualitative treatment of the Newton-Laplace correction for the speed of sound in a gas — you are not expected to derive $B=\gamma P$ from thermodynamics, only to understand and apply it. You are expected to use $v = f\lambda$ together with a given (or computed) value of $v$ , and to compare speeds of sound across different media using given elastic moduli and densities. A full thermodynamic derivation of the adiabatic bulk modulus, and any treatment of the Doppler effect, are outside this year's prescription; where either topic brushes against this one, it is flagged and left for later.

The physical picture

Picture a long line of masses connected by springs, laid along the direction the wave travels — this is a mechanical model of any elastic medium, solid, liquid or gas. Push the first mass; it compresses the spring ahead of it, which pushes the next mass, and so on down the line. The disturbance — a moving zone of compression followed by a zone of rarefaction (stretching) — is a longitudinal wave, and it is exactly what a sound wave is doing inside air, water or a solid rod.

Figure

A row of particles along a horizontal line showing alternating compressed and spread out regions labelled compression and rarefaction, with an arrow marking the direction of wave travel and the wavelength marked between two neighbouring compressions

For a longitudinal wave in any medium, the speed is set by two properties of that medium:

v = \sqrt{\frac{B}{\rho}}

where: $B$ is the bulk modulus of the medium, in pascals (Pa) — a measure of how strongly it resists being squeezed into a smaller volume; and $\rho$ is the density of the medium, in kilograms per cubic metre. A larger $B$ (a stiffer medium) means each layer snaps back and shoves the next layer along faster, so $v$ rises. A larger $\rho$ (a heavier medium) means each layer is more sluggish to accelerate, so $v$ falls. Notice the structure: it is exactly $v = \sqrt{T/\mu}$ for a string, with the restoring quantity (bulk modulus instead of tension) on top and the inertia quantity (density instead of linear mass density) on the bottom. For sound specifically travelling through a gas like air, the bulk modulus itself depends on how the gas responds while it is being compressed — and that is where Newton's original reasoning went wrong, and where Laplace's fix comes in.

v = \sqrt{\frac{\gamma P}{\rho}}

where: $P$ is the pressure of the gas, $\rho$ is its density, and $\gamma = C_p/C_v$ is the ratio of specific heat at constant pressure to specific heat at constant volume (for air, $\gamma \approx 1.4$ ). This is the Newton-Laplace formula, and it is simply the general formula $v=\sqrt{B/\rho}$ with the adiabatic bulk modulus $B = \gamma P$ substituted in place of the isothermal one Newton originally used.

The physical picture continued: why isothermal was wrong

Newton assumed that as a sound wave compresses a small pocket of air, that pocket has time to exchange heat with its surroundings and stay at constant temperature — an isothermal process, for which the bulk modulus works out to simply $B = P$ (the gas pressure itself). Plugging this into $v = \sqrt{B/\rho}$ for air at standard conditions gives a speed of only about 280 m/s — roughly 16% below the experimentally measured value of about 340 m/s. Laplace's insight was that the compressions and rarefactions of a sound wave happen far too quickly — thousands of times a second — for heat to flow between the warmer compressed regions and the cooler rarefied regions. Each little parcel of gas heats up as it is compressed and cools as it expands, with essentially no time to share that heat with its neighbours. That is an adiabatic process, not an isothermal one, and for an adiabatic process the bulk modulus is larger: $B = \gamma P$ , where $\gamma > 1$ . Substituting this adiabatic bulk modulus gives the Newton-Laplace formula above, and it brings the calculated speed of sound in air to within about 2% of the measured value — a correction of roughly $\sqrt{1.4} \approx 1.18$ , exactly enough to close most of the 16% gap.

Derivation

Derivation AGetting from the general longitudinal wave-speed formula to the Newton-Laplace formula for sound in a gas

The goal here is not a full thermodynamic derivation of the adiabatic bulk modulus (that is beyond the Class 11 syllabus and is quoted, not derived) but to show clearly, step by step, exactly how the single substitution " $B = \gamma P$ instead of $B = P$ " turns the general elastic wave-speed formula into the specific, experimentally correct formula for the speed of sound in air, and why that one substitution is worth understanding rather than memorising as two unrelated equations.

Step 1. Start from the general result for any longitudinal wave in an elastic medium: $v = \sqrt{B/\rho}$ , where $B$ is whatever bulk modulus correctly describes how that medium resists compression under the conditions of the wave, and $\rho$ is its density. This formula itself is quoted at this level (its full derivation, tracking the force on a compressed element of the medium, uses calculus beyond this chapter's scope) — what matters here is applying it correctly.

Step 2. Identify what "resists compression" means for a gas specifically. For a gas, the bulk modulus is defined through how pressure changes as volume changes: a stiffer (harder to compress) gas is one where a small volume change produces a large pressure change. This relationship depends on what stays fixed while the gas is compressed — and that depends on how fast the compression happens.

Step 3. Apply Newton's assumption first. Newton assumed the compression in a sound wave happens slowly enough for the gas to stay at the same temperature as its surroundings throughout — an isothermal process. For an isothermal process the bulk modulus of an ideal gas works out to simply $B_{\text{isothermal}} = P$ , the gas pressure itself. Substituting into Step 1 gives Newton's formula, $v_{\text{Newton}} = \sqrt{P/\rho}$ .

Step 4. Apply Laplace's correction. Sound oscillations in air occur many hundreds to many thousands of times per second — far too fast for heat generated in a compressed pocket of gas to conduct away to the surrounding, cooler, rarefied pockets before the next compression arrives. The process is therefore adiabatic (no heat exchange), not isothermal. For an adiabatic process, the bulk modulus of an ideal gas is larger by a factor of $\gamma$ : $B_{\text{adiabatic}} = \gamma P$ , where $\gamma = C_p/C_v$ (about $1.4$ for the diatomic gases, nitrogen and oxygen, that make up most of air).

Step 5. Substitute the adiabatic bulk modulus back into the general formula from Step 1: $v = \sqrt{B_{\text{adiabatic}}/\rho} = \sqrt{\gamma P/\rho}$ . This is the Newton-Laplace formula. Since $\gamma \approx 1.4$ for air, this predicts a speed larger than Newton's isothermal estimate by a factor of $\sqrt{1.4} \approx 1.18$ , which is exactly the correction needed to bring the calculated speed of sound close to the experimentally measured value. Verified: $\gamma_{\text{air}} = 1.4$ , so $\sqrt{\gamma} \approx 1.183$ ; scaling Newton's isothermal estimate of about $278$ m/s by this factor gives about $329$ m/s, matching the Newton-Laplace worked example below.

Worked examples

Board level - speed of sound in steel

Steel has a bulk modulus of about $1.6\times10^{11}$ Pa and a density of $7800$ kg/m³. Estimate the speed of a longitudinal (sound) wave in steel using $v=\sqrt{B/\rho}$ , and compare it with the speed of sound in air (about $340$ m/s).

Step 1. Write down the general longitudinal-wave-speed formula for a solid: $v = \sqrt{B/\rho}$ , using the given bulk modulus and density directly (no $\gamma$ correction is needed here — that correction is specific to gases undergoing rapid, adiabatic compression; a solid rod does not need it at this level).

Step 2. Substitute the numbers. $v = \sqrt{\dfrac{1.6\times10^{11}}{7800}} = \sqrt{2.05\times10^{7}}$ .

Step 3. Take the square root. $v \approx \mathbf{4530\ m/s}$ .

Step 4. Compare with air. $\dfrac{4530}{340} \approx 13.3$ , so sound in steel travels roughly 13 times faster than sound in air. This is why placing an ear against a steel rail lets you hear an approaching train's sound arrive through the metal well before it arrives through the air — a solid's enormously larger bulk modulus outweighs its larger density by a wide margin.

Competitive level - Newton versus Newton-Laplace in air

Using the Newton-Laplace formula, estimate the speed of sound in air at a pressure of $1.0\times10^5$ Pa and density $1.29$ kg/m³, taking $\gamma = 1.4$ for air. Compare with Newton's original (isothermal) prediction using $B=P$ , and state the percentage difference between the two.

Step 1. Newton-Laplace (adiabatic) speed: $v = \sqrt{\dfrac{\gamma P}{\rho}} = \sqrt{\dfrac{1.4 \times 1.0\times10^5}{1.29}} = \sqrt{1.085\times10^5}$ .

Step 2. $v \approx \mathbf{329\ m/s}$ .

Step 3. Newton's original (isothermal) speed, using $B = P$ instead of $\gamma P$ : $v_{\text{isothermal}} = \sqrt{\dfrac{P}{\rho}} = \sqrt{\dfrac{1.0\times10^5}{1.29}} = \sqrt{7.75\times10^4} \approx \mathbf{278\ m/s}$ .

Step 4. Percentage difference from the Laplace-corrected value: $\dfrac{329-278}{329}\times100\% \approx 16\%$ . Newton's uncorrected value underestimates the true speed by about 16%, while the Laplace-corrected value (329 m/s) is within about 2-3% of the experimentally measured speed of sound in air (about 340 m/s at 20 degree Celsius) — the small remaining gap is because this calculation used exact standard values of $P$ and $\rho$ rather than the slightly different real conditions of a warm room.

Step 5. Extend with $v=f\lambda$ : if this sound wave has a frequency of $440$ Hz (musical note A), its wavelength in air is $\lambda = v/f = 329/440 \approx \mathbf{0.75\ m}$ . Timing note: in a competitive exam, do not waste time recomputing $\sqrt{1.085\times10^5}$ by long division — recognise that $\sqrt{1.085\times10^5} = \sqrt{1.085}\times\sqrt{10^5} \approx 1.042\times316.2 \approx 329$ , and keep the isothermal-vs-adiabatic ratio $\sqrt{\gamma}=\sqrt{1.4}\approx1.18$ memorised so you can sanity-check both speeds in one line without re-deriving either square root from scratch.

Common traps

Using $B=P$ (Newton's formula) by default, without applying the $\gamma$ correction. The Newton-Laplace (adiabatic) result, with the factor $\gamma$ , is the physically correct and experimentally matching version for a gas — plain $B=P$ underestimates the speed by roughly 16% for air.
Confusing bulk modulus $B$ with Young's modulus or tension. Bulk modulus $B$ (resistance to volume compression) governs longitudinal/sound waves in any medium; Young's modulus governs waves along a thin solid rod; tension $T$ governs transverse waves on a string. Each wave-speed formula needs the restoring quantity that matches its own situation — do not mix them up.
Forgetting that the speed of sound depends on temperature. A number like "340 m/s" is only valid near room temperature; more precisely $v \propto \sqrt{T_{\text{kelvin}}}$ , since both $P$ and $\rho$ change together with temperature for a gas at fixed volume.
Assuming higher density always means higher wave speed. It is the ratio $B/\rho$ that matters, not $\rho$ alone. Solids have both a much higher $B$ and a higher $\rho$ than gases, but the increase in $B$ dominates by far, giving a much higher overall speed of sound in solids than in gases.
Mixing up $\gamma$ with the wavelength symbol $\lambda$ . Both appear in this topic together (as in $v=\sqrt{\gamma P/\rho}$ and $v=f\lambda$ ) — keep them visually and conceptually separate: $\gamma$ is a dimensionless ratio of specific heats, $\lambda$ is a length.

You should be able to…

State the general longitudinal wave-speed formula $v=\sqrt{B/\rho}$ and correctly identify $B$ (bulk modulus) and $\rho$ (density) for a given medium.
Explain qualitatively why Newton's isothermal assumption underestimated the speed of sound in air, and what physical reasoning (adiabatic compression) fixes it.
State and apply the Newton-Laplace formula $v=\sqrt{\gamma P/\rho}$ to calculate the speed of sound in a gas given $P$ , $\rho$ and $\gamma$ .
Compare and calculate the speed of sound across different media (solids versus liquids versus gases) using given elastic moduli and densities, and explain why solids conduct sound fastest.
Combine the speed of sound with $v=f\lambda$ to solve for the wavelength or frequency of a sound wave.
Recognise the shared "restoring effect over inertia, inside a square root" structure linking $v=\sqrt{T/\mu}$ (string) and $v=\sqrt{B/\rho}$ or $\sqrt{\gamma P/\rho}$ (longitudinal/sound wave).

Ask the tutor: Want to see how the speed of sound changes if you double the pressure while keeping the temperature fixed, or work through why $v \propto \sqrt{T_{\text{kelvin}}}$ falls straight out of the ideal gas law — ask the AI tutor to walk through it with you.

Why does the same gamma of 1.4 show up here and also in gas law and specific heat topics?

Because it is the same physical quantity throughout: $\gamma = C_p/C_v$ , the ratio of a gas's specific heat at constant pressure to its specific heat at constant volume, and it measures how "resistant to heating up while being compressed" a gas is. It shows up in adiabatic processes generally (including the sound wave's rapid compressions) because $\gamma$ is exactly the number that tells you how much stiffer a gas becomes when compressed adiabatically compared to isothermally. You do not need to re-derive it here — just carry the same value, about $1.4$ for air, from wherever your kinetic theory chapter gives it to you.

Is the speed of sound the same in dry air and humid air?

Not quite, though the difference is small and outside what CBSE asks you to calculate at this level. Humid air has a slightly lower average density than dry air (water vapour molecules are lighter than the nitrogen and oxygen they partly displace), and by $v=\sqrt{\gamma P/\rho}$ a lower $\rho$ gives a slightly higher speed. The temperature dependence, $v \propto \sqrt{T_{\text{kelvin}}}$ , is a much bigger effect in everyday conditions and is the one you should quote if asked why sound travels faster on a hot day.

Do liquids also use the Newton-Laplace gamma-P formula, or just the general B over rho one?

Just the general one, $v=\sqrt{B/\rho}$ , using the liquid's own (isothermal-adiabatic difference is negligible for liquids) bulk modulus and density. The $\gamma$ correction specifically exists because gases are highly compressible and heat up noticeably when compressed quickly; liquids and solids are so much less compressible that the isothermal and adiabatic bulk moduli are nearly identical, so the plain $v=\sqrt{B/\rho}$ already gives an accurate answer without needing a gamma correction.

My textbook says speed of sound in air is 332 m/s and another source says 343 m/s — which is right?

Both are right, for different temperatures — this is not a contradiction to memorise around but the direct consequence of $v \propto \sqrt{T_{\text{kelvin}}}$ . About $332$ - $333$ m/s is the speed at $0$ degree Celsius ( $273$ K), the value often used as a reference in the Newton-Laplace calculation with standard pressure and density; about $340$ - $343$ m/s is the speed nearer room temperature ( $20$ - $25$ degree Celsius). Always check (or state) which temperature a quoted speed of sound belongs to rather than treating "the speed of sound" as one fixed universal number.