BPSC 71st CCE Prelims General Studies · Question 124 of 150

Correct — A, Steel. Sound is a mechanical wave, and its speed in any medium is governed by the Newton-Laplace relation v = the square root of (elastic modulus divided by density). Elasticity is the restoring stiffness that snaps a compressed layer of the medium back and passes the disturbance on; density is the inertia that resists being moved. A stiff, light medium carries sound fast; a floppy, heavy one carries it slowly. Put numbers to the four options at ordinary room temperature and the order is not close: steel about 5,930 m/s, water 1,481 m/s at 20 °C, hydrogen roughly 1,270 to 1,330 m/s depending on temperature, and air 343 m/s at 20 °C (331 m/s at 0 °C). Steel wins by a factor of about 4 over the next fastest option, water and by about 17 over air. The reason steel wins looks paradoxical, because steel is by far the DENSEST of the four and density sits in the denominator. The resolution is that stiffness rises far faster than density as you go from a gas to a solid. Steel is roughly 6,400 times denser than air, but its bulk stiffness is on the order of a million times greater, so the ratio stiffness-to-density — and therefore the speed — still comes out far larger. You can check the formula rather than trust it. For a thin steel rod, Young's modulus of about 200 GPa over a density of 7,850 kg/m³ gives the square root of 2.55 × 10⁷, that is about 5,050 m/s, with the bulk longitudinal wave in a large steel body running faster still at about 5,930 m/s. For water, a bulk modulus of 2.2 GPa over 1,000 kg/m³ gives about 1,480 m/s, which is the tabulated value. For air, 1.4 × 101,325 Pa over 1.225 kg/m³ gives about 340 m/s. Three media, one formula, three correct answers — which is why the ordering solids faster than liquids faster than gases is a consequence and not something to be memorised. Everyday confirmation: put an ear to a railway rail and the sound of an approaching train reaches you through the steel well before it arrives through the air.

• (b)Water — Genuinely fast — about 1,481 m/s at 20 °C, some 4.3 times the speed in air, which is why sonar works, why whale calls carry for hundreds of kilometres and why a swimmer hears a struck stone almost instantly. But water is a liquid: it has bulk stiffness with a bulk modulus of about 2.2 GPa and no rigidity at all, against roughly 160 GPa for steel; its density is lower too, at 1,000 against 7,850 kg per cubic metre, but the stiffness gap is far the larger of the two effects, so its sound speed is a quarter of steel's.
• (c)Air — The slowest of the four and the easiest to eliminate: 343 m/s at 20 °C, 331 m/s at 0 °C, rising by about 0.6 m/s for every degree Celsius. Air is the reference medium for the Mach number, not a fast one. Gases are poor conductors of sound because their molecules are far apart and the medium is highly compressible, so each layer takes a long time to pass the disturbance on.
• (d)Hydrogen — The intelligent wrong answer, and it deserves respect rather than dismissal. Hydrogen is the FASTEST COMMON GAS — about 1,270 to 1,330 m/s, nearly four times air and close behind liquid water — because in a gas the speed is the square root of (gamma × R × T / M) and hydrogen has the smallest molar mass of any gas at 2.016 g/mol against air's 28.96. A candidate who knows that lighter gases carry sound faster is reasoning correctly and still loses the mark, because moving from gas to solid changes the stiffness by a millionfold and no gas can compete with steel.

Sound is a longitudinal mechanical wave: it travels by compressing and rarefying successive layers of a material medium, which is why it cannot cross a vacuum at all. Its speed is set by two competing properties of that medium, and the Newton-Laplace relation states them as v = the square root of (elastic modulus / density). The modulus takes a different name in each state of matter — Young's modulus Y for a wave along a thin solid rod, the bulk modulus K for a liquid or for the bulk of a solid, and for a gas the adiabatic value gamma × P, which turns the gas expression into v = the square root of (gamma × P / rho) or equivalently the square root of (gamma × R × T / M). That last form carries three consequences the exam tests repeatedly: the speed in a gas depends on absolute temperature and rises with it (in air by about 0.6 m/s per degree Celsius); it is INDEPENDENT of pressure, because raising the pressure of a gas raises its density in the same proportion; and it falls as the molar mass rises, so light gases are fast. The history matters too. Newton derived the gas formula in the Principia of 1687 but assumed the compressions were isothermal, and his figure came out roughly fifteen per cent below the measured speed. Laplace resolved the discrepancy in 1816 by pointing out that the compressions and rarefactions are so rapid that no heat has time to leak away — they are adiabatic — and inserting gamma, the ratio of specific heats, which for air is 1.40. Solids also carry a second kind of sound wave that fluids cannot: transverse or shear waves, because only a solid resists a change of shape. That is why an earthquake sends both P-waves and S-waves through rock but only P-waves through the Earth's liquid outer core.

Reason to this answer in two steps and it takes five seconds. Step one: classify the options by state of matter. Steel is a solid; water is a liquid; air and hydrogen are gases. Step two: apply the ordering that follows from v = the square root of (stiffness / density) — solids faster than liquids faster than gases — and the only solid on the list is the answer. Everything else is checking. The single discriminating fact is that stiffness, not density, decides. Candidates who go wrong usually reason from density, note that steel is by far the heaviest of the four, see density sitting in the denominator, and eliminate the right answer. The counter is quantitative: from air to steel the density rises about six thousand fold but the stiffness rises about a millionfold, so the quotient — and the speed — go up by a factor of roughly seventeen. Now the harder half of this question, which is what makes it worth more than one mark. Hydrogen is not a throwaway option. Within the family of gases hydrogen genuinely IS the fastest, at about 1,300 m/s against air's 343, because at a given temperature the speed varies as the square root of (gamma / M) and hydrogen's molar mass of 2.016 g/mol is the smallest there is. It even runs close to liquid water at 1,481 m/s, and it would overtake room-temperature water only around 90-100 °C, since the speed in a gas rises as the square root of absolute temperature — the one place where the tidy 'solids beat liquids beat gases' ordering nearly breaks down, because that ordering compares typical values and not extremes. Helium behaves the same way for the same reason, at about 1,000 m/s; it is slower than hydrogen despite a larger gamma of 1.66, because the molar mass term dominates, and it is why an inhaled helium voice turns squeaky. So the correct mental note is not 'hydrogen is wrong' but 'hydrogen is the right answer to a different question — the fastest GAS — while the question asked for the fastest MEDIUM, and a solid is on the list'.

• Representative speeds of sound at ordinary temperature: steel about 5,930 m/s, water 1,481 m/s at 20 °C, hydrogen about 1,270-1,330 m/s, air 343 m/s at 20 °C and 331 m/s at 0 °C. Steel is roughly 17 times faster than air and about 4.5 times faster than hydrogen.
• The Newton-Laplace relation v = square root of (elastic modulus / density) reproduces all of these. Steel: Young's modulus about 200 GPa over 7,850 kg/m³ gives about 5,050 m/s for a thin rod. Water: bulk modulus 2.2 GPa over 1,000 kg/m³ gives about 1,480 m/s. Air: 1.4 × 101,325 Pa over 1.225 kg/m³ gives about 340 m/s.
• In a gas v equals the square root of (gamma × R × T / M), so at a fixed temperature the speed varies as the square root of (gamma / M). Hydrogen's molar mass is 2.016 g/mol against air's 28.96, a ratio whose square root is about 3.8 — which is exactly how much faster sound is in hydrogen than in air.
• Helium, with a molar mass of 4.003 and a higher gamma of 1.66, carries sound at about 1,000 m/s — slower than hydrogen, because the molar mass term outweighs the specific-heat ratio. The raised speed shifts the resonant frequencies of the vocal tract upward, which is why a helium-filled voice sounds squeaky.
• Newton computed the speed of sound in air in the Principia (1687) assuming isothermal compressions and came out roughly fifteen per cent below measurement; Laplace corrected it in 1816 by treating the compressions as adiabatic and inserting gamma = 1.40 for air, which brought theory and experiment together.
• Sound requires a material medium and cannot travel through a vacuum. In air its speed rises about 0.6 m/s per degree Celsius, is independent of pressure at constant temperature, and increases slightly with humidity because water vapour has a molar mass of 18 against dry air's 28.96. Only solids also carry transverse (shear) waves, which is why S-waves stop at the Earth's liquid outer core.

• Reasoning only from density: steel is the densest option here and density sits in the denominator, yet steel is still fastest because stiffness rises far more steeply than density from a gas to a solid
• Over-generalising the true rule that light gases carry sound faster, and therefore choosing hydrogen — hydrogen is indeed the fastest common GAS at about 1,300 m/s, but the question asks for the fastest medium of any kind and a solid is on the list
• Believing that the speed of sound in air rises with pressure — it does not, because pressure and density rise together; it is the temperature, not the pressure, that changes it

BPSC asks this as a five-word one-liner and hides the difficulty in the option list, slipping in hydrogen so that a candidate who has learnt the gas rule but not the medium rule still loses the mark. UPSC set the identical question in 2006 with air at 0 °C, air at 100 °C, water and wood as the options, again resolving to the solid; but its more characteristic form is conceptual — in 2007 it asked in assertion-reason format whether a jet at Mach 1 travels faster at 15 km than at sea level, which needs the fact that the speed of sound depends on temperature, and in the same year it asked whether sound travels in rocks as longitudinal waves only.

• practice — not a real PYQ

  Among the following gases, at the same temperature, sound travels fastest in

  • (a)Oxygen
  • (b)Nitrogen
  • (c)Carbon dioxide
  • (d)Hydrogen

  Answer(d) Hydrogen — in a gas the speed of sound varies as the square root of (gamma / M), so the lightest gas is the fastest. Hydrogen's molar mass is 2.016 g/mol against 32 for oxygen, 28 for nitrogen and 44 for carbon dioxide, giving about 1,300 m/s in hydrogen and the slowest speed of the four in carbon dioxide.

• practice — not a real PYQ

  Newton's calculation of the speed of sound in air was about 15% lower than the measured value. Laplace corrected it by assuming that the compressions and rarefactions in a sound wave are

  • (a)isothermal
  • (b)adiabatic
  • (c)isobaric
  • (d)isochoric

  Answer(b) adiabatic — the compressions are too rapid for heat to escape, so the relevant elasticity is gamma × P rather than P, and inserting gamma = 1.40 for air raises the computed speed to the measured 331 m/s at 0 °C.