In which medium, the speed of sound is maximum?

Question

UnlockIAS · Accepted Answer

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.

Why the other options are wrong — (b) 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) 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) 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.

Concept: 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.

State PCS 2025 Q124 — Science & Technology PYQ with Answer

State PCS 2025 Q124 — Science & Technology PYQ with Answer