NDA (II) 2016 General Ability Test · Question 108 of 150

Correct — D, the speed depends on both the elastic and the inertia properties of the medium. The general result for a mechanical wave is v = √(E/ρ), where E is the relevant elastic modulus of the medium and ρ is its density. Both quantities sit in that one expression, so neither can be dropped. The physical reading is straightforward: elasticity is what makes a disturbed layer of the medium spring back and pass the disturbance on, so a stiffer medium carries the wave faster, while inertia, measured by the density, is what resists being set in motion, so a heavier medium carries it slower. Sound in steel travels at roughly 5,000 metres per second and in air at about 340, and the difference comes from steel's enormously greater stiffness outweighing its greater density.

• (a)The speed of sound waves in a medium depends upon the elastic property of the medium but not on inertia property — Half the truth. Elasticity does appear in v = √(E/ρ), but so does the density in the denominator. If inertia were irrelevant, hydrogen and oxygen at the same temperature and pressure would carry sound at the same speed; in fact sound travels about four times faster in hydrogen, purely because hydrogen is lighter.
• (b)The speed of sound waves in a medium depends upon the inertia property of the medium but not on elastic property — The other half, and equally incomplete. Density alone cannot explain why sound moves faster in water than in air even though water is far denser — water's bulk modulus is larger by a much bigger factor than its density is.
• (c)The speed of sound waves in a medium depends neither on its elastic property nor on its inertia property — This would make the speed of sound the same in every material, which contradicts ordinary observation — a distant hammer blow on a rail is heard through the rail well before it arrives through the air. Sound is a mechanical wave and cannot be independent of the medium; it does not even travel through a vacuum.

Sound is a longitudinal mechanical wave: the particles of the medium oscillate along the direction in which the wave travels, producing alternate compressions and rarefactions. Its speed is set by the medium, not by the source, and is given by v = √(E/ρ). For a solid rod the modulus used is Young's modulus, for liquids and gases it is the bulk modulus. Newton first computed the speed in air assuming the compressions were isothermal and got an answer about 15 per cent too low; Laplace corrected it by pointing out that the compressions are so rapid that they are adiabatic, giving v = √(γP/ρ).

The most useful check on this idea is what happens when a gas is heated. Raising the temperature of a gas at constant pressure lowers its density while leaving the pressure unchanged, so the speed rises — roughly 0.61 metres per second for every degree Celsius in air, from about 331 metres per second at 0 °C. Notice also what the speed does not depend on: not the frequency, not the loudness, and — for a gas — not the pressure by itself, because raising the pressure at fixed temperature raises the density in exactly the same proportion and the ratio in the square root is unchanged.

• The speed of a mechanical wave in a medium is v = √(E/ρ), with E an elastic modulus and ρ the density.
• Sound travels fastest in solids, slower in liquids and slowest in gases — roughly 5,000 m/s in steel, 1,500 m/s in water and 340 m/s in air at room temperature.
• The Newton-Laplace formula for a gas is v = √(γP/ρ), where γ is the ratio of the specific heats.
• In a gas the speed rises with temperature but does not change with pressure alone, and it does not depend on the frequency or the loudness of the sound.

• Assuming that a denser medium always carries sound more slowly — density is only the denominator, and the modulus usually rises faster.
• Thinking the speed of sound in a gas rises with pressure; at fixed temperature it does not change at all.

The GAT sets one sound question almost every year, most often on what the speed does or does not depend on, so learn v = √(E/ρ) as a two-part story rather than a formula.

• practice — not a real PYQ

  Sound travels fastest in

  • (a)vacuum
  • (b)air
  • (c)water
  • (d)steel

  Answer(d) steel — solids have much larger elastic moduli than liquids or gases, and sound does not travel through a vacuum at all.

• practice — not a real PYQ

  The speed of sound in air at a fixed temperature is doubled if

  • (a)the pressure of the air is doubled
  • (b)the frequency of the sound is doubled
  • (c)the loudness of the sound is doubled
  • (d)none of the above happens

  Answer(d) none of the above happens — at fixed temperature pressure and density rise together and cancel, while frequency and loudness never affect the speed.