Questions: Sound Speed: Temperature and Medium Dependence

Question 1

Sound travels through steel (~5,000 m/s) much faster than through air (~340 m/s). What is the primary reason?

• A. Steel is denser than air, and denser media carry sound faster
• B. Steel has a much higher elastic modulus relative to its density than air does
• C. Steel molecules are closer together, so vibrations transfer more quickly
• D. Sound travels faster in solids because solids are at a higher temperature than air

Answer: (B)

Sound speed is v = √(elastic modulus / density). Steel is far denser than air, which alone would slow sound down. But steel's elastic modulus — its resistance to compression — is enormously larger than air's, and this increase dominates. The ratio (modulus/density) is much larger for steel than for air, so sound is faster in steel. Proximity of molecules (option C) is a folk explanation that doesn't hold up — what matters is not spacing but the mechanical stiffness of the medium relative to its mass.

Question 2

Air temperature rises from 0°C to 100°C (273 K to 373 K). By approximately what factor does the speed of sound in air increase?

• A. By a factor of about 1.04 — speed increases very slightly with temperature
• B. By a factor of about 1.17 — speed increases as the square root of absolute temperature
• C. By a factor of about 1.37 — speed increases proportionally to absolute temperature
• D. By a factor of about 2 — sound speed doubles when temperature doubles in Celsius

Answer: (B)

Speed is proportional to √T (absolute temperature). The ratio is √(373/273) ≈ √1.366 ≈ 1.17, so sound is about 17% faster. The key error in option C is treating speed as proportional to T rather than √T. Option D conflates Celsius ratios with Kelvin ratios: 'doubling' from 0°C to 100°C is not a temperature doubling in absolute terms — 0°C is 273 K, so the ratio is 373/273 ≈ 1.37, not 2.

Question 3

The statement 'sound travels faster in denser materials' is generally true.

• T. True
• F. False

Answer: False

This is one of the most persistent misconceptions in wave physics. Sound speed depends on the ratio of elastic modulus to density, not on density alone. Lead is much denser than aluminum, yet sound travels faster in aluminum because aluminum's elastic modulus is higher relative to its density. Air has very low density, yet sound is slow in it because its elastic modulus (bulk modulus) is also tiny. Density in the denominator of v = √(modulus/density) actually works against speed — it is the stiffness term in the numerator that must overcome density to produce fast propagation.

Question 4

Humid air carries sound faster than dry air, even though water is heavier than nitrogen.

• T. True
• F. False

Answer: True

This seems counterintuitive but is correct. Water vapor molecules (molar mass 18 g/mol) are lighter than the N₂ (28 g/mol) and O₂ (32 g/mol) they displace in humid air. Because sound speed in a gas is v = √(γRT/M) and M (molar mass) is in the denominator, replacing heavier molecules with lighter water vapor lowers the effective molar mass and increases sound speed. The effect is small (~0.3% at full saturation) but real. The misconception — that heavier water makes sound slower — ignores that individual water molecules are actually lighter than the diatomic molecules they replace.

Question 5

Why does the formula for sound speed in a gas include the square root of absolute temperature rather than just temperature, and what physical phenomenon does this reflect?

Model answer: Sound speed in a gas is v = √(γRT/M). The square root of temperature appears because sound speed is proportional to the average molecular speed (or more precisely, the root-mean-square molecular speed), which itself scales as √T from the kinetic theory of gases. Temperature measures the average kinetic energy (½mv² ∝ T), so molecular speed scales as √T. Hotter molecules hit their neighbors harder and more frequently, propagating pressure disturbances faster — but the relationship is sublinear (square root) because it is speed, not energy, that governs propagation.

The √T dependence is not an arbitrary formula feature; it reflects the microscopic mechanism of sound propagation. A compression wave moves at a speed set by how fast energy can be handed off between neighboring molecules. That transfer rate scales with molecular kinetic speed, which is √(kT/m) by kinetic theory. This is why doubling absolute temperature increases sound speed by only ~41% (√2 ≈ 1.41), not by 100%.