Questions: Sound Wave Speed and Temperature Dependence

Question 1

A student claims that compressing air to twice its pressure should make sound travel faster, because the air is more 'energetic.' What does the formula v = √(γRT/M) say about this?

• A. The student is correct — higher pressure increases molecular collisions, raising speed
• B. Partially correct — higher pressure increases speed, but only by √2
• C. Incorrect — pressure does not appear in the formula; doubling pressure also doubles density, and these effects cancel exactly in an ideal gas
• D. Incorrect — only molar mass M matters; pressure and temperature are irrelevant at constant volume

Answer: (C)

The formula v = √(γRT/M) contains no pressure term. For an ideal gas, the elastic modulus of the gas (the 'stiffness') is proportional to pressure γP, while density is also proportional to pressure ρ = PM/(RT). When you form the ratio stiffness/density, pressure cancels, leaving v = √(γRT/M). This is why sound speed in air is the same at sea level and at altitude (if temperature is the same), despite very different pressures.

Question 2

In which medium does sound travel fastest?

• A. Air at high pressure, because compressed air has more molecules to transmit the disturbance
• B. Water, because it is much denser than air
• C. Steel, because its elastic stiffness far outweighs the penalty of its higher density
• D. Helium gas, because its low molar mass makes molecules lighter and faster

Answer: (C)

Sound speed = √(stiffness/inertia). Steel has an elastic modulus roughly a million times larger than air; even though it is also ~7,000 times denser, stiffness wins decisively, giving ~5,100 m/s. Water (~1,480 m/s) beats air for the same reason. Helium is faster than air (~970 m/s vs ~343 m/s) because of its low molar mass, but this still falls far short of solids. The key insight is that stiffness, not density, is the dominant factor.

Question 3

Sound travels faster at higher temperatures because warmer gas molecules move faster and transmit pressure disturbances more quickly through the medium.

• T. True
• F. False

Answer: True

This is the correct physical intuition. Temperature is a measure of average molecular kinetic energy; higher temperature means molecules are moving faster, colliding more frequently, and propagating pressure pulses more rapidly. The formula v = √(γRT/M) confirms this: v is proportional to √T. In air, speed rises by roughly 0.6 m/s per °C.

Question 4

Because steel is far denser than water, sound travels more slowly in steel than in water.

• T. True
• F. False

Answer: False

Sound speed depends on the ratio of elastic stiffness to inertia (density). Steel is indeed denser than water, but its elastic modulus (stiffness) is enormously larger — roughly 200 GPa vs. ~2 GPa for water. The stiffness advantage overwhelms the density penalty: sound travels at ~5,100 m/s in steel versus ~1,480 m/s in water. This counterintuitive result is why denser does not mean slower for sound.

Question 5

Why does doubling the air pressure at constant temperature not change the speed of sound?

Model answer: Doubling pressure also doubles the density of the air (from PV = nRT, at constant T and V, density ∝ P). The elastic restoring force scales with pressure (the bulk modulus of an ideal gas is γP), while the inertia scales with density (also ∝ P). In the formula v = √(bulk modulus / density), both numerator and denominator double and the ratio stays constant. Speed depends on temperature and molecular properties, not on pressure.

This is a key test of whether students understand the formula mechanically versus conceptually. Many students expect denser or more compressed air to carry sound better, but the cancellation is exact for ideal gases. The only way to change sound speed is to change temperature (which changes molecular velocities) or the gas itself (different γ or M).