Sound travels about 4.4 times faster in water (~1500 m/s) than in air (~340 m/s), even though water is approximately 800 times denser than air. The best explanation is:
ASound waves in water are transverse while in air they are longitudinal, making them faster
BWater's bulk modulus is approximately 10,000 times greater than air's, more than compensating for the higher density
CDenser media always carry sound faster because more mass is available to transmit the wave
DWater molecules are closer together, so each vibration travels a shorter distance between collisions
Wave speed = √(bulk modulus / density). Water is ~800× denser than air, which alone would make it slower by √800 ≈ 28×. But water's bulk modulus is ~10,000× greater than air's, which alone would make it faster by √10000 = 100×. The ratio favors water: √(10000/800) ≈ 3.5, consistent with the observed 4.4× difference. Option C states the common misconception — density alone does not determine speed. Sound waves in both water and air are longitudinal.
Question 2 Multiple Choice
A new elastic material has twice the bulk modulus and four times the density of steel. Compared to steel, the wave speed in this material is:
ATwice the speed of steel (modulus doubled)
B1/√2 times the speed of steel (approximately 0.71×)
CThe same speed as steel (factors cancel)
DFour times the speed of steel (density quadrupled)
v = √(B/ρ). If the new material has B' = 2B and ρ' = 4ρ, then v' = √(2B/4ρ) = √(B/2ρ) = v/√2 ≈ 0.71v. Doubling the modulus increases speed by √2; quadrupling density decreases speed by 2. The net effect is a factor of √2/2 = 1/√2. This illustrates that both properties matter and must be considered together — doubling the modulus does not simply double the speed.
Question 3 True / False
Whether a medium transmits waves faster or slower than another medium depends on the ratio of elastic modulus to density — not on either property alone.
TTrue
FFalse
Answer: True
This is the central insight: v = √(stiffness/density). A medium can be denser yet faster (like water vs. air) if its stiffness increases proportionally more. A medium can be lighter yet slower if it is also much less stiff. Neither density nor stiffness alone predicts wave speed; only their ratio does. This is why intuitions like 'denser = slower' or 'stiffer = faster' are incomplete without the other factor.
Question 4 True / False
A denser medium generally transmits sound more slowly than a less dense medium.
TTrue
FFalse
Answer: False
This is the classic misconception. Density is in the denominator of v = √(B/ρ), so higher density does decrease speed — all else being equal. But all else is rarely equal. Water is ~800× denser than air yet transmits sound ~4.4× faster, because its bulk modulus is ~10,000× greater. Steel is denser than water yet transmits sound even faster (~5100 m/s) because its elastic modulus is orders of magnitude higher. The correct statement is: higher density decreases wave speed, but this effect can be overwhelmed by a sufficiently higher elastic modulus.
Question 5 Short Answer
Explain the physical logic behind the formula v = √(stiffness/density). Why does higher stiffness increase wave speed, and why does higher density decrease it?
Think about your answer, then reveal below.
Model answer: A wave propagates when each layer of the medium disturbs the next. A stiffer medium transmits the restoring force more forcefully to its neighbors, so the disturbance moves along faster — stiffness is in the numerator. A denser medium has greater inertia per unit volume, so each layer accelerates more sluggishly in response to the same restoring force — density is in the denominator. The formula v = √(stiffness/density) captures this competition: speed increases with stiffness and decreases with density, with both appearing under a square root because the relationship derives from Newton's second law.
The square root comes from dimensional analysis and the underlying wave equation derivation (applying F = ma to an infinitesimal slice of the medium). Neither factor acts independently — doubling stiffness multiplies speed by √2, while doubling density divides speed by √2. The same structural formula v = √(elastic property / inertial property) appears for all mechanical wave types (sound in fluids, waves on strings, seismic waves), making it a universal pattern worth internalizing.