How does the velocity of sound waves in air vary with temperature, assuming normal atmospheric conditions?
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AConstant at all temperatures
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BDirectly proportional to temperature
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CInversely proportional to absolute temperature
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DDirectly proportional to the square root of absolute temperature
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EIndependent of humidity and temperature
Answer
Correct Answer: Directly proportional to the square root of absolute temperature
Explanation
Introduction / Context:The speed of sound in gases is a fundamental acoustic property that influences microphone calibration, room acoustics, sonar, and atmospheric acoustics. Temperature dependence is particularly important for precise measurement and system design.
Given Data / Assumptions:
- Ideal-gas approximation for air.
- Neglect humidity and wind for the basic relation.
Concept / Approach:
For an ideal gas, speed of sound c = sqrt(γ * R * T / M), where γ is ratio of specific heats, R is the universal gas constant, M is molar mass, and T is absolute temperature. Thus, c ∝ sqrt(T).
Step-by-Step Solution:
Write relationship: c = sqrt(γ * R * T / M).Hold γ, R, and M constant for air → c ∝ sqrt(T).Therefore, as T increases, c increases with the square root of T.Verification / Alternative check:
Empirical formula: c ≈ 331 m/s + 0.6 m/s per °C at moderate temperatures; this approximates the sqrt(T) dependence around standard conditions.
Why Other Options Are Wrong:
- Constant: contradicts both theory and measurement.
- Directly proportional to T or inversely proportional to T: incorrect functional dependence.
- Independent of humidity/temperature: humidity has a small effect; temperature a major one.
Common Pitfalls:
Assuming a linear dependence across all temperatures; the linear relation is only a local approximation.
Final Answer:
Directly proportional to the square root of absolute temperature