Which of the following represents the correct graphical variation of velocity of sound and absolute temperature?

1. Nature of Sound: Longitudinal Mechanical Waves (basic)

To understand sound, we must first recognize it as a mechanical wave. Unlike light, which can travel through the emptiness of space, sound requires a material medium—be it a solid, liquid, or gas—to propagate. This is because sound travels by vibrating the molecules of the medium itself. In physics, we classify sound as a longitudinal wave. This means the particles of the medium oscillate back and forth parallel to the direction in which the wave is moving. As these particles push against one another, they create alternating regions of high pressure called compressions and low pressure called rarefactions Physical Geography by PMF IAS, Earths Magnetic Field, p.64.

The speed at which sound travels is not constant; it depends heavily on the properties of the medium, specifically its elasticity and density. While we often think density is the primary driver, it is actually the elasticity (the ability of a material to return to its original shape) that often plays a larger role. For instance, sound travels fastest in solids, then liquids, and slowest in gases (Solids > Liquids > Gases) because solids are highly elastic and can transmit vibrations more efficiently Physical Geography by PMF IAS, Earths Interior, p.60. A fascinating real-world application of this is found in seismology: P-waves (Primary waves) are essentially sound waves traveling through the Earth's interior, moving much faster than the transverse S-waves because of their longitudinal nature Physical Geography by PMF IAS, Earths Interior, p.61.

In the specific case of an ideal gas (like air), the speed of sound (v) is governed by the absolute temperature (T). Mathematically, this is expressed as v = √(γRT/M), where γ (ratio of specific heats), R (gas constant), and M (molar mass) remain constant for a specific gas. This tells us that the velocity of sound is directly proportional to the square root of the absolute temperature (v ∝ √T). As the air warms up, the molecules move faster and collide more frequently, allowing the sound wave to skip from one molecule to the next more rapidly. However, because this is a square-root relationship, the speed increases quickly at first but then the rate of increase slows down as temperature continues to rise.

Wave Type	Particle Motion	Example	Medium Required?
Longitudinal	Parallel to wave direction	Sound Waves, P-waves	Yes (Mechanical)
Transverse	Perpendicular to wave direction	Light Waves, S-waves	No (for Light) / Yes (for S-waves)

Sources: Physical Geography by PMF IAS, Earths Magnetic Field, p.64; Physical Geography by PMF IAS, Earths Interior, p.60; Physical Geography by PMF IAS, Earths Interior, p.61; Physical Geography by PMF IAS, Earths Interior, p.62

2. Speed of Sound in Different Media (basic)

To understand how sound travels, we must first look at the particulate nature of matter. Sound is a mechanical wave that moves through a medium by the compression and rarefaction of its particles (Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64). In solids, particles are "closely packed" and held together by "very strong interparticle interactions" (Science, Class VIII . NCERT, Particulate Nature of Matter, p.113). This proximity allows vibrations to be passed from one particle to the next almost instantly. Consequently, sound travels fastest in solids, slower in liquids, and slowest in gases where particles are far apart.

Beyond the state of matter, two physical properties determine speed: density and elasticity. Elasticity refers to how quickly a medium returns to its original shape after being deformed. While we often think density is the only factor, elasticity is often more dominant. For instance, sound travels faster in iron than in mercury; even though mercury is denser, iron is much more elastic, allowing for a more efficient transfer of sound energy (Physical Geography by PMF IAS, Earths Interior, p.61).

In gases like air, temperature plays a critical role. As temperature rises, particles gain kinetic energy and move more vigorously. The speed of sound (v) is directly proportional to the square root of the absolute temperature (T), expressed as v ∝ √T. This relationship means that as the air warms up, sound travels faster. At absolute zero (0 K), molecular motion theoretically ceases, and sound can no longer be transmitted. On a graph, this relationship appears as a parabolic curve, where the speed increases rapidly at first and then the rate of increase gradually slows down.

Medium Type	Speed Trend	Primary Reason
Solids	Fastest	Strong interparticle forces and high elasticity.
Liquids	Intermediate	Particles are close but can move past each other.
Gases	Slowest	Large interparticle spaces and low elasticity.

Sources: Science, Class VIII . NCERT (Revised ed 2025), Particulate Nature of Matter, p.113; Physical Geography by PMF IAS, Earths Interior, p.61; Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64

3. Atmospheric Factors Influencing Sound (intermediate)

In our journey through acoustics, it is vital to understand that sound is not a fixed traveler; it is deeply influenced by the medium it moves through. In the context of our atmosphere, three primary factors dictate how fast sound moves: temperature, humidity, and wind. Understanding these allows us to predict how sound behaves in different climates and altitudes.

Temperature is the most dominant factor. In an ideal gas like air, the speed of sound (v) is directly proportional to the square root of its absolute temperature (T), expressed by the formula v = √(γRT/M). This means that as the temperature rises, the kinetic energy of the molecules increases, allowing them to transmit vibrations more rapidly. If you were to plot this on a graph, you would see a parabolic curve: velocity increases as the square root of temperature, starting from absolute zero (0 K), where molecular motion—and thus sound transmission—theoretically ceases. While the speed increases with heat, the rate of that increase gradually slows down as it follows this square root relationship.

Humidity often confuses students because we tend to think of "humid" air as thick or heavy. However, water vapor (H₂O) actually has a lower molar mass than the nitrogen (N₂) and oxygen (O₂) that make up the bulk of our atmosphere. When humidity increases, these lighter water molecules displace the heavier nitrogen and oxygen molecules, making the air less dense. Since sound travels faster through less dense gases (at constant pressure), sound actually moves faster on a humid or rainy day than on a dry one. We measure this moisture as Relative Humidity, where 100% signifies saturated air Exploring Society: India and Beyond, Social Science-Class VII, Understanding the Weather, p.38.

Finally, we must consider Wind. Wind is simply "air in motion" Certificate Physical and Human Geography, GC Leong, Weather, p.121. Because sound waves are carried by the air, the wind's velocity is added to or subtracted from the speed of sound. If the wind is blowing in the same direction as the sound, the sound will travel faster relative to a stationary observer. Conversely, if you are shouting against the wind, the sound is slowed down. While we often focus on density, it is important to remember that the elasticity of a medium is also a critical factor in wave velocity, often overriding density in materials like solids Physical Geography by PMF IAS, Earths Interior, p.61.

Factor	Change in Factor	Effect on Sound Speed
Temperature	Increase	Increases (v ∝ √T)
Humidity	Increase	Increases (Air becomes less dense)
Wind	With Sound Direction	Increases

Sources: Exploring Society: India and Beyond, Social Science-Class VII, Understanding the Weather, p.38; Certificate Physical and Human Geography, GC Leong, Weather, p.121; Physical Geography by PMF IAS, Earths Interior, p.61

4. Kinetic Theory and Absolute Temperature (intermediate)

To understand how sound travels through air, we must first look at the Kinetic Theory of Matter. In a gas, particles are not fixed; they move freely in all directions because they possess enough energy to overcome the attractive forces that keep solids and liquids together Science, Class VIII, Particulate Nature of Matter, p.112. When we talk about Temperature, we are essentially measuring the average kinetic energy of these particles. If you heat a gas, you are giving its particles more energy, causing them to move and vibrate faster.

Since sound is a mechanical wave that relies on the collisions of particles to propagate, the speed at which these particles move directly dictates how fast sound can travel. In an ideal gas, the speed of sound (v) is mathematically related to the Absolute Temperature (T), measured in Kelvin. The formula is v = √(γRT/M), where γ (gamma) is the ratio of specific heats, R is the universal gas constant, and M is the molar mass of the gas. For a specific gas like air, everything in that formula is constant except for the temperature. This leads us to a fundamental principle: The speed of sound is directly proportional to the square root of its absolute temperature (v ∝ √T).

This "square root" relationship has two very important implications for your conceptual clarity:

• The Parabolic Curve: If you plot the speed of sound against temperature on a graph, it doesn't form a straight line. Instead, it forms a curve that starts steep and gradually flattens out. This means that while sound speed increases as it gets hotter, the rate of that increase slows down at very high temperatures.
• The Origin (0 Kelvin): At Absolute Zero (0 K), theoretical molecular motion ceases entirely Science, Class VIII, Particulate Nature of Matter, p.115. Because there is no particle motion or vibration to pass the energy along, sound cannot travel. This is why our graph starts at the origin (0,0).

Condition	Molecular Behavior	Effect on Sound Speed
Cold Air	Particles move slowly; fewer collisions.	Lower velocity; sound travels slower.
Warm Air	Particles move rapidly; frequent collisions.	Higher velocity; sound travels faster.

Sources: Science, Class VIII (NCERT), Particulate Nature of Matter, p.112; Science, Class VIII (NCERT), Particulate Nature of Matter, p.115; Science, Class VIII (NCERT), Particulate Nature of Matter, p.106

5. Laplace’s Correction and the v-T Formula (exam-level)

To understand the speed of sound in air, we must first look at Laplace’s Correction. Sir Isaac Newton originally calculated the speed of sound assuming the process was isothermal (constant temperature). However, Laplace corrected this by noting that sound waves compress and rarefy air so rapidly that heat does not have time to flow out of or into the system. This means the process is actually adiabatic. This correction introduced the term γ (gamma)—the ratio of specific heats ($C_p/C_v$)—into the formula, which for air is approximately 1.4. While the speed of waves in solids like iron is influenced heavily by elasticity Physical Geography by PMF IAS, Earths Interior, p.61, in gases, the primary drivers are temperature and molecular structure.

The resulting v-T Formula for an ideal gas is v = √(γRT/M), where R is the universal gas constant, T is the absolute temperature (in Kelvin), and M is the molar mass of the gas. This equation reveals a fundamental physical law: the speed of sound is directly proportional to the square root of its absolute temperature (v ∝ √T). Unlike the linear relationship seen in Ohm’s Law where the graph is a straight line Science Class X, Electricity, p.176, the relationship between $v$ and $T$ is sub-linear (parabolic). This means that as temperature increases, the speed of sound increases, but the rate of that increase gradually slows down.

If we were to plot this on a v-T graph, the curve would start at the origin (0 K), where molecular motion theoretically stops and sound can no longer propagate. As the temperature rises, the curve climbs steeply at first and then flattens out, following the square root function. This explains why sound travels faster on a hot summer day than on a cold winter night—at 0°C (273 K), the speed is roughly 331 m/s, but it increases by about 0.6 m/s for every degree Celsius rise in temperature.

Sources: Physical Geography by PMF IAS, Earths Interior, p.61; Science Class X (NCERT 2025 ed.), Electricity, p.176

6. Graphical Analysis: Linear vs. Square Root Growth (exam-level)

In physics and economics alike, understanding how one variable responds to another is the key to mastering complex systems. When we look at the speed of sound (v) in an ideal gas, we find it isn't a simple 1:1 relationship with temperature. Instead, it follows a square root relationship, expressed by the formula v = √(γRT/M). Here, γ (gamma), R (gas constant), and M (molar mass) are constants for a specific gas. This means that while speed increases as temperature rises, it does so at a diminishing rate. To visualize this, we must contrast it with linear growth. In a linear graph, such as the one representing autonomous consumption or the line of perfect equality, the relationship is a straight line where the slope (rate of change) remains constant Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.58. However, the speed of sound graph (plotting velocity against absolute temperature in Kelvin) is sub-linear. Starting from the origin (0 K), where molecular motion theoretically stops and sound cannot travel, the curve rises steeply at first and then begins to flatten out. This is technically a parabolic curve—resembling the arched profile of natural landforms like parabolic dunes Physical Geography by PMF IAS, Major Landforms and Cycle of Erosion, p.238.

Feature	Linear Growth (v ∝ T)	Square Root Growth (v ∝ √T)
Graph Shape	Straight line	Curve (Parabolic/Sub-linear)
Slope (Rate)	Constant	Decreasing (flattens over time)
Example	Perfect Income Equality Indian Economy, Nitin Singhania (ed 2nd 2021-22), Poverty, Inequality and Unemployment, p.45	Speed of Sound vs. Temperature

Understanding this distinction is vital for analytical clarity. If the relationship were linear, doubling the temperature would double the speed. But because it is a square root relationship, you would actually need to quadruple the absolute temperature to double the speed of sound. This 'flattening' of the curve is the visual signature of any square root growth process.

Sources: Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.58; Physical Geography by PMF IAS, Major Landforms and Cycle of Erosion, p.238; Indian Economy, Nitin Singhania (ed 2nd 2021-22), Poverty, Inequality and Unemployment, p.45

7. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental properties of longitudinal waves and the Laplace correction, this question asks you to synthesize those building blocks into a visual representation. You have learned that the speed of sound is not constant but depends on the medium's elasticity and density. By applying the ideal gas law to the speed formula, we derive the crucial relationship: v = √(γRT/M). As an aspiring civil servant, you must recognize that for a specific gas, the velocity is directly proportional to the square root of its absolute temperature (v ∝ √T), a concept extensively detailed in NASA Glenn Research Center.

To identify the correct graph, use mathematical visualization: a square root function (y = √x) starts at the origin (0,0) because at 0 Kelvin, molecular motion ceases and sound cannot propagate. As temperature increases, the velocity increases, but the rate of increase gradually slows down, creating a curve that is concave-downward towards the temperature axis. Graph III perfectly captures this sub-linear, parabolic growth. This is the hallmark of a square root relationship, distinguishing it from a straight line where the rate of change would remain constant regardless of the temperature.

UPSC frequently uses "distractor" graphs to test your precision. Option (A) is a common trap representing a linear relationship (v ∝ T), which students often choose if they forget the square root sign. Option (B) might suggest that velocity is independent of temperature, while Option (D) often depicts an exponential or inverse relationship, both of which contradict the kinetic theory of gases. Understanding that Graph (C) III is the only one representing a decreasing slope while moving away from the origin is the key to avoiding these conceptual pitfalls, as noted in ScienceDirect.