Short Answer:

Sonic velocity is the speed at which a small pressure wave or sound wave travels through a fluid medium such as air, gas, or liquid. It depends on the elastic properties and density of the medium. In simple terms, it represents the rate at which disturbances or sound signals move from one point to another.

Mathematically, the sonic velocity is expressed as:

where is the ratio of specific heats, is the gas constant, and is the absolute temperature. It increases with temperature and decreases with heavier gases.

Detailed Explanation:

Sonic Velocity

The sonic velocity, also known as the speed of sound, is a fundamental property of a medium that describes how fast a pressure disturbance or sound wave travels through it. When a disturbance, such as a vibration or fluctuation in pressure, is created in a fluid, it propagates through the medium as a longitudinal wave by successive compression and rarefaction of particles.

In gases, sonic velocity depends primarily on temperature and the nature of the gas. In liquids and solids, it also depends on the material’s elasticity and density. Sonic velocity plays a vital role in many engineering applications, especially in aerodynamics, acoustics, thermodynamics, and compressible fluid flow.

Definition

The sonic velocity is defined as:

“The velocity at which small pressure or sound waves travel through a medium.”

This velocity represents how quickly information or disturbances are transmitted through the substance. The particles of the medium do not move long distances; rather, they oscillate around their mean position, transmitting energy in the form of waves.

The symbol generally used for sonic velocity is or , and it is measured in meters per second (m/s).

Derivation of Sonic Velocity in a Gas

Consider a small pressure wave traveling through a gas. The propagation of this wave is governed by the relationship between pressure and density changes in the medium.

From the fundamental thermodynamic relationship, the square of sonic velocity is given by:

where,

= pressure,
= density, and
subscript indicates the process is adiabatic (no heat exchange).

For an ideal gas, the equation of state is:

Differentiating this for adiabatic conditions, we use the polytropic relation:

Taking derivative,

Substituting this into the earlier equation,

Hence, the expression for sonic velocity becomes:

Using the ideal gas law , the above equation can also be written as:

Factors Affecting Sonic Velocity

Temperature:
- The speed of sound increases with temperature because higher temperatures increase the kinetic energy of molecules.
- For gases, .
Density and Pressure:
- In gases at a given temperature, the effect of pressure is negligible since both pressure and density vary proportionally.
Nature of the Gas (γ and R):
- The value of (ratio of specific heats) and (gas constant) determines the speed of sound.
- For example, lighter gases like hydrogen have higher sonic velocities than heavier gases like carbon dioxide.
Medium Type:
- Sound travels faster in solids than in liquids and gases due to higher elasticity.
  - Typical speeds:
    - Air (at 20°C): 343 m/s
    - Water: 1480 m/s
    - Steel: 5000 m/s
Humidity:
- In air, humidity increases sonic velocity slightly because water vapor is less dense than dry air.

Significance of Sonic Velocity

Determines Flow Regime:
- Sonic velocity helps classify flow as subsonic, sonic, or supersonic based on the Mach number ().
Design of Aerodynamic Components:
- Used in designing nozzles, diffusers, and airfoils to handle compressible flow efficiently.
Acoustic Applications:
- Essential in sound wave propagation analysis, acoustic engineering, and sonar systems.
Thermodynamic Calculations:
- The value of is used in energy and momentum equations in compressible flow.
Measurement of Material Properties:
- In solids, sonic velocity is used to determine elastic modulus and detect material flaws using ultrasonic testing.

Example Calculation

For air at 20°C,

Then,

Thus, the sonic velocity in air at 20°C is approximately 343 m/s.

Practical Engineering Importance

In Aerospace Engineering: Determines the critical velocity for aircraft design.
In Mechanical Systems: Used in the analysis of flow through compressors, turbines, and nozzles.
In Civil and Marine Engineering: Helps in underwater sound communication and pipeline inspection.
In Ultrasonic Testing: Used for non-destructive material testing by measuring wave reflection.

Conclusion

The sonic velocity is the speed at which pressure or sound waves propagate through a medium. It depends on the medium’s thermodynamic and elastic properties, particularly its temperature, density, and specific heat ratio. In gases, it is given by , indicating that sonic velocity increases with temperature and decreases with molecular weight. Understanding sonic velocity is vital in compressible flow analysis, aerodynamic design, acoustics, and material testing, making it a fundamental concept in both mechanical and aerospace engineering.