Mechanical Waves

SYLLABUS

Speed of wave motion; Velocity of sound in solid and liquid, Velocity of sound in gas, Laplace’s correction, Effect of temperature, pressure, humidity on velocity of sound.

What is This Chapter About?

When you speak, your vocal cords vibrate and create pressure disturbances in the air. These disturbances travel outward as sound waves — mechanical waves that require a medium to propagate. By the time they reach a listener’s ear, they have carried energy across the room without any air molecule making the entire journey.

Mechanical Waves are waves that propagate through a physical medium — solid, liquid, or gas — by means of oscillations of the medium’s particles. Chapter 7 of NEB Grade 12 Physics focuses specifically on the speed of mechanical waves, with particular emphasis on sound in different media. Carrying 4 teaching hours, this chapter provides the quantitative tools needed to understand acoustic phenomena and sets the stage for Chapter 8 (Waves in Pipes and Strings) and Chapter 9 (Acoustic Phenomena).

Key Concepts You Will Learn

Speed of Wave Motion

The speed of a mechanical wave through a medium depends on two factors: the elasticity of the medium (how strongly it resists deformation and tries to restore itself) and the inertia of the medium (its density, which determines how reluctant it is to start moving). In general: [latex]v = \sqrt{\frac{\text{elastic property}}{\text{inertia property}}}[/latex]. A stiffer medium transmits waves faster; a denser medium transmits waves more slowly.

Velocity of Sound in Solids and Liquids

For a solid rod, the speed of a longitudinal wave (compression wave) is: [latex]v = \sqrt{\frac{Y}{\rho}}[/latex], where Y is Young’s modulus and [latex]\rho[/latex] is the density of the solid. For a liquid, the corresponding expression uses the bulk modulus B: [latex]v = \sqrt{\frac{B}{\rho}}[/latex]. Solids are generally much stiffer than liquids or gases, which is why sound travels faster in steel than in water, and faster in water than in air.

Velocity of Sound in Gas — Newton’s Formula

Newton assumed that sound propagates through air by isothermal compression and rarefaction — meaning the temperature of the gas remains constant during the pressure variations of the wave. Using the isothermal bulk modulus [latex]B = P[/latex] (where P is the equilibrium pressure of the gas), Newton derived: [latex]v = \sqrt{\frac{P}{\rho}}[/latex]. At standard atmospheric pressure and room temperature, this gives approximately 280 m/s for air — significantly lower than the experimentally measured value of about 332 m/s. Newton’s formula underestimates the speed because his isothermal assumption is incorrect.

Laplace’s Correction

The French mathematician Laplace corrected Newton’s error by pointing out that sound propagation is actually an adiabatic process — the compressions and rarefactions occur so rapidly that there is no time for heat exchange with the surroundings. Using the adiabatic bulk modulus [latex]B = \gamma P[/latex], where [latex]\gamma = \frac{C_p}{C_v}[/latex] is the ratio of specific heat capacities, Laplace obtained: [latex]v = \sqrt{\frac{\gamma P}{\rho}}[/latex]. For air, [latex]\gamma \approx 1.4[/latex], giving approximately 332 m/s — in excellent agreement with experiment. Laplace’s correction is a beautiful example of how a single physical insight transforms a flawed theory into an accurate one.

Effect of Temperature on Speed of Sound

Using the ideal gas law, the speed of sound in a gas can be written as: [latex]v = \sqrt{\frac{\gamma RT}{M}}[/latex], where R is the universal gas constant, T is the absolute temperature, and M is the molar mass of the gas. This shows that speed of sound is proportional to [latex]\sqrt{T}[/latex] — it increases with increasing temperature. At 0°C (273 K), sound travels at about 332 m/s in air, increasing by approximately 0.61 m/s for every degree Celsius rise in temperature.

Effect of Pressure and Humidity

At constant temperature, changing pressure does not affect the speed of sound in an ideal gas — because both P and [latex]\rho[/latex] change proportionally and their ratio remains constant. This often surprises students who expect denser air at higher pressure to transmit sound differently.

Humidity increases the speed of sound slightly. Water vapour (molar mass 18 g/mol) is lighter than dry air (molar mass approximately 29 g/mol), so humid air is effectively less dense than dry air at the same temperature and pressure. Since speed increases with decreasing density, sound travels slightly faster in humid air.

Why This Chapter Matters for NEB Exams

Mechanical Waves is examined consistently in NEB Grade 12 papers. Newton’s formula for speed of sound, Laplace’s correction, and the derivation of [latex]v = \sqrt{\frac{\gamma P}{\rho}}[/latex] are standard long-answer questions. The effect of temperature, pressure, and humidity on the speed of sound appear as short-answer and numerical questions. Numerical problems asking students to calculate the speed of sound at a given temperature or compare speeds in different media are common.

Important Topics to Focus On

• General formula for wave speed: [latex]v = \sqrt{\frac{\text{elastic modulus}}{\text{density}}}[/latex]
• Speed of sound in solids: [latex]v = \sqrt{\frac{Y}{\rho}}[/latex]
• Speed of sound in liquids: [latex]v = \sqrt{\frac{B}{\rho}}[/latex]
• Newton’s formula for speed of sound in gas: [latex]v = \sqrt{\frac{P}{\rho}}[/latex] and why it fails
• Laplace’s correction and adiabatic propagation: [latex]v = \sqrt{\frac{\gamma P}{\rho}}[/latex]
• Temperature dependence: [latex]v \propto \sqrt{T}[/latex]
• Why pressure does not affect speed at constant temperature
• Effect of humidity — why sound is faster in humid air
• Numerical problems on speed of sound under various conditions

Connection to Other Chapters

Mechanical Waves builds directly on Chapter 6 (Wave Motion) — it applies the general wave concepts to sound specifically. It connects forward to Chapter 8 (Waves in Pipes and Strings), where the speed of sound in air determines the frequencies of harmonics in organ pipes and strings. Chapter 9 (Acoustic Phenomena) uses sound wave properties for Doppler effect calculations. The adiabatic nature of sound propagation connects back to Chapter 4 (First Law of Thermodynamics) and Chapter 5 (Second Law), giving thermodynamics a direct application in wave physics.