2026-01-19 · Wave-Motion-Notes

MECHANICAL WAVE

Mechanical Waves - Complete Notes (7.1-7.6)

7.1 Speed of Wave Motion

The speed of wave motion depends on the elastic properties and inertial properties of the medium through which the wave travels. Understanding these factors is essential for calculating wave velocities in different media.

General Principle

Fundamental Relationship:

The velocity of a wave in any medium depends on:

Elastic Property: How quickly the medium restores itself after deformation (restoring force)
Inertial Property: The resistance to motion (mass/density)

v = √(Elastic Property / Inertial Property)

Or more specifically:
v = √(E / ρ)

Where:
• E = elastic modulus (Pa or N/m²)
• ρ = density of medium (kg/m³)
• v = wave velocity (m/s)

Physical Interpretation

Why v ∝ √(Elasticity)?

Greater elasticity → stronger restoring force
Particles return to equilibrium faster
Disturbance propagates more quickly
Result: Higher wave velocity

Why v ∝ 1/√(Density)?

Greater density → more inertia (resistance to motion)
Particles accelerate more slowly
Disturbance propagates more slowly
Result: Lower wave velocity

Different Types of Elastic Moduli

Elastic Modulus	Definition	Formula	Applies To
Young's Modulus (Y)	Resistance to longitudinal strain (stretching/compression)	Y = (F/A) / (ΔL/L)	Solids (longitudinal waves in rods)
Bulk Modulus (K)	Resistance to volume change under uniform pressure	K = -V(ΔP/ΔV)	Solids & Liquids (volume compression)
Rigidity Modulus (η)	Resistance to shear deformation (shape change)	η = (F/A) / θ	Solids (transverse waves)
Adiabatic Bulk Modulus (γP)	Bulk modulus under adiabatic conditions	K_adiabatic = γP	Gases (sound waves)

Wave Velocities in Different Media

General Formulas:

1. Transverse Waves in Solids (strings, rods):

v = √(η / ρ)
or for strings: v = √(T / μ)

Where:
• η = rigidity modulus
• T = tension in string (N)
• μ = mass per unit length (kg/m)

2. Longitudinal Waves in Solids (bulk):

v = √(Y / ρ)

Where Y = Young's modulus

3. Longitudinal Waves in Liquids:

v = √(K / ρ)

Where K = Bulk modulus

4. Longitudinal Waves in Gases:

v = √(γP / ρ) = √(γRT / M)

Where:
• γ = adiabatic index (C_p/C_v)
• P = pressure
• R = universal gas constant
• T = absolute temperature
• M = molar mass

Factors Affecting Wave Speed

Key Factors:

1. Nature of Medium:

Different materials have different elastic properties
Steel: v ≈ 5000 m/s (very elastic)
Water: v ≈ 1500 m/s
Air: v ≈ 343 m/s (less elastic)

2. Temperature (in gases):

Higher temperature → higher velocity
v ∝ √T

3. Density:

Higher density → lower velocity
v ∝ 1/√ρ

4. Pressure (in gases at constant temperature):

No effect! (ρ ∝ P, so √(P/ρ) remains constant)

Wave Velocities in Different Media (Approximate Values)

Why Sound is Faster in Solids?

Although solids are denser (which would slow sound), they are MUCH more elastic (which speeds up sound).

Medium	Density Effect	Elasticity Effect	Net Result
Gas (Air)	Very low ✓	Very low ✗	Slow (~343 m/s)
Liquid (Water)	Medium ✗	Medium ✓	Medium (~1500 m/s)
Solid (Steel)	High ✗✗	Very High ✓✓✓	Fast (~5000 m/s)

Key Insight: The elastic property increases MORE dramatically than density when going from gases to solids, so elasticity dominates!

Important for NEB Exams:

General formula: v = √(Elastic Property / Inertial Property)
Basic form: v = √(E/ρ) where E is appropriate elastic modulus
Higher elasticity → faster wave propagation
Higher density → slower wave propagation
Different moduli for different wave types (Y, K, η, γP)
Sound fastest in solids, slowest in gases (elasticity dominates)
In gases: v ∝ √T (temperature dependent)
Pressure has no effect on sound speed in gases at constant T

7.2 Velocity of Sound in Solid and Liquid

Sound waves travel as longitudinal waves in both solids and liquids. The velocity depends on the elastic modulus and density of the medium.

Velocity of Sound in Solids

For Longitudinal Waves in Solid Rods/Bars:

v = √(Y / ρ)

Where:
• v = velocity of sound in solid (m/s)
• Y = Young's modulus of elasticity (Pa or N/m²)
• ρ = density of solid (kg/m³)

Young's Modulus (Y):

Measures resistance to longitudinal strain (stretching/compression)
Y = (Stress) / (Strain) = (F/A) / (ΔL/L)
Higher Y → stiffer material → faster sound

Derivation for Velocity in Solid

Step-by-Step Derivation:

Step 1: Consider a solid rod

When a compression pulse travels through a solid rod of cross-section A and density ρ.

Step 2: Dimensional Analysis

Velocity must depend on:

Elastic property: Young's modulus Y (units: Pa = N/m² = kg/(m·s²))
Inertial property: Density ρ (units: kg/m³)

Step 3: Form the expression

v ∝ Y^a × ρ^b

Dimensional analysis:

[L/T] = [M/(L·T²)]^a × [M/L³]^b

[M⁰L¹T⁻¹] = [M^a+b L^-a-3b T^-2a]

Comparing powers:

M: 0 = a + b → a = -b
T: -1 = -2a → a = 1/2
Therefore: b = -1/2

Step 4: Final Result

v = k × Y^1/2 × ρ^-1/2

Where k = 1 (determined by detailed wave theory)

v = √(Y / ρ)

Velocity of Sound in Liquids

For Longitudinal Waves in Liquids:

v = √(K / ρ)

Where:
• v = velocity of sound in liquid (m/s)
• K = Bulk modulus of elasticity (Pa or N/m²)
• ρ = density of liquid (kg/m³)

Bulk Modulus (K):

Measures resistance to volume change under pressure
K = -V(ΔP/ΔV) = -(ΔP)/(ΔV/V)
Liquids cannot support shear stress (no transverse waves)
Only longitudinal waves can propagate

Why Different Moduli?

Understanding the Choice of Elastic Modulus:

In Solids (use Young's modulus Y):

Solids can support both compression/extension AND shear
For longitudinal waves in thin rods: lateral expansion is free
Young's modulus describes longitudinal elasticity
Appropriate for sound in rods, bars, beams

In Liquids (use Bulk modulus K):

Liquids cannot support shear stress
Only volume compression/expansion occurs
Bulk modulus describes volume elasticity
No transverse waves in liquids (only longitudinal)

Note on Extended Solids:

For sound in extended solid media (not thin rods), the formula becomes more complex and involves both bulk and shear moduli. For NEB exams, focus on v = √(Y/ρ) for solids.

Comparison: Solids vs Liquids

Property	Solids	Liquids
Formula	v = √(Y/ρ)	v = √(K/ρ)
Elastic Modulus	Young's modulus (Y)	Bulk modulus (K)
Type of Strain	Longitudinal (length change)	Volume change
Typical Velocity	3000-6000 m/s	1000-1500 m/s
Transverse Waves	Yes (can propagate)	No (cannot propagate)
Longitudinal Waves	Yes	Yes
Example	Steel: ~5200 m/s	Water: ~1480 m/s

Typical Values

Sound Velocities in Common Materials:

SOLIDS:

Material	Y (×10¹⁰ Pa)	ρ (kg/m³)	v (m/s)
Aluminum	7.0	2700	~5100
Steel	20.0	7800	~5100
Copper	13.0	8900	~3800
Glass	7.0	2500	~5300

LIQUIDS:

Material	K (×10⁹ Pa)	ρ (kg/m³)	v (m/s)
Water (20°C)	2.2	1000	~1480
Sea Water	2.3	1025	~1500
Mercury	28.5	13600	~1450
Ethanol	1.0	790	~1160

      Key Observations:
      Solids generally faster than liquids because Young's modulus >> Bulk modulus (relative to density difference)
Steel vs Aluminum: Steel has higher Y but also higher ρ; effects nearly cancel, giving similar velocities
Mercury: Despite very high bulk modulus, very high density results in moderate sound speed
Temperature effect: Slight increase in v with temperature (K increases slightly, ρ decreases slightly)

     

Sound Wave Propagation in Solid vs Liquid

Important for NEB Exams:

Solid formula: v = √(Y/ρ) using Young's modulus
Liquid formula: v = √(K/ρ) using Bulk modulus
Both formulas follow v = √(Elasticity/Density) pattern
Solids: Can support longitudinal AND transverse waves
Liquids: Only longitudinal waves (no shear resistance)
Sound generally faster in solids than liquids (higher elasticity)
Know typical values: Steel ~5000 m/s, Water ~1500 m/s
Both Y and K have units of pressure (Pa or N/m²)

7.3 Velocity of Sound in Gas

Sound propagates through gases as longitudinal pressure waves. The derivation of the velocity formula involves understanding whether the compressions and rarefactions occur under isothermal or adiabatic conditions.

Newton's Formula (Isothermal Assumption)

Newton's Original Formula:

Isaac Newton assumed that compressions and rarefactions in sound waves occur isothermally (at constant temperature).

For isothermal process:

The bulk modulus K = P (pressure itself)

Using v = √(K/ρ):

v = √(P / ρ)

Newton's Formula (INCORRECT)

Where:
• P = pressure of gas (Pa)
• ρ = density of gas (kg/m³)

Derivation of Newton's Formula

Step-by-Step Derivation:

Step 1: Isothermal Bulk Modulus

For an ideal gas undergoing isothermal compression:

PV = constant (Boyle's law at constant T)

Differentiating: P dV + V dP = 0

P dV = -V dP

dP = -P (dV/V)

Step 2: Define Bulk Modulus

K = -V(dP/dV) = -(dP)/(dV/V)

K = P (for isothermal process)

Step 3: Apply Wave Velocity Formula

v = √(K/ρ)

v = √(P/ρ)

Step 4: Alternative form using ideal gas law

From PV = nRT and ρ = m/V = nM/V:

P = ρRT/M

Therefore:

v = √(P/ρ) = √(RT/M)

Where:
• R = universal gas constant (8.314 J/(mol·K))
• T = absolute temperature (K)
• M = molar mass of gas (kg/mol)

Problem with Newton's Formula

Why Newton's Formula Failed:

Experimental Observation:

At 0°C and 1 atm, the experimental velocity of sound in air = 332 m/s

Newton's prediction:

Using v = √(P/ρ):

P = 1.013 × 10⁵ Pa
ρ = 1.293 kg/m³
v = √(1.013 × 10⁵ / 1.293) = 280 m/s

Discrepancy:

Calculated: 280 m/s vs Experimental: 332 m/s

Newton's formula gives ~16% error!

Reason for Failure:

Newton assumed isothermal compressions (constant temperature)
In reality, sound waves involve rapid compressions and rarefactions
Too fast for heat exchange with surroundings
Process is actually adiabatic (no heat transfer)!

Laplace Correction

Laplace's Improvement (1816):

Pierre-Simon Laplace corrected Newton's formula by recognizing that sound propagation in gases is an ADIABATIC process, not isothermal.

Key Insight:

Sound waves are too rapid for heat exchange
Compressions → temperature increase (adiabatic heating)
Rarefactions → temperature decrease (adiabatic cooling)
No time for thermal equilibrium with surroundings

For adiabatic process:

PV^γ = constant

Where γ = C_p/C_v (adiabatic index)

Derivation of Laplace Formula

Complete Derivation:

Step 1: Adiabatic Bulk Modulus

For adiabatic process: PV^γ = constant

Differentiating:

P·γ·V^γ-1·dV + V^γ·dP = 0

γ·P·dV + V·dP = 0

dP = -γ·P·(dV/V)

Step 2: Calculate Bulk Modulus

K_adiabatic = -V(dP/dV) = -(dP)/(dV/V)

K_adiabatic = γP

Step 3: Apply to Velocity Formula

v = √(K/ρ)

v = √(γP/ρ)

v = √(γP / ρ)

Laplace's Corrected Formula (CORRECT)

Where:
• γ = C_p/C_v (adiabatic index)
• P = pressure of gas
• ρ = density of gas

Step 4: Alternative Form Using Ideal Gas Law

From P = ρRT/M:

v = √(γRT / M)

Alternative Laplace Formula

Where:
• γ = adiabatic index
• R = 8.314 J/(mol·K)
• T = absolute temperature (K)
• M = molar mass (kg/mol)

Verification of Laplace Formula

Testing Laplace's Correction:

For air at 0°C (273 K):

γ = 1.4 (for diatomic gases like O₂, N₂)
P = 1.013 × 10⁵ Pa
ρ = 1.293 kg/m³

Calculation:

v = √(γP/ρ)

v = √(1.4 × 1.013 × 10⁵ / 1.293)

v = √(109,644)

v = 331 m/s

Laplace prediction: 331 m/s ≈ Experimental: 332 m/s ✓

Success! Laplace's correction perfectly matches experimental results!

Comparison: Newton vs Laplace

Aspect	Newton's Formula	Laplace's Formula
Assumption	Isothermal process (constant T)	Adiabatic process (no heat exchange)
Bulk Modulus	K = P	K = γP
Formula	v = √(P/ρ)	v = √(γP/ρ)
Alternative Form	v = √(RT/M)	v = √(γRT/M)
Predicted v (air, 0°C)	~280 m/s	~331 m/s
Experimental v	332 m/s	332 m/s
Accuracy	16% error ✗	< 1% error ✓
Status	Incorrect (historical)	Correct (accepted)

Values of γ for Different Gases

Adiabatic Index (γ = C_p/C_v):

Gas Type	Examples	γ	Reasoning
Monatomic	He, Ar, Ne	1.67	3 degrees of freedom (translation only)
Diatomic	O₂, N₂, H₂, Air	1.40	5 degrees of freedom (3 trans + 2 rot)
Polyatomic	CO₂, NH₃, CH₄	1.33	6+ degrees of freedom (more complex)

For most calculations involving air, use γ = 1.4

Isothermal vs Adiabatic Compressions in Sound Waves

Important for NEB Exams:

Newton's formula: v = √(P/ρ) [WRONG - assumed isothermal]
Laplace's formula: v = √(γP/ρ) [CORRECT - adiabatic process]
Alternative form: v = √(γRT/M)
Sound waves are too rapid for heat exchange → adiabatic
Laplace correction factor: γ = C_p/C_v
For air: γ = 1.4, v ≈ 331 m/s at 0°C
Newton's prediction: 16% too low
Monatomic: γ=1.67, Diatomic: γ=1.40, Polyatomic: γ=1.33

7.4 Laplace Correction in Detail

The Laplace correction is the modification made to Newton's formula for the velocity of sound in gases by introducing the adiabatic index γ. This section explores the correction in depth.

What is Laplace Correction?

Definition of Laplace Correction:

Laplace correction is the introduction of the adiabatic index (γ) into Newton's formula to account for the adiabatic nature of sound wave propagation in gases.

Newton's Formula: v = √(P/ρ)

↓ LAPLACE CORRECTION ↓

Corrected Formula: v = √(γP/ρ)

The correction factor is γ = C_p/C_v

Physical Meaning of γ

Understanding the Adiabatic Index:

γ (gamma) = C_p/C_v

Where:

C_p: Molar heat capacity at constant pressure
C_v: Molar heat capacity at constant volume

Physical Significance:

γ > 1 always (because C_p > C_v)
Measures how much extra energy is needed to heat gas at constant pressure vs constant volume
Related to degrees of freedom of gas molecules
Determines how pressure changes with volume in adiabatic process

Why C_p > C_v?

When heated at constant pressure, gas must do work to expand. This requires extra energy beyond just increasing temperature.

C_p - C_v = R

Where R = 8.314 J/(mol·K)

Values of γ Based on Molecular Structure

Theoretical Values from Kinetic Theory:

1. Monatomic Gases (f = 3):

Degrees of freedom: f = 3 (only translational)
C_v = (f/2)R = (3/2)R
C_p = C_v + R = (5/2)R
γ = C_p/C_v = (5/2)R / (3/2)R = 5/3 = 1.67
Examples: He, Ar, Ne, all noble gases

2. Diatomic Gases (f = 5):

Degrees of freedom: f = 5 (3 trans + 2 rot)
C_v = (5/2)R
C_p = (7/2)R
γ = 7/5 = 1.40
Examples: O₂, N₂, H₂, CO, air (~1.4)

3. Polyatomic Gases (f ≥ 6):

Degrees of freedom: f ≥ 6 (3 trans + 3 rot + vibrational)
C_v = (6/2)R or more
γ ≈ 1.33
Examples: CO₂, H₂O, NH₃, CH₄

Gas	Type	γ (theoretical)	γ (experimental)
Helium (He)	Monatomic	1.67	1.66
Argon (Ar)	Monatomic	1.67	1.67
Hydrogen (H₂)	Diatomic	1.40	1.41
Nitrogen (N₂)	Diatomic	1.40	1.40
Oxygen (O₂)	Diatomic	1.40	1.40
Air	Mixture (mainly diatomic)	1.40	1.40
Carbon dioxide (CO₂)	Polyatomic	1.33	1.30
Methane (CH₄)	Polyatomic	1.33	1.31

Mathematical Justification

Why the Correction Works:

Isothermal vs Adiabatic Bulk Modulus:

For Isothermal Process (Newton):

PV = constant

Differentiating: P dV + V dP = 0

K_isothermal = -V(dP/dV) = P

For Adiabatic Process (Laplace):

PV^γ = constant

Differentiating: P·γ·V^γ-1·dV + V^γ·dP = 0

K_adiabatic = -V(dP/dV) = γP

Ratio:

K_adiabatic / K_isothermal = γP / P = γ

Therefore:
v_adiabatic / v_isothermal = √γ

Numerical Example for Air (γ = 1.4):

v_Laplace / v_Newton = √1.4 ≈ 1.183

Laplace's value is about 18.3% higher than Newton's!

Experimental Verification

Testing the Correction:

Setup: Measure velocity of sound in air at 0°C

Method	Formula	Predicted v (m/s)	Error
Experimental	Direct measurement	332	—
Newton	√(P/ρ)	280	-16%
Laplace	√(γP/ρ), γ=1.4	331	-0.3%

Calculation Details:

At 0°C and 1 atm:

P = 1.013 × 10⁵ Pa
ρ = 1.293 kg/m³
γ = 1.4 (for air)

Newton: v = √(1.013×10⁵/1.293) = 280 m/s

Laplace: v = √(1.4×1.013×10⁵/1.293) = 331 m/s

Experiment: v = 332 m/s

Laplace's correction is spectacularly accurate!

Why the Process is Adiabatic

Time Scale Analysis:

Sound Wave Frequency Range:

Audible range: 20 Hz to 20,000 Hz
Typical: ~1000 Hz

Period of Oscillation:

T = 1/f ≈ 0.001 s = 1 millisecond

Heat Diffusion Time:

Time for heat to travel distance of wavelength λ:

t_heat ~ λ²/D (D = thermal diffusivity)

For air: D ≈ 2 × 10⁻⁵ m²/s

λ ≈ 0.34 m (at 1000 Hz)

t_heat ~ (0.34)² / (2×10⁻⁵) ~ 5800 s ≈ 1.6 hours!

Conclusion:

t_heat >> T_sound

Heat diffusion (hours) >> Sound period (milliseconds)

Therefore: Process is ADIABATIC (no time for heat transfer)

Summary of Laplace Correction:

1. Problem: Newton assumed isothermal compressions → predicted v too low

2. Solution: Laplace recognized adiabatic compressions → introduced γ

3. Correction Factor: γ = C_p/C_v (always > 1)

4. Effect: Increases predicted velocity by factor of √γ

5. Values:

Monatomic: γ = 1.67
Diatomic (air): γ = 1.4
Polyatomic: γ = 1.33

6. Result: Perfect agreement with experiment!

v = √(γP/ρ) = √(γRT/M)

Important for NEB Exams:

Laplace correction: Introducing γ into Newton's formula
γ = C_p/C_v (adiabatic index, ratio of heat capacities)
Correction needed because sound waves are adiabatic, not isothermal
γ > 1 always (typically 1.33 to 1.67)
For air: γ = 1.4 (remember this!)
Increases velocity by √γ ≈ 1.18 for air
Monatomic: γ=1.67, Diatomic: γ=1.40, Polyatomic: γ=1.33
Process is adiabatic because sound waves too fast for heat exchange

7.5 Effect of Temperature, Pressure, and Humidity on Velocity of Sound

The velocity of sound in gases is affected by various physical conditions. Understanding these effects is crucial for practical applications and theoretical understanding.

Effect of Temperature

Temperature Dependence:

Starting from Laplace formula:

v = √(γRT/M)

Since γ, R, and M are constants for a given gas:

v ∝ √T

Velocity is directly proportional to
the square root of absolute temperature

Physical Meaning:

Higher temperature → faster molecular motion
Faster molecules → quicker transmission of disturbance
Sound travels faster in warm air than cold air

Deriving Temperature Effect

Complete Derivation:

Let:

v₀ = velocity at temperature T₀ (usually 0°C = 273 K)
v_t = velocity at temperature T

Since v ∝ √T:

v₀/v_t = √(T₀/T)

Therefore:

v_t = v₀ × √(T/T₀)

Using Celsius temperature:

Let t°C be temperature in Celsius

T = (273 + t) K

T₀ = 273 K (at 0°C)

v_t = v₀ × √[(273 + t)/273]

v_t = v₀ × √[1 + t/273]

For small t (binomial approximation):

√(1 + x) ≈ 1 + x/2 when x << 1

v_t ≈ v₀[1 + t/(2×273)]

v_t ≈ v₀[1 + t/546]

v_t = v₀(1 + t/546)

Or: v_t ≈ v₀ + 0.61t

Where:
• v_t = velocity at t°C
• v₀ = velocity at 0°C (≈ 331 m/s for air)
• t = temperature in Celsius

Alternative form:

v_t ≈ 331 + 0.6t m/s

(Specifically for air at t°C)

Effect of Pressure (at constant temperature)

Pressure Dependence:

From Laplace formula:

v = √(γP/ρ)

From ideal gas law:

PV = nRT

P = (m/MV)RT = (ρ/M)RT

P/ρ = RT/M

Substituting into velocity formula:

v = √[γ(P/ρ)] = √(γRT/M)

At constant T: v is INDEPENDENT of P

v = √(γRT/M) contains no P term!

Physical Explanation:

Increasing pressure → increases both elastic restoring force AND inertia
P increases but ρ increases proportionally: P/ρ = RT/M (constant at fixed T)
These effects exactly cancel out!
Sound travels at same speed at sea level and mountain top (if same T)

Mathematical Proof: Pressure Independence

Detailed Proof:

Consider two states at same temperature T:

State 1: Pressure P₁, density ρ₁
State 2: Pressure P₂, density ρ₂

From ideal gas law at constant T:

P₁/ρ₁ = RT/M

P₂/ρ₂ = RT/M

Therefore: P₁/ρ₁ = P₂/ρ₂

Velocity at each state:

v₁ = √(γP₁/ρ₁)

v₂ = √(γP₂/ρ₂)

Since P₁/ρ₁ = P₂/ρ₂:

v₁ = v₂

Conclusion: Velocity is same regardless of pressure (at constant T)

Effect of Humidity

Humidity (Moisture) Dependence:

Humid air = mixture of dry air + water vapor

Key Insight:

v = √(γRT/M)

Velocity depends on molar mass M

Molar masses:

Dry air: M_air ≈ 29 g/mol
Water vapor: M_H₂O = 18 g/mol

When humidity increases:

Water vapor (lighter) replaces some air molecules (heavier)
Average molar mass M_mixture decreases
Since v ∝ 1/√M → velocity increases

v_humid > v_dry

Sound travels FASTER in humid air
(at same temperature and pressure)

Physical Meaning:

Humid air is less dense (lighter molecules)
Lower inertia → faster propagation
Effect is small but measurable (~0.1-0.3% increase)

Quantitative Analysis of Humidity Effect

Calculating Effective Molar Mass:

Let:

x = mole fraction of water vapor
(1-x) = mole fraction of dry air

Effective molar mass:

M_eff = x·M_H₂O + (1-x)·M_air

M_eff = 18x + 29(1-x)

M_eff = 29 - 11x

Velocity ratio:

v_humid/v_dry = √(M_dry/M_humid)

v_humid/v_dry = √[29/(29-11x)]

Example: At 100% relative humidity and 30°C, x ≈ 0.04

M_eff = 29 - 11(0.04) = 28.56

v_humid/v_dry = √(29/28.56) ≈ 1.008

Result: About 0.8% faster in humid air

Summary Table: Effects on Sound Velocity

Factor	Effect	Relation	Physical Reason
Temperature ↑	Velocity ↑	v ∝ √T	Faster molecular motion
Pressure ↑ (at constant T)	No effect	v independent of P	P and ρ increase proportionally; effects cancel
Humidity ↑	Velocity ↑ (slightly)	v ∝ 1/√M	Water vapor lighter than air; reduces average M
Density ↑ (other factors constant)	Velocity ↓	v ∝ 1/√ρ	More inertia to overcome
Molar mass ↑	Velocity ↓	v ∝ 1/√M	Heavier molecules move slower

Practical Examples

Real-World Applications:

1. Thunder Delay in Summer vs Winter:

Summer (30°C): v ≈ 349 m/s
Winter (0°C): v ≈ 331 m/s
Same lightning distance appears farther in winter (longer delay)

2. Mountain vs Sea Level (same temperature):

Mountain: Lower pressure (but also lower density)
Sea level: Higher pressure (but also higher density)
Result: Same sound velocity! (P and ρ change proportionally)

3. Humid Day vs Dry Day:

Humid air: Slightly faster sound (~0.5% increase)
Makes sound travel very slightly farther
Effect usually negligible compared to temperature

4. Different Gases at Same Conditions:

Helium (M=4): v ≈ 970 m/s (very light, high γ)
Air (M=29): v ≈ 343 m/s
CO₂ (M=44): v ≈ 260 m/s (heavy, lower γ)

Effect of Temperature on Sound Velocity in Air

Important for NEB Exams:

Temperature: v ∝ √T, or v_t = 331 + 0.6t m/s (for air)
Pressure (at constant T): NO effect on velocity
Humidity: Slight increase (water vapor lighter than air)
v ∝ 1/√M (lighter gases → faster sound)
Temperature effect most significant in practice
Each °C increase → ~0.6 m/s increase in air
Pressure independence: P/ρ = RT/M (constant at fixed T)
Know formula: v = √(γRT/M) to derive all effects

7.6 Numerical Problems: Velocity of Sound

This section covers solved numerical problems related to calculating velocity of sound in various media and conditions.

Problem 1: Basic Velocity Calculation in Gas

Problem:

Calculate the velocity of sound in air at 27°C and 1 atm pressure. Given: R = 8.314 J/(mol·K), M = 29 × 10⁻³ kg/mol, γ = 1.4

Solution:

Given:

T = 27°C = 27 + 273 = 300 K
R = 8.314 J/(mol·K)
M = 29 × 10⁻³ kg/mol
γ = 1.4

Formula:

v = √(γRT/M)

Calculation:

v = √[(1.4 × 8.314 × 300) / (29 × 10⁻³)]

v = √[3491.88 / 0.029]

v = √(120,409.66)

v = 347 m/s

ANSWER: Velocity of sound in air at 27°C = 347 m/s

Problem 2: Temperature Effect

Problem:

The velocity of sound in air at 0°C is 331 m/s. Calculate the velocity of sound at:

(a) 20°C

(b) 40°C

Solution:

Given:

v₀ = 331 m/s at 0°C

Formula:

v_t = v₀ + 0.6t

Or: v_t = 331 + 0.6t

(a) At 20°C:

v₂₀ = 331 + 0.6(20)

v₂₀ = 331 + 12

v₂₀ = 343 m/s

(b) At 40°C:

v₄₀ = 331 + 0.6(40)

v₄₀ = 331 + 24

v₄₀ = 355 m/s

Alternative method using √T:

v_t/v₀ = √(T/T₀)

For 20°C: v₂₀ = 331 × √(293/273) = 331 × 1.036 = 343 m/s ✓

ANSWERS: (a) v₂₀ = 343 m/s, (b) v₄₀ = 355 m/s

Problem 3: Velocity in Different Gases

Problem:

At 0°C and 1 atm, calculate the velocity of sound in:

(a) Hydrogen (H₂): M = 2 g/mol, γ = 1.4

(b) Helium (He): M = 4 g/mol, γ = 1.67

Given: R = 8.314 J/(mol·K)

Solution:

Given:

T = 0°C = 273 K
R = 8.314 J/(mol·K)

Formula:

v = √(γRT/M)

(a) Hydrogen (H₂):

M = 2 × 10⁻³ kg/mol, γ = 1.4

v = √[(1.4 × 8.314 × 273) / (2 × 10⁻³)]

v = √[3173.13 / 0.002]

v = √(1,586,565)

v = 1259 m/s

(b) Helium (He):

M = 4 × 10⁻³ kg/mol, γ = 1.67

v = √[(1.67 × 8.314 × 273) / (4 × 10⁻³)]

v = √[3790.19 / 0.004]

v = √(947,547)

v = 973 m/s

(c) Carbon dioxide (CO₂):

M = 44 × 10⁻³ kg/mol, γ = 1.3

v = √[(1.3 × 8.314 × 273) / (44 × 10⁻³)]

v = √[2949.44 / 0.044]

v = √(67,032.7)

v = 259 m/s

ANSWERS: (a) H₂: 1259 m/s, (b) He: 973 m/s, (c) CO₂: 259 m/s

Note: Lighter gases have faster sound velocities!

Problem 4: Velocity in Solid

Problem:

Calculate the velocity of sound in steel. Given: Young's modulus Y = 2.0 × 10¹¹ Pa, density ρ = 7800 kg/m³

Solution:

Given:

Y = 2.0 × 10¹¹ Pa
ρ = 7800 kg/m³

Formula for solid:

v = √(Y/ρ)

Calculation:

v = √[(2.0 × 10¹¹) / 7800]

v = √(25,641,026)

v = 5064 m/s

v ≈ 5100 m/s

ANSWER: Velocity of sound in steel ≈ 5100 m/s

Problem 5: Velocity in Liquid

Problem:

Calculate the velocity of sound in water. Given: Bulk modulus K = 2.2 × 10⁹ Pa, density ρ = 1000 kg/m³

Solution:

Given:

K = 2.2 × 10⁹ Pa
ρ = 1000 kg/m³

Formula for liquid:

v = √(K/ρ)

Calculation:

v = √[(2.2 × 10⁹) / 1000]

v = √(2,200,000)

v = 1483 m/s

v ≈ 1480 m/s

ANSWER: Velocity of sound in water ≈ 1480 m/s

Problem 6: Newton vs Laplace Comparison

Problem:

At 0°C and 1 atm, calculate the velocity of sound in air using:

(a) Newton's formula

(b) Laplace's formula

Given: P = 1.013 × 10⁵ Pa, ρ = 1.293 kg/m³, γ = 1.4

Solution:

Given:

P = 1.013 × 10⁵ Pa
ρ = 1.293 kg/m³
γ = 1.4

(a) Newton's formula:

v_Newton = √(P/ρ)

v_Newton = √[(1.013 × 10⁵) / 1.293]

v_Newton = √(78,331.5)

v_Newton = 280 m/s

(b) Laplace's formula:

v_Laplace = √(γP/ρ)

v_Laplace = √[(1.4 × 1.013 × 10⁵) / 1.293]

v_Laplace = √(109,664.1)

v_Laplace = 331 m/s

(c) Percentage error:

Error = [(Experimental - Newton) / Experimental] × 100%

Taking experimental ≈ Laplace = 331 m/s:

Error = [(331 - 280) / 331] × 100%

Error = (51/331) × 100%

Error = 15.4% ≈ 16%

ANSWERS: (a) Newton: 280 m/s, (b) Laplace: 331 m/s, (c) Error ≈ 16%

Laplace's correction (factor of √1.4 ≈ 1.18) is essential!

Problem 7: Finding Temperature from Velocity

Problem:

The velocity of sound in air is measured to be 349 m/s. At what temperature was this measurement made? (Assume v₀ = 331 m/s at 0°C)

Solution:

Given:

v_t = 349 m/s
v₀ = 331 m/s

Formula:

v_t = v₀ + 0.6t

Solving for t:

349 = 331 + 0.6t

349 - 331 = 0.6t

18 = 0.6t

t = 18/0.6

t = 30°C

Verification using √T method:

v_t/v₀ = √(T/T₀)

349/331 = √(T/273)

1.054 = √(T/273)

1.111 = T/273

T = 303 K = 30°C ✓

ANSWER: Temperature = 30°C

Problem 8: Ratio of Velocities

Problem:

Find the ratio of velocity of sound in hydrogen to that in oxygen at the same temperature. Given: M_H₂ = 2 g/mol, M_O₂ = 32 g/mol, γ_H₂ = γ_O₂ = 1.4

Solution:

Given:

M_H₂ = 2 g/mol
M_O₂ = 32 g/mol
γ_H₂ = γ_O₂ = 1.4
Same temperature T

Formula:

v = √(γRT/M)

For hydrogen:

v_H₂ = √(γRT/M_H₂)

For oxygen:

v_O₂ = √(γRT/M_O₂)

Ratio:

v_H₂/v_O₂ = √(M_O₂/M_H₂)

(γ, R, T are same and cancel out)

v_H₂/v_O₂ = √(32/2)

v_H₂/v_O₂ = √16

v_H₂/v_O₂ = 4

ANSWER: v_H₂ : v_O₂ = 4 : 1

Sound travels 4 times faster in hydrogen than oxygen!

Problem-Solving Strategy:

For Gases:

Use v = √(γRT/M) for general calculations
Use v = √(γP/ρ) when P and ρ given directly
Use v_t = 331 + 0.6t for air at Celsius temperatures
Remember: γ = 1.4 for air and diatomic gases

For Solids:

Use v = √(Y/ρ) with Young's modulus
Typical values: ~5000 m/s for metals

For Liquids:

Use v = √(K/ρ) with Bulk modulus
Water: ~1500 m/s

Common Mistakes to Avoid:

Convert temperature to Kelvin (T = t + 273)
Convert molar mass to kg/mol (divide g/mol by 1000)
Don't confuse Newton's and Laplace's formulas
Remember R = 8.314 J/(mol·K), not 8.314 J/(g·K)

Important for NEB Exams:

Gas: v = √(γRT/M) or v = √(γP/ρ)
Solid: v = √(Y/ρ)
Liquid: v = √(K/ρ)
Air temperature: v = 331 + 0.6t m/s
Always convert to SI units first
For ratios: cancel common terms (γ, R, T)
Laplace correction: multiply Newton's result by √γ
Know typical values: Air ~340 m/s, Water ~1500 m/s, Steel ~5000 m/s

MECHANICAL WAVE

MECHANICAL WAVES

7.1 Speed of Wave Motion

General Principle

Fundamental Relationship:

Physical Interpretation

Why v ∝ √(Elasticity)?

Why v ∝ 1/√(Density)?

Different Types of Elastic Moduli

Wave Velocities in Different Media

General Formulas:

Factors Affecting Wave Speed

Key Factors:

Why Sound is Faster in Solids?

Important for NEB Exams:

7.2 Velocity of Sound in Solid and Liquid

Velocity of Sound in Solids

For Longitudinal Waves in Solid Rods/Bars:

Derivation for Velocity in Solid

Step-by-Step Derivation:

Velocity of Sound in Liquids

For Longitudinal Waves in Liquids:

Why Different Moduli?

Understanding the Choice of Elastic Modulus:

Comparison: Solids vs Liquids

Typical Values

Sound Velocities in Common Materials:

Key Observations:

Important for NEB Exams:

7.3 Velocity of Sound in Gas

Newton's Formula (Isothermal Assumption)

Newton's Original Formula:

Derivation of Newton's Formula

Step-by-Step Derivation:

Problem with Newton's Formula

Why Newton's Formula Failed:

Laplace Correction

Laplace's Improvement (1816):

Derivation of Laplace Formula

Complete Derivation:

Verification of Laplace Formula

Testing Laplace's Correction:

Comparison: Newton vs Laplace

Values of γ for Different Gases

Adiabatic Index (γ = Cp/Cv):

Important for NEB Exams:

7.4 Laplace Correction in Detail

What is Laplace Correction?

Definition of Laplace Correction:

Physical Meaning of γ

Understanding the Adiabatic Index:

Values of γ Based on Molecular Structure

Theoretical Values from Kinetic Theory:

Mathematical Justification

Why the Correction Works:

Experimental Verification

Testing the Correction:

Why the Process is Adiabatic

Time Scale Analysis:

Summary of Laplace Correction:

Important for NEB Exams:

7.5 Effect of Temperature, Pressure, and Humidity on Velocity of Sound

Effect of Temperature

Temperature Dependence:

Deriving Temperature Effect

Complete Derivation:

Effect of Pressure (at constant temperature)

Pressure Dependence:

Mathematical Proof: Pressure Independence

Detailed Proof:

Effect of Humidity

Humidity (Moisture) Dependence:

Quantitative Analysis of Humidity Effect

Calculating Effective Molar Mass:

Summary Table: Effects on Sound Velocity

Practical Examples

Real-World Applications:

Important for NEB Exams:

7.6 Numerical Problems: Velocity of Sound

Problem 1: Basic Velocity Calculation in Gas

Adiabatic Index (γ = C_p/C_v):