Sound Waves Physics: Complete Guide to Beats & Doppler Effect Formulas with Examples

Mastering Beats Phenomenon and Doppler Effect with Mathematical Derivations and Practical Applications

Sound Waves Physics Beats Phenomenon Doppler Effect Wave Superposition Frequency Calculation Reading Time: 18 min

📋 Table of Contents

• 1. Introduction to Sound Waves
• 2. Beats Phenomenon
  • 2.1 Mathematical Derivation of Beats
  • 2.2 Beat Frequency Formula
  • 2.3 Practical Applications of Beats
• 3. Doppler Effect Fundamentals
  • 3.1 Understanding Apparent Frequency
• 4. Doppler Effect Cases
  • 4.1 Source Moving Towards Stationary Observer
  • 4.2 Source Moving Away from Stationary Observer
  • 4.3 Observer Moving Towards Stationary Source
  • 4.4 Observer Moving Away from Stationary Source
  • 4.5 Both Source and Observer Moving
• 5. Practical Applications
• 6. Problem Solving Examples
• Frequently Asked Questions

📜 Historical Background

The study of sound waves and their properties has evolved through centuries of scientific discovery:

• Christian Doppler (1842): First proposed the Doppler effect for both light and sound waves
• Buys Ballot (1845): Experimentally verified Doppler's theory using sound waves from a moving train
• Lord Rayleigh (1877): Published "The Theory of Sound" with comprehensive treatment of wave phenomena
• Wallace Clement Sabine (1900): Founded modern architectural acoustics

These developments established the mathematical foundation for understanding sound wave behavior in various conditions.

Introduction to Sound Waves

🔊 What are Sound Waves?

Sound waves are longitudinal mechanical waves that propagate through elastic media by means of compressions and rarefactions. These waves require a material medium for propagation and cannot travel through vacuum.

The key characteristics of sound waves include:

• Frequency (f): Number of oscillations per second, measured in Hertz (Hz)
• Wavelength (λ): Distance between successive compressions or rarefactions
• Amplitude: Maximum displacement of particles from equilibrium position
• Speed (v): Depends on the medium (approximately 343 m/s in air at 20°C)

📝 Fundamental Wave Relationship

The fundamental relationship connecting wave speed, frequency, and wavelength is:

\[ v = f \lambda \]

This equation is fundamental to understanding both beats and the Doppler effect in sound waves.

Beats Phenomenon

🎵 What are Beats?

Beats are the periodic alterations in the intensity of sound between minimum and maximum loudness. This phenomenon arises due to the superposition of two sound waves of the same amplitude but slightly different frequencies.

When two such waves interfere, they produce a resultant wave whose amplitude varies periodically with time, creating the characteristic "wah-wah" sound of beats.

⚙️ Visualizing Beats

Beats Formation

Wave 1 + Wave 2 = Beats Pattern

Working Principle: When two sound waves with slightly different frequencies superpose:

1. They periodically come into phase (constructive interference)
2. They periodically go out of phase (destructive interference)
3. This creates periodic variations in amplitude
4. The ear perceives these as periodic variations in loudness

Practical Example: When tuning two guitar strings, beats are heard when their frequencies are close but not identical. The beat frequency disappears when the strings are perfectly tuned.

Mathematical Derivation of Beats

🧮 Mathematical Analysis of Beats

Step 1: Individual Wave Equations

Consider two sound waves with equal amplitude but different frequencies:

\[ \Delta P_1(t) = \Delta P_m \sin \omega_1 t \]

\[ \Delta P_2(t) = \Delta P_m \sin \omega_2 t \]

where ΔP represents pressure variation and ω represents angular frequency

Step 2: Superposition Principle

By the superposition principle, the resultant pressure is:

\[ \Delta P = \Delta P_1(t) + \Delta P_2(t) \]

\[ = \Delta P_m \sin \omega_1 t + \Delta P_m \sin \omega_2 t \]

\[ = \Delta P_m (\sin \omega_1 t + \sin \omega_2 t) \]

Step 3: Trigonometric Identity

Using the trigonometric identity for sum of sines:

\[ \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \]

\[ \Delta P = 2 \Delta P_m \cos \left( \frac{\omega_1 - \omega_2}{2} \right) t \cdot \sin \left( \frac{\omega_1 + \omega_2}{2} \right) t \]

Step 4: Simplified Form

Let:

\[ \omega_{amp} = \frac{\omega_1 - \omega_2}{2} \]

\[ \omega_{av} = \frac{\omega_1 + \omega_2}{2} \]

\[ \Delta P = 2 \Delta P_m \cos \omega_{amp} t \cdot \sin \omega_{av} t \]

Beat Frequency Formula

🧮 Deriving Beat Frequency

Step 1: Amplitude Variation

The amplitude varies as:

\[ A(t) = 2 \Delta P_m \cos \omega_{amp} t \]

This amplitude varies between maximum (2ΔPₘ) and minimum (0) with frequency ωₐₘₚ

Step 2: Intensity Variation

Since intensity is proportional to the square of amplitude:

\[ I \propto A^2 \]

The intensity reaches maximum when |cos ωₐₘₚt| = 1

Step 3: Beat Frequency

The amplitude frequency is ωₐₘₚ, but intensity maxima occur twice per amplitude cycle:

\[ \omega_{beat} = 2 \omega_{amp} \]

\[ = 2 \left( \frac{\omega_1 - \omega_2}{2} \right) \]

\[ = \omega_1 - \omega_2 \]

Step 4: Final Formula

Converting to linear frequency:

\[ 2\pi f_{beat} = 2\pi f_1 - 2\pi f_2 \]

\[ f_{beat} = f_1 - f_2 \]

💡 Key Insight

The number of beats produced per second is equal to the absolute difference between the frequencies of the two interfering waves. This simple relationship makes beats extremely useful for frequency measurement and instrument tuning.

Practical Applications of Beats

🎸 Musical Instrument Tuning

Musicians use beats to tune instruments. When two strings are close in frequency, beats are heard. The tuner adjusts until beats disappear, indicating identical frequencies.

📡 Radar and Sonar

In Doppler radar and sonar systems, the beat frequency between transmitted and reflected waves is used to determine the speed of moving objects.

🔧 Machinery Monitoring

Engineers use beat phenomena to detect small frequency variations in rotating machinery, helping identify wear or misalignment before failure occurs.

Doppler Effect Fundamentals

🚗 What is the Doppler Effect?

The Doppler effect is the apparent change in the frequency (pitch) of sound due to relative motion between the source and the observer. This phenomenon is commonly experienced when a vehicle with a siren passes by - the pitch appears higher as it approaches and lower as it recedes.

📝 Key Observations

• When source moves toward observer: Apparent frequency increases
• When source moves away from observer: Apparent frequency decreases
• When observer moves toward source: Apparent frequency increases
• When observer moves away from source: Apparent frequency decreases
• When both move: Combined effect depends on relative motion

Understanding Apparent Frequency

🔬 Fundamental Concept

The apparent frequency change occurs because:

• For moving source: Wavefronts are compressed or expanded, changing wavelength
• For moving observer: More or fewer wavefronts are encountered per second, changing perceived frequency

The fundamental relationship is:

\[ f' = \frac{v \pm v_o}{v \pm v_s} f \]

where f' is apparent frequency, f is actual frequency, v is sound speed, vₒ is observer speed, and vₛ is source speed.

Doppler Effect Cases

Case 1: Source Moving Towards Stationary Observer

🧮 Mathematical Analysis

Step 1: Wavelength Compression

When source moves toward observer at speed uₛ:

\[ \Delta \lambda = \frac{u_s}{f} \]

\[ \lambda_A = \lambda - \Delta \lambda \]

\[ = \frac{v}{f} - \frac{u_s}{f} \]

\[ = \frac{v - u_s}{f} \]

Step 2: Apparent Frequency

\[ f_A = \frac{v}{\lambda_A} \]

\[ = \frac{v}{\left( \frac{v - u_s}{f} \right)} \]

\[ = \left( \frac{v}{v - u_s} \right) f \]

Step 3: Result

Since v/(v - uₛ) > 1, we have fₐ > f

Conclusion: Apparent frequency increases when source moves toward stationary observer

Case 2: Source Moving Away from Stationary Observer

🧮 Mathematical Analysis

Step 1: Wavelength Expansion

When source moves away from observer at speed uₛ:

\[ \Delta \lambda = \frac{u_s}{f} \]

\[ \lambda_B = \lambda + \Delta \lambda \]

\[ = \frac{v}{f} + \frac{u_s}{f} \]

\[ = \frac{v + u_s}{f} \]

Step 2: Apparent Frequency

\[ f_B = \frac{v}{\lambda_B} \]

\[ = \frac{v}{\left( \frac{v + u_s}{f} \right)} \]

\[ = \left( \frac{v}{v + u_s} \right) f \]

Step 3: Result

Since v/(v + uₛ) < 1, we have fʙ < f

Conclusion: Apparent frequency decreases when source moves away from stationary observer

Case 3: Observer Moving Towards Stationary Source

🧮 Mathematical Analysis

Step 1: Relative Velocity

When observer moves toward source at speed uₒ:

\[ \text{Relative velocity} = v + u_o \]

Step 2: Apparent Frequency

\[ f_c = \frac{v + u_o}{\lambda} \]

\[ = \frac{v + u_o}{\left( \frac{v}{f} \right)} \]

\[ = \left( \frac{v + u_o}{v} \right) f \]

Step 3: Result

Since (v + uₒ)/v > 1, we have f꜀ > f

Conclusion: Apparent frequency increases when observer moves toward stationary source

Case 4: Observer Moving Away from Stationary Source

🧮 Mathematical Analysis

Step 1: Relative Velocity

When observer moves away from source at speed uₒ:

\[ \text{Relative velocity} = v - u_o \]

Step 2: Apparent Frequency

\[ f_D = \frac{v - u_o}{\lambda} \]

\[ = \frac{v - u_o}{\left( \frac{v}{f} \right)} \]

\[ = \left( \frac{v - u_o}{v} \right) f \]

Step 3: Result

Since (v - uₒ)/v < 1, we have fᴅ < f

Conclusion: Apparent frequency decreases when observer moves away from stationary source

Case 5: Both Source and Observer Moving

🧮 Combined Effect Analysis

Step 1: General Formula

When both source and observer are moving:

\[ f''' = \left( \frac{v \pm u_o}{v \pm u_s} \right) f \]

Step 2: Moving Toward Each Other

\[ f''' = \left( \frac{v + u_o}{v - u_s} \right) f \]

Maximum frequency increase occurs when both approach each other

Step 3: Moving Away From Each Other

\[ f''' = \left( \frac{v - u_o}{v + u_s} \right) f \]

Maximum frequency decrease occurs when both recede from each other

💡 Sign Convention Rule

Remember the sign convention:

• + in numerator: Observer moving toward source
• - in numerator: Observer moving away from source
• - in denominator: Source moving toward observer
• + in denominator: Source moving away from observer

Practical Applications

🚑 Emergency Vehicle Sirens

The characteristic pitch change of ambulance or police sirens is a classic example of the Doppler effect, helping listeners determine if the vehicle is approaching or receding.

🌤️ Weather Radar

Doppler radar measures the frequency shift of reflected microwaves to determine wind speed and direction, crucial for weather forecasting and storm tracking.

🚗 Speed Detection

Police radar guns use the Doppler effect to measure vehicle speeds by detecting frequency changes in reflected radio waves.

🩺 Medical Ultrasound

Doppler ultrasound measures blood flow velocity by detecting frequency shifts in reflected sound waves, helping diagnose cardiovascular conditions.

🌌 Astronomy

The redshift of light from distant galaxies (cosmological Doppler effect) provides evidence for the expansion of the universe.

🎵 Music and Audio

Audio engineers use Doppler effect principles in sound design and special effects to create realistic moving sound sources.

Problem Solving Examples

Example 1: Beat Frequency Calculation

Two tuning forks have frequencies of 256 Hz and 260 Hz respectively. When sounded together, what beat frequency is heard?

Given:

\[ f_1 = 256 \, \text{Hz} \]

\[ f_2 = 260 \, \text{Hz} \]

Beat frequency formula:

\[ f_{beat} = |f_1 - f_2| \]

\[ = |256 - 260| \]

\[ = 4 \, \text{Hz} \]

Answer: 4 beats per second are heard

Example 2: Doppler Effect - Moving Source

A police car siren has frequency 1000 Hz. If the car moves toward a stationary observer at 30 m/s, and sound speed is 340 m/s, what frequency does the observer hear?

Given:

\[ f = 1000 \, \text{Hz} \]

\[ u_s = 30 \, \text{m/s} \]

\[ v = 340 \, \text{m/s} \]

Observer stationary: \( u_o = 0 \)

Source moving toward observer:

\[ f' = \left( \frac{v}{v - u_s} \right) f \]

\[ = \left( \frac{340}{340 - 30} \right) \times 1000 \]

\[ = \left( \frac{340}{310} \right) \times 1000 \]

\[ = 1.0968 \times 1000 \]

\[ = 1096.8 \, \text{Hz} \]

Answer: The observer hears 1096.8 Hz

Example 3: Doppler Effect - Moving Observer

A stationary source emits sound at 500 Hz. If an observer moves toward the source at 20 m/s, and sound speed is 340 m/s, what frequency does the observer hear?

Given:

\[ f = 500 \, \text{Hz} \]

\[ u_o = 20 \, \text{m/s} \]

\[ v = 340 \, \text{m/s} \]

Source stationary: \( u_s = 0 \)

Observer moving toward source:

\[ f' = \left( \frac{v + u_o}{v} \right) f \]

\[ = \left( \frac{340 + 20}{340} \right) \times 500 \]

\[ = \left( \frac{360}{340} \right) \times 500 \]

\[ = 1.0588 \times 500 \]

\[ = 529.4 \, \text{Hz} \]

Answer: The observer hears 529.4 Hz

Frequently Asked Questions

❓ Why does the beat frequency equal the difference between the two frequencies?

The beat frequency equals the difference between the two interfering frequencies because of the way the waves constructively and destructively interfere over time:

• When two waves with frequencies f₁ and f₂ interfere, the resultant wave has a frequency equal to the average (f₁ + f₂)/2
• The amplitude of this resultant wave varies with frequency |f₁ - f₂|/2
• Since intensity (which we perceive as loudness) is proportional to the square of amplitude, the intensity varies with frequency |f₁ - f₂|
• Our ears detect these intensity variations as beats

This is why we hear |f₁ - f₂| beats per second when two sound waves of frequencies f₁ and f₂ interfere.

❓ Does the Doppler effect work the same way for light as it does for sound?

While both sound and light exhibit Doppler effects, there are important differences:

• Sound: Requires a medium, and the formulas depend on both source and observer motion relative to the medium
• Light: Requires no medium, and the relativistic Doppler effect formulas are different
• Sound: Frequency changes are different depending on whether source or observer is moving
• Light: There's no distinction between source motion and observer motion due to relativity

The relativistic Doppler effect for light is given by:

\[ f' = f \sqrt{\frac{1 - v/c}{1 + v/c}} \quad \text{(source moving away)} \]

\[ f' = f \sqrt{\frac{1 + v/c}{1 - v/c}} \quad \text{(source moving toward)} \]

where c is the speed of light and v is the relative speed between source and observer.

❓ Can the Doppler effect make a sound completely inaudible?

While the Doppler effect can significantly change the perceived frequency of sound, it cannot make a sound completely inaudible through frequency shift alone. However:

• For very high relative speeds, the frequency can shift beyond the audible range (20 Hz - 20 kHz)
• The intensity of sound decreases with distance due to the inverse square law
• For supersonic motion, shock waves create additional effects

For example, if a source emitting 100 Hz sound moves away from an observer at 0.9 times the speed of sound:

\[ f' = \left( \frac{v}{v + u_s} \right) f = \left( \frac{340}{340 + 306} \right) \times 100 \approx 52.6 \, \text{Hz} \]

The sound is still audible, though at a lower pitch. To shift completely out of the audible range would require extreme relative speeds not typically encountered in everyday situations.

📚 Master Sound Waves Physics

Understanding beats and the Doppler effect is fundamental to wave physics, with applications ranging from music to medical imaging to astronomy. These concepts demonstrate how wave behavior changes with relative motion and interference.

Read More: Physics HRK Notes on Sound Waves

© House of Physics | HRK Physics Chapter 20: Sound Waves - Beats and Doppler Effect

Based on Halliday, Resnick, and Krane's "Physics" with additional insights from university physics curriculum

House of Physics | Contact: aliphy2008@gmail.com