WAVES – NOTES
1. Introduction
• Wave: A disturbance that propagates energy without transporting matter.
• Types of Waves:
o Mechanical Waves – Require a medium (e.g., sound, water waves).
o Electromagnetic Waves – Do not require a medium (e.g., light, radio waves).
2. Types of Mechanical Waves
1. Transverse Waves:
a. Particles oscillate perpendicular to wave propagation.
b. Example: Light waves, water waves.
c. Crest (high point) & Trough (low point).
d. Formula: v=fλv = f \lambdav=fλ Where:
i. vvv = Wave speed
ii. fff = Frequency
iii. λ\lambdaλ = Wavelength
2. Longitudinal Waves:
a. Particles oscillate parallel to wave propagation.
b. Example: Sound waves.
c. Compression (high pressure) & Rarefaction (low pressure).
3. Characteristics of a Wave
• Wavelength (λ\lambdaλ) – Distance between two consecutive crests or
compressions.
• Frequency (ff f) – Number of oscillations per second (Unit: Hz).
• Time Period (TTT) – Time for one complete oscillation. T=1fT = \frac{1}{f}T=f1
• Wave Speed (vvv) – Speed at which the wave propagates. v=fλv = f \lambdav=fλ
4. Sound Waves
• Speed of Sound:
o Fastest in solids, slower in liquids, slowest in gases.
o In air at 25°C: 343 m/s.
• Factors Affecting Speed of Sound:
o Increases with temperature, humidity.
o Decreases with density in gases.
Newton’s Formula (Speed of Sound in Gas)
v=Pρv = \sqrt{\frac{P}{\rho}}v=ρP
Laplace’s Correction:
v=γPρv = \sqrt{\gamma \frac{P}{\rho}}v=γρP
Where γ\gammaγ = Adiabatic index.
5. Reflection & Refraction of Waves
• Reflection: Wave bounces back when it hits a barrier.
• Refraction: Wave changes direction and speed when moving to a different medium.
• Laws of Reflection:
o Angle of incidence = Angle of reflection.
6. Principle of Superposition
• When two or more waves meet, the resultant displacement is the sum of individual
displacements.
7. Standing Waves (Stationary Waves)
• Formed by interference of two identical waves traveling in opposite directions.
• No energy transfer.
• Nodes: Points of zero displacement.
• Antinodes: Points of maximum displacement.
Standing Wave in a String (Fixed Ends)
λn=2Ln,fn=nv2L\lambda_n = \frac{2L}{n}, \quad f_n = \frac{n v}{2L}λn =n2L ,fn =2Lnv
Where nnn = 1, 2, 3… (harmonics).
Standing Wave in an Open Pipe
λn=2Ln,fn=nv2L\lambda_n = \frac{2L}{n}, \quad f_n = \frac{n v}{2L}λn =n2L ,fn =2Lnv
Standing Wave in a Closed Pipe
λn=4Ln,fn=nv4L\lambda_n = \frac{4L}{n}, \quad f_n = \frac{n v}{4L}λn =n4L ,fn =4Lnv
(Only odd harmonics exist).
8. Doppler Effect
• Change in frequency due to motion of source or observer.
• Formula: f′=fv±vov∓vsf' = f \frac{v \pm v_o}{v \mp v_s}f′=fv∓vs v±vo Where:
o f′f'f′ = Apparent frequency
o fff = Actual frequency
o vvv = Speed of sound
o vov_ovo = Speed of observer
o vsv_svs = Speed of source
Cases:
• Source moving towards observer → Frequency increases.
• Source moving away from observer → Frequency decreases.
9. Beats
• Definition: When two waves of nearly equal frequency interfere, they produce
periodic variations in loudness called beats.
• Formula: fbeat=∣f1−f2∣f_{\text{beat}} = |f_1 - f_2|fbeat =∣f1 −f2 ∣
• Example: Tuning musical instruments using beats.
