It is disturbance or vibrating that travel through a medium (like air, water etc.) carrying energy from 1 point to another without transfer of energy.

Generally a waves require medium for propagation but not all wave.

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Mechanical waves- Those waves which require medium for propagate. ex- sound waves require air and water waves water.

Non-mechanical waves- Those waves which does not require any medium for propagate(or travel) and can travel through through vacuum. ex- light waves and electromagnetic waves.

Matter waves- They associated with particles of matter that showing wave-like behavior due to their motion. This concept is based on quantum mechanics, where particles like electrons or protons can behave like waves. It is also called de Broglie waves.

Some scientist who studied about waves are Christiaan Huygens(1629-1695), Isaac Newton and Robert Hooke.

Pulse- It is a single, short disturbance that travel through a medium.

TRANSVERSE AND LONGITUDINAL WAVES-

  • Longitudinal waves- The particles of the medium move back and forth in same direction as the waves is traveling. ex- sound waves.
  • Transverse waves- The particle of the medium move up and down at a right angle to the direction the waves is traveling. ex- water waves, sound waves.
  • In both waves, no any material of medium move.
  • The waves on surface of liquid (like water) is of 2 types-
    • Capillary waves- It is small rippling waves on a liquid ‘s surface covered by surface tension.
    • Gravity waves- It is surface waves which restore by gravity. The range of wavelength of gravity waves is 1 meter to many meters.
  • The ocean waves is both longitudinal and transverse.
  • Also longitudinal and transverse waves travel different speeds in same medium.
  • Also the waves of ocean because the gravity of moon which pull water toward moon.

DISPLACEMENT RELATION IN PROGRESSIVE WAVES-

  • A sinusoidal wave is a smooth, repetitive wave that follows the sine function (y = sin ⁡x). It has:
    • Amplitude: Height of the wave.
    • Frequency: How often it repeats in a second.
    • Wavelength: Distance of one full wave.
    • Time Period: Time to complete one wave.
  • It is mostly used as signal waves.
  • The displacement relation for a progressive wave is:-
    • y(x,t) = A sin(kx − ωt + ϕ)
      • Where, y(x,t) is displacement at position x and time t, A is amplitude (maximum displacement), k is wave number (2π/λ), ω is angular frequency (2πf), ϕ is initial phase.
    • It shows that how the wave’s displacement changes with position and time.

AMPLITUDE OF A WAVE-

  • Amplitude- It is the maximum displacement of a particle from its mean (equilibrium) position in a wave.
  • The displacement can be positive and negative
  • It represents the wave’s height and determines the wave’s energy.
  • Higher amplitude means more energy.
  • Amplitude (A) is often given directly as the maximum displacement in wave equations:-
    • y(x,t) = A sin (kx − ωt + ϕ), where A is amplitude.

PHASE OF A WAVE-

  • Phase- It is the position of a particle in its oscillation cycle at a given point and time.
  • It tells whether the particle is at the crest, trough, or in-between.
  • Phase is measured in radians or degrees.
  • The phase of a wave is the argument of the sine or cosine function in the wave equation:-
    • Phase = (kx − ωt + ϕ), where k is Wave number (2π/λ), ω is Angular frequency (2πf), ϕ is Initial phase.

WAVELENGTH OF A WAVE-

  • Wavelength(λ)- It is the minimum distance between 2 points having same phase.
  • It is denoted by λ.
  • It is the length of one complete cycle of the wave and is usually measured in meters (m).
  • The wavelength determines the wave’s frequency and energy.

ANGULAR WAVE NUMBER-

  • Angular wave number (k)– It is the number of wavelengths per unit distance.
  • k = 2π / ​λ, where λ is the wavelength.
  • It is denoted by k.
  • It is SI unit is radians per meter (rad/m) or Rad m-1.

TIME PERIOD OF A WAVE-

  • Time period(T)- Time taken for one complete wave cycle.
  • It SI unit is second(s).
  • T = 1 / v, where v is frequency.

ANGULAR FREQUENCY OF A WAVES-

  • Angular frequency(ω)-It is a rate of change of phase of the wave.
  • ω = 2π / T
  • Its SI unit rad / sec.

FREQUENCY OF A WAVE-

  • Frequency(v)- It is a number of wave cycles per second.
  • v = 1 / T
  • Its is measured in Hertz(Hz).

SPEED OF A WAVES-

  • It is the distance the travel by the waves as per unit time.
  • v = f λ, where v is speed, f is frequency and λ is wavelength.

RANSVERSE WAVE SPEED ON STRETCHED STRING-

  • The speed of mechanical waves is determined by restoring force setup in the medium when it is disturbed and mass density(inertial properties).
  • In case of string, the restoring force provide by tension (T), inertial property by mass density(μ), mass is m and string divided by length l.
  • The dimension of μ is [M1L-1T0].
  • v = C (T / μ)1/2, where C is undetermined constant of dimension analysis
  • Finally, the speed of transverse wave on a stretched string is-
    • v = (T/μ)1/2.
  • The speed v only depend on property of medium(T) and μ (T is a property of stretched string due to external force).
  • It does not depend on wavelength and frequency.

LONGITUDIONAL WAVES ‘s SPEED-

  • Sound waves also longitudinal waves.

1. Speed of Longitudinal Wave (Sound in a Medium):

  • Nature of Wave:
    • Longitudinal waves cause particles in the medium to oscillate in the same direction as the wave propagation.
    • Example: Sound waves propagate as compressions (high pressure) and rarefactions (low pressure).
  • Elastic Property – Bulk Modulus (B):
    • Describes how resistant a medium is to compression.
    • Defined as B=−ΔP/(ΔV/V)​, where:
      • ΔP: Change in pressure
      • ΔV/V: Fractional volume change
    • SI Unit: Pascal (Pa)
  • Inertial Property – Mass Density (ρ):
    • Density of the medium, with dimensions [M1L−3T0].
    • Governs the inertia of the medium.
  • Speed Formula Derivation:
    • From dimensional analysis, V ∝ (B/ρ)1/2​​​​.
    • Exact derivation shows v = (B/ρ)1/2​​.

2. Speed of Sound in Solids and Gases:

  • In Solids:
    • Elastic property: Young’s modulus (Y).
    • Speed formula: v= (Y/ρ)1/2
  • In Gases:
    • Elastic property: Bulk modulus (B) equals the pressure (P) for isothermal changes.
    • Speed formula: vv= (P/ρ)1/2​​.

3. Laplace Correction for Gases:

  • Newton assumed isothermal changes; Laplace corrected it for adiabatic processes:
    • For adiabatic conditions:Bad​= γP, where γ= Cp/Cv (specific heat ratio).
    • Correct speed formula: v= (γP​​/ρ)1/2.

4. Comparison of Speed in Different Media:

  • Solids > Liquids > Gases:
    • Solids: High bulk modulus dominates despite higher density.
    • Gases: Lower density but much lower modulus reduces speed.

5. Principle of Superposition of Waves:

  • Definition: When two or more waves overlap, the resultant displacement is the algebraic sum of individual displacements:
    •   y(x,t)= y1​(x,t)+y2​(x,t).
  • Interference:
    • Constructive: Amplitudes add up (in phase).
    • Destructive: Amplitudes cancel out (out of phase).

6. Reflection of Waves:

  • At rigid boundaries:
    • Wave reflects with a phase change of π\piπ (inverted).
  • At free boundaries:
    • Wave reflects with no phase change.

7. Standing Waves and Natural Modes:

  • Formed by the superposition of two waves traveling in opposite directions.
  • Equation: y(x,t)=2a sin⁡(kx)cos⁡(ωt)
    • Nodes: Zero amplitude.
    • Antinodes: Maximum amplitude.
  • Normal Modes:
    • Wavelengths: λ= 2L/n​, n= 1,2,3…
    • Frequencies: νn​= nv​/2L.
  • Fundamental frequency (n=1): First harmonic.
  • Higher harmonics correspond to n=2,3,… .

8. Applications:

  • Musical Instruments: Utilize principles of superposition and harmonics.
  • Speed in Air Estimation:
    • Example: At STP (γ= 7/5, P= 101325 Pa, ρ= 1.29 kg/m3), speed ≈ 331 m/s.

Extra Knowledge for Competitive Questions:

  1. Dimensional Analysis Tips:
    • Speed of wave: [M0L1T−1]
    • Bulk modulus: [M1L−1T-2]
    • Density: [M1L−3T0]
  2. Factors Affecting Wave Speed:
    • Medium properties (density, elasticity).
    • For gases, temperature influences speed: V∝(T)1/2​.
  3. Conceptual Pointers:
    • Rigid boundary reflection = π phase shift.
    • Free boundary reflection = no phase shift.
    • Nodes and antinodes: Key to standing wave patterns.
  4. Harmonics in Real Systems:
    • Plucking a string closer to an end emphasizes higher harmonics.
    • Pipe organs: Resonance depends on boundary conditions (open vs closed).
  5. Critical Equations to Remember:
    • Speed in solids: v= (Y/ρ)1/2
    • Speed in ideal gases (adiabatic): v=(ργ/P)1/2​​.

1. What Are Beats?

  • Definition: Beats occur when two sound waves of nearly equal frequencies interfere.
  • What We Hear:
    • A single sound of average frequency (mean of the two frequencies).
    • Variations in intensity (louder and softer sounds), called waxing and waning.
  • Beat Frequency: The frequency of intensity changes equals the difference between the two wave frequencies. νbeat​= ν1​−ν2​

2. Illustration of Beats

  • Example:
    • Two waves with frequencies 11 Hz and 9 Hz produce beats at νbeat​=2Hz.
  • Explanation:
    • The sound alternates between loud and soft twice per second (waxing and waning).

3. Application in Music

  • Musicians and Artists Use Beats:
    • Tuning instruments.
    • Identifying harmonic inconsistencies.
  • Practical Example:
    • Adjusting the tension in a string changes its frequency, helping tune the instrument.

4. Concept of Musical Pillars

  • Unique Feature: Certain Indian temple pillars produce musical notes when tapped.
  • Why?
    • Vibrations depend on:
      1. Material elasticity: Ability of the stone to deform and return.
      2. Density: Mass per unit volume of the stone.
      3. Shape: Carving and dimensions of the pillar.
  • Types of Musical Pillars:
    1. Shruti Pillars: Produce basic musical notes (Sa, Re, Ga…).
    2. Gana Thoongal: Create raga-based tunes.
    3. Laya Thoongal: Generate rhythmic beats.

Example Temples:

  • Nellaiappar Temple, Tamil Nadu.
  • Temples at Hampi and Kanyakumari.

5. Physics of Waves in Brief

  • Mechanical Waves: Require a medium (like air or water) for propagation.
  • Types of Waves:
    1. Transverse Waves: Oscillations are perpendicular to wave direction (e.g., light waves).
    2. Longitudinal Waves: Oscillations are along the wave direction (e.g., sound waves).

6. Key Wave Formulas

  • General Wave Equation: y(x,t)=asin⁡(kx−ωt+ϕ)
    • a: Amplitude.
    • k: Wave number (2π/λ).
    • ω: Angular frequency (2πν).
    • ϕ: Phase constant.
  • Wave Properties:
    • Wavelength (λ): Distance between identical points in the wave.
    • Frequency (ν): Oscillations per second.
    • Period (T): Time for one oscillation (T=1/ν).
    • Wave Speed (v): v=νλ

7. Beats in Competitive Questions

  • How Beats Help Solve Problems:
    • Given: Frequencies of two waves (ν1​ and ν2).
    • Find: Beat frequency (νbeat)
    • Use: Beat frequency decreases when frequencies come closer after tuning.

Example Problem:

  • Frequencies of two strings are 427 Hz and 422 Hz.
  • Tension in one string is increased.
    • Result: Beat frequency decreases, confirming ν2​<ν1​.

8. Real-Life Insights

  • Sound Waves: Energy travels as compressions and rarefactions; matter itself does not move with the wave.
  • Applications of Beats:
    • Radar systems.
    • Medical imaging (ultrasound).
    • Noise-canceling technology.

9. Points to Remember

  • Wave vs. Matter Movement: Waves transfer energy, not matter.
  • Interference and Beats:
    • Constructive Interference: Waves add up (louder sound).
    • Destructive Interference: Waves cancel out (softer sound).

By understanding these foundational principles and applying related formulas, you can solve complex problems involving waves and beats effectively.

These all are the notes of chapter 14 in physics. And after some time you get important questions and NCERT solutions HERE. *#THANKS FOR VISITING, VISIT AGAIN#* 😊