Waves Class 11 Notes: CBSE Physics Chapter 14

Waves: Waves are a fundamental concept in physics, describing how energy and information propagate through various mediums. A wave is a disturbance that travels through space and matter, transferring energy from one point to another without causing any permanent displacement of the medium. Waves can be classified into different types based on their nature, such as mechanical waves, which require a medium to travel (like sound waves), and electromagnetic waves, which do not require a medium (like light waves). Understanding the behaviour of waves, including their speed, frequency, wavelength, and amplitude, is essential for studying a wide range of phenomena in both classical and modern physics

Types of Waves: 

1. Mechanical Waves:

• Definition: Waves that require a medium (solid, liquid, or gas) to propagate

• Examples: Sound waves, water waves, seismic waves. 

Subtypes: 

• Transverse Waves: The particles of the medium move perpendicular to the direction of wave propagation.

• Example: Waves on a string, electromagnetic waves (when considering them as a wave model). 

• Longitudinal Waves: The particles of the medium move parallel to the direction of wave propagation.

• Example: Sound waves in air, compressional waves in a slinky.

2. Electromagnetic Waves: 

• Definition: Waves that do not require a medium and can propagate through a vacuum.

• Examples: Light waves, radio waves, X-rays, microwaves. 

• Characteristics: Consists of oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation. 

3. Surface Waves:

• Definition: Waves that travel along the surface of a medium, with characteristics of both transverse and longitudinal waves.

• Examples: Ocean waves, and ripples on water.

• Motion: Particles move in a circular or elliptical path, combining both perpendicular and parallel motion to the direction of wave travel.

Matter Waves:

• Definition: Waves associated with particles of matter, as described by quantum mechanics

• Examples: De Broglie waves (electron waves).

• Characteristics: Reflect the wave-particle duality of matter, where particles exhibit wave-like properties under certain conditions.

5. Standing Waves:

• Definition: Waves that remain in a constant position, typically formed by the interference of two waves travelling in opposite directions.

• Examples: Vibrations in a guitar string, sound waves in a pipe.

• Characteristics: Consist of nodes (points of no displacement) and antinodes (points of maximum displacement).

6. Progressive Waves:

• Definition: Waves that move or propagate through a medium, transferring energy from one point to another.

• Examples: Light waves, and sound waves.

• Characteristics: Can be either transverse or longitudinal, and continue to move through the medium without being confined.

TRANSVERSE AND LONGITUDINAL WAVES:

Transverse and Longitudinal Waves: Detailed Explanation and Differences

1. Transverse Wave

Definition:

In transverse waves, the particles of the medium move perpendicular to the direction of wave propagation. This means that if the wave is moving horizontally, the particles of the medium are oscillating vertically.

Characteristics:

• Crests and Troughs: The highest points of the wave are called crests, and the lowest points are called troughs.

• Perpendicular Motion: The particle displacement is perpendicular to the direction of energy transfer.

• Examples: Light waves, waves on a string, electromagnetic waves, and surface waves on water.

Mathematical Representation:

The displacement of particles in a transverse wave can be described by the equation

$y(x, t) = A \sin(kx - \omega t + \phi)$

Where,

y(x,t) is the displacement at position x and time t.

A is the amplitude of the wave.

k is the wave number ($k = \frac{2\pi}{\lambda}$​).

$\omega$ is the angular frequency ($\omega = 2\pi f$).

$\ phi$ is the phase constant.

2. Longitudinal Waves:

Definition: In longitudinal waves, the particles of the medium move parallel to the direction of wave propagation. Here, the particles oscillate back and forth in the same direction as the wave.

Characteristics:

• Compressions and Rarefactions: Areas where particles are close together are called  compressions, and areas where they are spread out are called rarefactions.

• Parallel Motion: The particle displacement is parallel to the direction of energy transfer.

• Examples: Sound waves in air, waves in a slinky, and seismic P-waves.

Mathematical Representation:

The displacement of particles in a longitudinal wave can be described by a similar sinusoidal function:

s(x,t)=$s_o$​ cos(kx−ωt+ϕ)

Here’s a breakdown of the terms:

• s(x,t) represents the displacement of the wave at position x and time t.

• $ s_0$ is the amplitude of the wave (maximum displacement).

• k is the wave number, related to the wavelength $\lambda$ by $k = \frac{2\pi}{\lambda}$

• $\omega$ is the angular frequency, related to the frequency f by $ω=2πf\omega = 2\pi f$f.

• t is time.

• x is the position along the direction of the wave propagation.

• $\phi$ is the phase constant, which determines the initial phase of the wave.

Key Differences Between Transverse and Longitudinal Waves

Displacement Relation in a Progressive Wave:

A progressive wave is a wave that travels or propagates through a medium, transferring energy from one point to another without transferring matter. The displacement of particles in the medium due to the wave can be described by a mathematical relation known as the displacement equation.

1. General Form of the Displacement Equation:

For a sinusoidal wave travelling in the positive x-direction, the displacement y (x ,t ) at a point x at time t is given by: 

$y(x, t) = A \sin(kx - \omega t + \phi)$

Where,

y(x,t) is the displacement at position x and time t.

A is the amplitude of the wave.

k is the wave number ($k = \frac{2\pi}{\lambda}$​).

$\omega$ is the angular frequency ($\omega = 2\pi f$).

$\ phi$ is the phase constant.

2. Explanation of the Terms: 

• Amplitude A: The amplitude is the maximum displacement of the particles from their equilibrium position. It represents the height of the wave crest or the depth of the trough.

• Wave Number k: - The wave number k gives the number of wavelengths per unit distance. It is related to the wavelength  $\lambda$ by the relation $k = \frac{2\pi}{\lambda}$​. 

• Angular Frequency $\omega$:- The angular frequency $\omega$ is related to the time period T and the frequency f of the wave. It represents how rapidly the wave oscillates in time.

• Phase$(kx - \omega t + \phi)$: The phase of the wave indicates the state of the oscillation at a particular position and time. It determines the position of the wave within its cycle. The term kx - $\omega$ t shows the wave’s progression in space and time. The phase constant shifts the wave along the x-axis.

Amplitude and Phase in Waves

Amplitude and Phase are two fundamental concepts that describe the characteristics of 

waves, including their behaviour and how they interact with each other.

1. Amplitude

Definition: The amplitude of a wave is the maximum displacement of particles in the medium from their equilibrium position due to the passage of the wave. It is a measure of the wave's intensity or strength.

Representation:

In the displacement equation of a sinusoidal wave:

$y(x, t) = A \sin(kx - \omega t + \phi)$

Here, A represents the amplitude of the wave.

• Key Points: Magnitude: Amplitude is always a positive quantity and is measured in units of displacement (e.g., meters in mechanical waves). 

• Effect on Energy: The energy carried by a wave is proportional to the square of its amplitude. A wave with a larger amplitude carries more energy.
  Physical Interpretation: For a sound wave, the amplitude determines the loudness; for a light wave, it determines the brightness. 

• Example: In a transverse wave on a string, the amplitude is the maximum height of the wave crest or the depth of the trough relative to the equilibrium position.

2. Phase 

• Definition: The phase of a wave refers to the position of a point within the wave cycle. It indicates the state of oscillation of the wave at a specific point in space and time.

• Phase Angle: The phase of a wave is usually expressed in terms of a phase angle, which is measured in radians or degrees.

In the Displacement Equation

$y(x, t) = A \sin(kx - \omega t + \phi)$

• Wavelength ($\lambda$): The distance between two successive peaks (high points) or troughs (low points) of the wave.

• Frequency (f): The number of waves that pass a given point in one second, measured in Hertz (Hz).

• Unit: Frequency is measured in hertz (Hz), where 1 Hz equals one cycle per second. - Relation to Period: Frequency is the reciprocal of the period:

• Amplitude (A): The maximum height of the wave from its equilibrium position, related to the wave's energy.

• Wave Speed (v): How fast the wave moves through the medium, calculated by the Formula: $v = f \lambda$

Reflection of Waves:

Reflection of waves is a fundamental phenomenon that occurs when a wave encounters a boundary or an obstacle and is reflected back into the original medium instead of being transmitted through or absorbed by the boundary. This principle applies to various types of waves, including light, sound, water, and seismic waves

Key Concepts in Wave Reflection:

1. Phase Change on Reflection:

• Fixed-End Reflection: When a wave reflects off a fixed boundary, it inverts, meaning a crest reflects as a trough and vice versa, resulting in a 180-degree phase change

• Free-End Reflection: When a wave reflects off a free boundary, it does not invert, and there is no phase change.

2. Superposition of Waves:

When incident and reflected waves overlap, they superpose, which can result in constructive or destructive interference, depending on their phases.

Principles of Superposition

• Interference: When two or more waves overlap, their effects combine. This can lead to:

• Constructive Interference: When waves add up to make a larger wave.

• Destructive Interference: When waves cancel each other out.

3. Standing Waves:

When two waves of the same frequency and amplitude travel in opposite directions, they can interfere to form standing waves. These are characterised by nodes (points of no displacement) and antinodes (points of maximum displacement).

Speed of a Transverse Wave on a Stretched String

The speed of a transverse wave travelling along a stretched string depends on the physical properties of the string, particularly its tension and mass per unit length. This relationship is crucial in understanding wave motion in mediums like strings, ropes, and even some musical instruments.

4. Sound Waves

• Properties of Sound Waves: Sound waves are longitudinal waves that travel through air, water, and solids.

• Speed of Sound: Depends on the medium. For example, sound travels faster in water than in air. The speed of sound in air is approximately 343 m/s.

• Doppler Effect: The change in frequency or wavelength of a wave in relation to an observer moving relative to the wave source. Example: The pitch of an ambulance siren changes as it moves past you.

5. Quantum Mechanics: 

The principle of superposition is also fundamental in quantum mechanics, where the wavefunctions of particles superpose to determine the probability distribution of a particle's position or momentum.

Standing Waves and Normal Modes 

Standing waves and normal modes are fundamental concepts in wave physics that describe specific patterns of wave behaviour when waves reflect and interfere within a confined medium. These concepts are critical in understanding phenomena in musical instruments, resonant cavities, and various physical systems. 

Standing Waves: 

Definition: 

A standing wave is a wave pattern that results from the interference of two waves of the same frequency and amplitude travelling in opposite directions. Unlike travelling waves, standing waves do not move through the medium; instead, they create stationary patterns of constructive and destructive interference.

Formation:

Standing waves are formed when a wave reflects on itself after encountering a boundary, such as the end of a string or the walls of a cavity. The reflected wave interferes with the incoming wave, leading to a pattern of nodes and antinodes.

Beats 

Beats are a phenomenon that occurs when two sound waves of slightly different frequencies interfere with each other. The result is a new sound wave that fluctuates in amplitude, creating a pulsing effect that is heard as periodic variations in loudness. This is commonly referred to as "beats."

Explanation of Beats:

1. Interference of Waves: - When two sound waves of frequencies $f_1$ and $ f_2$ (where  $f_1$   is slightly different from $ f_2$ ) are played together, they interfere with each other. - The interference can be constructive (when the waves are in phase) or destructive (when the waves are out of phase).

2. Formation of Beats:

• Constructive Interference: Occurs when the crests and troughs of the two waves align, resulting in a louder sound (maximum amplitude). 

• Destructive Interference: Occurs when the crest of one wave aligns with the trough of the other, resulting in a softer sound (minimum amplitude). 

• This alternating pattern of constructive and destructive interference causes the sound to alternately increase and decrease in volume, producing the best effect.

3. Beat Frequency:

The beat frequency is the rate at which the amplitude of the sound wave fluctuates, or the number of beats heard per second. 

Applications of Beats: 

1. Tuning Musical Instruments: Musicians often use beats to tune instruments. By playing a reference note and adjusting the instrument until the beats slow down and disappear, the instrument can be tuned to match the reference frequency.

2. Sound Engineering: Beats are used in sound engineering to create effects or to measure differences in frequencies between two sound sources. 

3. Doppler Effect Analysis: Beats can be observed in scenarios involving the Doppler effect, where the frequency of sound changes due to relative motion between the source and the observer

Waves Class 11 Notes Physics - Basic Subjective Questions

Section – A (1 Mark Questions)

1. Why can the transverse waves not be produced in air?

Ans. For air, the modulus of rigidity is zero or it does not possess property of possession. Therefore, transverse Waves cannot be produced.

2.  Why are all stringed instruments provided with hollow boxes?

Ans. The stringed instruments are provided with a hollow box called sound box. When the strings are set into vibration, forced vibrations are produced in the sound box. Since sound box has a large area, it sets a large volume of air into vibration. This produces a loud sound of the same frequency of that of the string.

3.  Can beats be produced in two light sources of nearly equal frequencies?

Ans. No, because the emission of light is a random and rapid phenomenon and instead of beats, we get uniform intensity.

4. A stationary boat is rocked by waves whose crests are 100m apart velocity is 25m/s. Find the time after which the boat bounces up every time.

Ans. $v=n\lambda$

$v=\dfrac{\lambda }{T}\Rightarrow T=\dfrac{\lambda }{v}$

$v=\dfrac{100}{25}=4s$

5. A long string having mass density as 0.01kg/m is subjected to a tension of 64N. Then find the speed of the transverse wave on the string.

Ans. $v=\sqrt{\dfrac{T}{\mu }}=\sqrt{\dfrac{60}{0\cdot 01}}$

$v=\sqrt{6400}=80m/s$

Section – B (2 Marks Questions)

6. Three harmonic waves of same frequency (f) and intensity $I_{0}$ having initial phase angles $0,\dfrac{\pi }{4},\dfrac{\pi }{4}$ rad respectively. When they are superimposed, find the resultant intensity.

Ans. Amplitude can be added using vector addition

A_{resultant}=(\sqrt{2}+1)A

Since $1\varpropto A^{2}$ ,Where I is intensity 

Therefore, $I_{res}=\left ( \sqrt{2}+1 \right )^{2}I_{0}=5\cdot 8I_{0}(Approx)$

7. Guitar strings X and Y striking the note ‘Ga’ are a little out of tune and give beats at 6 Hz. When the string X is slightly loosened and the beat frequency becomes 3 Hz. Given that the original frequency of X is 324 Hz, find the frequency of Y.

Ans. Given,

Frequency of X, $f_{x}=324Hz$

Frequency of Y = fy

Beat’s frequency, n = 6 Hz

Also,

$n=\left | f_{x}\pm f_{y} \right |$

$6=324\pm f_{y}$

$\Rightarrow f_{y}=330\;Hz$ or $318\;Hz$

As frequency drops with a decrease in tension in the string, thus $f_{y}$ cannot be 330 Hz

$\Rightarrow f_{y}=318\;Hz$

8. A narrow sound pulse (for example, a short pip by a whistle) is sent across a medium.

(a) Does the pulse have a definite (i) frequency, (ii) wavelength, (iii) speed of propagation?

(b) If the pulse rate is 1 after every 20 s, (that is the whistle is blown for a split of second after every 20 s), is the frequency of the note produces by the whistle equal to 1/20 or 0.05 Hz?

Ans.

(a) The speed of propagation is definite; it is equal to the speed of the sound in air. The wavelength and frequency will not be definite.

(b) The frequency of the note produced by a whistle is not 1/20 = 0.05 Hz. However, 0.05 Hz is the frequency of repetition of the short pip of the whistle.

9. A transverse wave passes through a string with the equation y = 10 sin $\pi$ (0.02 x - 2.00t) where x is in meter and t in second. Then find the maximum velocity of the particle in wave motion.

Ans. $y=10sin\pi \left ( 0\cdot 02x-2\cdot 00t \right )$

In order to derive maximum velocity,

$\dfrac{dy}{dt}=-20\pi cos\pi \left ( 0\cdot 02x-2\cdot 00t \right )$

$v=-20\pi cos\pi \left ( 0\cdot 02x-2\cdot 00t \right )$

$v_{max}=20\pi$

$v_{max}=63ms^{-1}$

10. Two tuning forks P and Q when set vibrating, given 4 beats/s. If a prong of the fork P is filed, the beats are reduced to 2 beats/s. What is frequency of P, if that of Q is 250 Hz ?

Ans. If fork P is filed then frequency increases, increased frequency is given by

${P}'-Q=2$

${P}'-250=2$

${P}'=252Hz$

Therefore, initial frequency of P is lower than 252Hz and at that time 4 beats are heard

Q-P=4

250-P=4

P=246Hz

Important formula in Class 11 Physics Chapter 14 Waves

1. Wave Speed: $v = f \lambda$

2. Frequency and Wavelength Relation: $\lambda = \frac{v}{f}$

3. Period of a Wave: The time taken for one complete cycle of the wave.
$T = \frac{1}{f}$​

Important Topics of Class 11 Physics Chapter 14 Waves 

Importance of Class 11 Physics Waves Notes

• Waves are an essential part of Physics and understanding them helps in many areas such as sound, light, and even quantum mechanics.

• These notes break down complex concepts into simple, easy-to-understand explanations.

• They help in reinforcing the fundamental principles of waves, which are crucial for solving problems and performing well in exams.

• The notes provide key formulas and examples that simplify the application of concepts to real-life scenarios.

• Studying these notes gives a clear understanding of the types of waves, their behavior, and their impact on the world around us.

Tips for Learning the Class 11 Chapter 14 Physics Waves

• Start by understanding the basic types of waves, both transverse and longitudinal, and how they differ.

• Practice the key formulas related to wave speed, frequency, and wavelength, as these are crucial for solving problems.

• Use visual aids like graphs and diagrams to see how wave displacement changes with time and position.

• Work through as many practice problems as possible to reinforce your understanding of wave behavior.

• Relate the concepts to everyday examples like sound waves and light waves to get a better grasp of their applications.

Conclusion 

Class 11 Chapter 14 on Waves is an important topic that introduces key concepts about wave motion and its applications. By studying the different types of waves, their properties, and behaviors, you build a strong foundation in physics. These notes help simplify the subject by providing clear explanations, formulas, and practical examples. Consistent practice and regular review will ensure a solid understanding of the chapter, which is essential for both exams and real-world applications of wave theory.

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