CBSE Class 11th Chapter Waves

by Abhinav Kapoor
by April,08,2024
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CBSE Class 11th Chapter Waves
Waves: Without any real substance movement, a wave is a type of disturbance that moves across a material medium as a result of the medium's particles moving repeatedly and periodically around their mean positions.

Characteristics of waves:
Wave characteristics include the following:

(i) The medium's particles that a wave travels through vibrate relatively little about their mean locations; nonetheless, the particles do not shift indefinitely in the wave's direction of propagation.
(ii) Along or perpendicular to the wave's path of travel, each subsequent particle in the medium moves in a manner very similar to that of its predecessors.
(iii) Only energy is transferred during wave motion; medium penetration is not affected.

There are three primary categories of waves: (a) elastic or mechanical waves, (b) electromagnetic waves, and (c) matter waves.

Mechanical waves:

Only in a material medium are mechanical waves able to be generated or transmitted. The principles of motion established by Newton apply to these waves. For instance, sound waves, waves on strings, and waves on the surface of water.

Electromagnetic Waves:

These waves can be produced and carried through any material medium, including vacuums, without the need for a material medium. Electromagnetic waves include, but are not limited to, radio waves, microwaves, ultraviolet light, and visible light.

Matter waves:

These waves are connected to moving matter particles such as protons, neutrons, and electrons. There are two kinds of mechanical waves:
(i) motion of transverse waves, (ii) motion of longitudinal waves.

Motion of transverse waves:
The medium's particles vibrate in transverse waves in a direction perpendicular to the wave's propagation direction. Transverse waves include electromagnetic waves, waves on strings, and waves on the surface of water. Electromagnetic waves, which include light waves, are characterized by a traveling disturbance that results from the oscillation of electric and magnetic fields at right angles to the direction of the wave. This disturbance is not caused by particle vibrations.

Motion of longitudinal waves:
Particles in the medium vibrate to and fro about their mean location along the direction of energy propagation in these kinds of waves. Another name for these is pressure waves. Longitudinal mechanical waves make up sound waves.

Wavelength:
The wavelength (λ) of a medium particle is the distance traveled by the disturbance in one vibration. One definition of wavelength in the context of a transverse wave is the separation between two subsequent crests or troughs. The wavelength (λ) of a longitudinal wave is the length of time that separates the centers of two compressions (or refractions).

Wave Velocity:
The time rate at which waves propagate in a given medium is known as the wave velocity. It is not the same as the velocity of particles. Wave velocity is dependent on the medium's composition.
Wavelength (λ) x frequency (v) = wave velocity (υ).

Amplitude:
The amplitude of a wave is the maximum displacement of the particles of the medium from their mean position.

Frequency:
The number of vibrations made by a particle in one second is called Frequency. It is represented by v. Its unit is hertz (Hz) v =1/T.

Time Period:
A time period is the amount of time it takes a particle to complete one vibration.
The formula for T is given in seconds. T = 1/v
The velocity of transverse waves in a stretched string is given by

where μ, commonly known as the string's linear mass density, is the mass per unit length and T is the string's tension. μ has a SI unit of kg m-1.

where E is the modulus of elasticity of the medium and ρ is the density of the medium. In case of solids, if E is Young’s modulus of elasticity (Y), then

Velocity of Sound in Air:
According to Newton, when sound waves travel in air or in a gaseous media, the change is taking place isothermally and hence, it is found that

According to Newton's formula, the speed of sound in air at STP conditions is 280 ms-1. The numbers found by experimentation, however, are 332 ms-1.
According to Laplace, because gases are thermal insulators and because compressions and refractions occur sporadically at high frequencies, changes occur during the transmission of sound waves under adiabatic conditions.

Factors Influencing the Velocity of Sound:
In every gaseous medium, a wide range of parameters, including density, pressure, temperature, humidity, wind speed, and others, can influence the sound velocity.
(i) The square root of the gas's density and the sound's velocity in that gas are inversely related.
(ii) As long as the temperature doesn't change, the sound's velocity is unaffected by changes in gas pressure.
(iii) The square root of a gas's absolute temperature determines the velocity of sound in that gas.
(iv) Sound travels faster via damp air than it does through dry air.
(v) The velocity of sound is v + w cos θ, where w is the wavelength, if the wind is flowing at an angle θ to the direction that sound propagates.

General Equation of Progressive Waves:
A wave is said to be progressive if it moves in a certain direction without attenuating and with a constant amplitude.Since displacement in wave motion is a function of both space and time, the displacement relation can be represented as a combined function of position and time as follows:
y (x,t) = A sin (kx — ωt + Ф)
An alternative to the sine function is the cosine function. The four constants in this case are A, K, ω, and Τ, which stand for the wave's amplitude, angular wave number, angular frequency, and starting phase angle.

Relationship between phase and path difference:

Both a free border and a stiff boundary can reflect a wave motion. A traveling wave has phase reversal in reflection at a stiff barrier or closed end, whereas no phase shift occurs in reflection at an open border.

The Principle of Superposition of Waves:
The net displacement at a given time is the algebraic total of the displacements attributable to each wave at that particular time when any number of waves meet concurrently at a location in a medium.

Standing waves or stationary waves:
A new set of waves is created when two sets of progressive wave trains of the same type (both longitudinal and transverse) that are traveling at the same speed down the same straight line in opposite directions and have the same amplitude, time period, frequency, and wave length superimpose. We refer to these as standing waves or stationary waves.

Progressive Waves:

The disruption spreads further, passing from one particle to another. Every particle vibrates in the same way as the one before it, but at a different time.
The waves have a defined velocity and take the shape of sine/cosine functions, or crests and troughs.
Each particle has the same amplitude, which it reaches at its own pace based on the wave's trajectory.
Every particle has a phase that alternates continuously between 0 and 2π.
No particle is ever completely at rest. The particles briefly rest twice during each vibration. Particles arrive at this place at varying periods.
As the wave moves forward, each particle reaches its maximum velocity at the same time.
Energy flows consistently across all planes in the wave's direction of propagation. A wave has half potential energy and half kinetic energy on average.

Stationary Waves:
1. The wave is neither moving forward or backward, indicating that the disturbance is stable. Every particle has unique vibrational properties.
2. Twice during each vibration, the waves resemble a sine/cosine function that shrinks to a straight line. It never gets farther.
3. The amplitude assigned to each particle is fixed. While some always have the maximum amplitude (antodes), others always have zero amplitude (nodes). Every participant consumes this at the exact same time.
4. In one half of the wave, every particle has a fixed phase, whereas in the other half, every particle concurrently has the same phase in the opposite direction.
5. All other particles have a maximum displacement that they all reach simultaneously, except for those particles (called nodes) that are always at rest. Twice throughout each vibration, these particles are momentarily at rest simultaneously.
6. Every particle instantaneously achieves its unique velocity assigned to it based on its position. In a single wave form, two particles, or nodes, always have zero velocities.
7. Energy does not move in any direction on any plane. Every particle has a specific amount of energy assigned to it. At some point, they all reach their RE values, and at some point, all energy turns into KB.

In the simplest mode of vibration, nodes form at the fixed ends and an antinode forms at the center point when a stationary wave is set up in a string of length l that is fixed at both ends. The basic mode of vibration's (or first harmonic's) frequency is provided by

The Law of Length:
The fundamental frequency v is inversely proportional to,the length L of the stretched string.

The Law of Mass:
The fundamental frequency is inversely proportional to the square root of mass per unit length of the given string when L and T are kept constants.

Beats: Beats are the regular increase and fall in sound intensity caused by the superposition of two approximately equal-frequency waves traveling in the same direction and along the same path.
A beat is defined as a peak and fall in sound intensity, while beat frequency is the number of beats per second. Given v1 and v2, the frequencies of the two interfering waves, with v1 being bigger than v2, are expressed as follows: vb = (v1-v2).

Doppler Effect:
The apparent frequencies of sound heard by the listener differ from the actual frequencies of sound released by the source anytime there is a relative motion between the source and the listener, according to Doppler's effect.

Here v = the true frequency of the wave emitted by the source, v = the speed of sound through the medium, v0 is the velocity of the observer relative to the medium, and vs is the velocity of the source relative to the medium. In using this formula, velocities in the direction OS (i.e., from the observer towards the source) are treated as positive, and those opposite it are taken as negative.

FAQ-
Q.1 What are waves?
Ans. Waves are disturbances that transfer energy without transferring matter. They propagate through a medium or space, carrying energy from one place to another.
Q.2 What are the different types of waves?
Ans. There are several types of waves, including mechanical waves (such as sound waves and seismic waves) that require a medium to travel, and electromagnetic waves (such as light waves and radio waves) that can travel through a vacuum.
Q.3 How are waves classified?
Ans. Waves can be classified based on various characteristics, including their medium of propagation, direction of particle displacement, and frequency. Common classifications include transverse waves, longitudinal waves, and surface waves.
Q.4 What is the difference between transverse and longitudinal waves?
Ans. Transverse waves are characterized by particle displacement perpendicular to the direction of wave propagation, while longitudinal waves involve particle displacement parallel to the direction of wave propagation.
Q.5 What is the wave frequency?
Ans. Frequency refers to the number of waves passing through a given point per unit of time. It is measured in hertz (Hz), where one hertz equals one wave per second.
Q.6 How is wave speed calculated?
Ans. Wave speed (v) is calculated by multiplying the wavelength (λ) of the wave by its frequency (f). Mathematically, it can be expressed as v = λ * f.