# Waves Class 11 Notes Physics Chapter 15

→ A wave is a form of disturbance that transmits energy from one place to another without the actual flow of matter as a whole.

→ Waves are of three types:

- Mechanical waves,
- e.m. waves,
- matter waves.

→ Water waves or sound waves are called mechanical or elastic waves as they require a material medium for their propagation.

→ A material medium possesses both elasticities as well as inertia.

→ Light waves don’t require any material medium for their propagation.

→ Light waves are electromagnetic waves or non-mechanical waves which can propagate through a vacuum.

→ Matter waves are associated with moving electrons, protons, neutrons and other fundamental particles and even atoms and molecules.

→ The matter is constituted by electrons, protons, neutrons and other fundamental particles.

→ The waves associated with matter particles are called matter waves.

→ Matter waves arise in the quantum mechanical description of nature.

→ Wave motion is a form of disturbance that is due to the repeated periodic vibrations of the particles of the medium about their mean positions.

→ The motion is handed over from one medium particle to another without any net transport of the medium during wave motion.

→ Mechanical waves are of two types

- transverse waves and
- longitudinal waves.

→ A wave is said to be a progressive or travelling wave if it travels from one point of the medium to another.

→ The waves on the surface of the water are of two types: capillary waves and gravity waves.

→ The restoring force that produces capillary waves is the surface tension of water.

→ The restoring force that produces gravity waves is the pull of gravity which tends to keep the water surface at its lowest level.

→ The oscillations of the particles in gravity waves are not confined to the surface only but extend with diminishing amplitude to the very bottom.

→ The particle motion in water waves involves a complicated motion, they not only move up and down but also back and forth.

→ The waves in an ocean are a combination of both longitudinal and transverse waves.

→ Transverse and longitudinal waves travel at different speeds in the same medium.

→ k is called propagation constant or angular wavenumber.

→ S.I. unit of k is radian (rad) per metre of rad m^{-1}.

→ The speed of transverse waves in a string is determined by two factors:

- Linear mass density i.e. mass per unit length (m),
- Tension (T) in the string.

→ Positions of zero amplitude are called nodes.

→ Positions of maximum amplitude are called antinodes.

→ Nodes and antinodes are separated by \(\frac{λ}{4}\).

→ Two successive nodes or antinodes are separated by \(\frac{λ}{2}\).

→ Audible sound waves have a frequency between 20 Hz to 20,000 Hz.

→ The equation of a simple harmonic wave travelling in the positive X-direction is given by

y = A sin (ωt – kx)

where ω = \(\frac{2 \pi}{\mathrm{T}}\) = 2πv

k = \(\frac{2 \pi}{\lambda}\)

→ The particle velocity in a wave is given by v = \(\frac{\mathrm{dy}}{\mathrm{dt}}\)

→ Wave velocity is given by C = \(\frac{\mathrm{dx}}{\mathrm{dt}}\).

→ A wave travelling in negative x-direction is given by

y = A sin (ωt + kx)

→ The speed of sound does not depend on the frequency or wavelength.

→ Sound waves are mechanical waves that can’t propagate in a vacuum.

→ Sound waves can’t travel in sawdust or dry sand because the medium is not continuous.

→ The damping of sound in wood is much larger as compared to that in metals.

→ The higher the frequency of sound greater is the pitch of the sound.

→ The voice of ladies and children is of higher pitch than that of men.

→ Unit of loudness is bell (B) = log\(\frac{\mathrm{I}}{\mathrm{I}_{0}}\).

→ The sound is reflected and refracted according to the same laws as the light does.

→ The wavelength for ultrasonics is very small, therefore they are not diffracted by the ordinary objects or holes etc.

→ The speed of mechanical waves is determined by the properties of the medium i.e. elasticity and inertia and not by the nature, intensity, amplitude or shape of the wave.

→ The velocity of sound is the largest in hydrogen among the gases.

→ Monosyllabic sound is produced in about 0.2 s.

→ The vibrations of the prongs of a tuning fork are transverse and that of the stem are longitudinal.

→ The point where the stem of the tuning fork is connected to the prongs is an antinode.

→ The ends of the prongs are also antinodes.

→ There is a node between them that is nearer to the stem than the ends of the prongs.

→ The speed of sound in the air is not affected by the changes in pressure.

→ For every 1°C rise in temperature, the speed of sound increases by 0. 61 ms^{-1}.

→ Due to a change in temperature, the wavelength of sound waves is affected.

→ Beats are not audible if the beat frequency is more than 10 Hz.

→ If the prong of a tuning fork is loaded near the stem its frequency increases and when it is filled near the stem, the frequency decreases.

→ The number of beats produced per second is equal to the difference in the frequencies of the superposing notes.

→ In the progressive wave, the crest and troughs or compressions and rarefactions move with the speed of the wave.

→ When there is no relative motion between the source and listener, the Doppler’s effect is not observed.

→ When a source of sound moves, it causes a change in the wavelength of k the sound received by the listener.

→ When the listener moves, it causes a change in the number of waves ( received by the listener.

→ If source and listener move in mutually perpendicular .directions, no Doppler’s effect is observed.

→ Not Doppler’s effect is produced when only the medium moves.

→ A musical sound consists of a quick, regular and periodic succession of compressions and rarefactions without a sudden change in amplitude.

→ Pitch, loudness and quality are the characteristics of musical sound.

→ Pitch depends on frequency, loudness depends on intensity and quality depends on the number and intensity of overtones.

→ Pitch increases with an increase in frequency.

→ The ratio of the frequencies of the two nodes is called the interval between them. e.g. interval between 256 and 512 Hertz is 1: 2.

→ Two nodes are said to be in unison if their frequencies are equal i. e. if the interval between them is 1: 1.

→ Some other common intervals found useful in producing musical founds are as follows: octave (1: 2), major tone (8: 9), minor tone (9: 10), semitone (15: 16).

→ The fundamental note is called the first harmonic.

→ If n, be the fundamental frequency, then 2n_{1}, 3n_{1}, 4n_{1}, …. are respectively called second, third, fourth,…. harmonics respectively.

→ Harmonics are the integral multiples of the fundamental frequency.

→ Overtones are the notes of frequency higher than the fundamental frequency actually produced by the instrument.

→ In the strings, all harmonics are produced.

→ In the open organ pipe, all the harmonics are produced.

→ In the closed organ pipe, only the odd harmonics are produced.

→ In an open organ pipe as well as the string the second harmonics is the first overtone.

→ In the closed organ pipe, the third harmonic is the first overtone.

→ The ratio of the frequencies of the overtones in an open organ pipe is 2: 3: 4: 5:…

→ The ratio of the frequency of the overtones in the closed organ pipe is 3: 5: 7: …..

→ The frequency of the notes produced by the organ pipe varies:

- directly as \(\sqrt{λ}\) , where λ, is a constant.
- ∝ \(\sqrt{T}\) where T is the absolute temperature of the gas.
- inversely as \(\sqrt{ρ}\) where ρ is the density of the gas.
- inversely as length (l) of the tube.

→ The sound produced by the open organ pipe is comparatively pleasant as compared to that produced by the closed organ pipe.

→ The rarefactions are the regions of decrease in density or pressure and compressions are the regions of increase in density or pressure in air, gas when wave propagates through it.

→ Two waves travelling along the same path in the same or opposite direction superpose.

→ Superposition of waves gives rise to the phenomenon of interference, stationary waves and beats.

→ Interference of waves: Superposition of two waves of the same frequency and same wavelength travelling in the same direction with the same speed results in interference of waves.

→ Constructive interference: Interference is said to be constructive if two waves of the same frequency travelling in the same direction with the same speed superpose on each other such that the resultant displacement is more than the individual displacements.

→ Destructive interference: If the resultant displacement due to the superposition of two waves is less than the individual displacements then it is called destructive interference.

→ The wavelength of a wave: It is defined as the distance between two consecutive points (i.e. two consecutive troughs or crests) in the same phase of wave motion.

→ The fundamental mode of the first harmonic: It is defined as the oscillation mode with the lowest frequency.

→ Infrasonics: Sound waves of frequency less than 20Hz are called infrasonics. They can’t be heard by the human ear.

→ Beats: They are defined as the periodic variations in the intensity of sound due to the superposition of two sound waves of slightly different frequencies.

→ Mechanical or elastic waves: The waves set up and propagated due to the presence of a material medium and its properties of elasticity and inertial are called mechanical waves.

→ Electromagnetic waves: They are defined as the waves set up by the variation in electric and magnetic fields of an oscillating charge.

→ Transverse wave: It is defined as the wave motion set up due to vibrations of medium particles perpendicular to the direction of propagation of the wave.

→ Longitudinal wave: It is defined as the wave motion set up due to the vibrations of the medium particles along the direction of wave propagation.

→ Phase (Φ): It is defined as the argument of sine or cosine function representing a wave. It is the angular periodic position of a wave.

→ Time period (T): It is defined as the time taken by the medium particles to complete one oscillation.

→ Velocity of wave motion (v): It is defined as the ratio of wavelength to the time period i.e. v = \(\frac{λ}{T}\) = vλ, (∵ v = \(\frac{1}{T}\))

→ Stationary wave: It is defined as the wave due to the superposition of two progressive waves of the same frequency and amplitude but travelling in the opposite directions along the same line. It is also called a standing wave.

→ Harmonics: The wave of frequencies having integral multiples of a fundamental frequency are called harmonics of the fundamental wave including itself.

→ Overtones: They are defined as the waves of frequencies having integral multiples of a fundamental frequency but excluding it.

→ The 2nd harmonics is the first overtone, the third harmonics is 2nd overtone and so on.

→ Taut string: It is defined as a string vibrating in any mode/modes fixed at one end and loaded at the other end.

→ Musical sound: It is defined as a sound having series of harmonic waves following each other rapidly at regular intervals of time without a Sudden change in their amplitude. It produces a pleasant effect on the ear of the listener.

→ Noise: It is defined as a sound having series of harmonic waves following each other at irregular intervals of time with a sudden change in their amplitude. It produces a displeasing effect on the ear of the listener.

→ The intensity of sound at a point (I): It is defined as the amount of energy passing per unit time per unit area held perpendicular to the incident sound waves at that point.

→ Temperature coefficient of the velocity of sound (α): It is defined as the change in velocity of sound per Kelvin change in temperature.

→ Capillary waves: They are defined as the ripples of a fairly short wavelength not more than a few centimetres.

→ Gravity waves: They are defined as waves that have wavelengths typically ranging from several metres to several hundred metres.

→ Superposition Principle: It states that when two or more waves of the same nature travel in a medium, then the resultant displacement at a point is the vector sum of the displacement due to the individual waves.

→ Threshold of hearing or zero levels (I_{0}): It is defined as the lowest intensity of sound that can be heard by the human ear. It is about 10^{-12} Wm^{-2} for a sound of frequency I KHz.

→ The loudness of a sound: It ¡s defined as the degree of sensation of sound produced ¡n the car. It distinguishes between a loud and a faint sound.

→ Weber Fechner Law : It states that the loudness of sound is proportional to the logarithm of its intensity i.e. L = log_{10} \(\left(\frac{\mathrm{I}}{\mathrm{I}_{0}}\right)\)

→ Bel (B): Loudness is said to be one bel if the intensity of sound is 10 times the threshold of hearing.

→ Pitch: It ¡s defined as that characteristic of musical sound which helps one to classify a note as a high note or low note.

→ Quality or Timber: Ills defined as that characteristic of musical sound which helps us to distinguish between.two sounds of the same intensity and pitch.

→ Musical scale: It consists of a series of flotes (frequencies) separated by definite and simple intervals so as to produce a musical effect when played in Succession.

→ Decibel (dB): \(\frac{1}{10}\)th of bel is called decible i.e. 1 dB = \(\frac{1}{10}\)B.

→ Keynote: The first note of the lowest frequency is called keynote. Octave: Two notes are said to be octave if the ratio of their frequencies is 2. It is also a musical scale called the diatomic scale which has 8 intervals (octave + 7 other intervals).

→ Shock wave: It is defined as the wave produced by a body moving with a speed greater than the speed of sound. Shock waves carry a large amount of energy and when strike a building rattling sound due to the vibration of the building is produced.

→ Mach number: It is defined as the ratio of the velocity of the body producing shock waves to the velocity of sound.

∴ Mach number = \(\frac{\mathrm{V}_{\mathrm{s}}}{\mathrm{v}}\)

→ Echo: It is defined as the repetition of the sound of short duration. It (echo) is heard if the minimum distance between the obstacle reflecting sound waves and the source of sound is 17 m.

→ Reverberation: It is defined as the persistence or prolongation of audible sound after the source has stopped emitting sound. It is due to multiple reflections of sound waves.

→ Reverberation time: It is defined as the time during which the intensity of sound falls to one million of its original value after the source has stopped producing it.

→ The acoustics of Building: It is that branch of science which deals with the design of big halls and auditoriums so that a speech delivered or music produced in them is distinctly and clearly heard at all places in the building.

**Important Formulae:**

→ Velocity of wave: v = vλ

v = frequency of oscillator generating the wave

λ = wavelength of the wave

v = velocity of wave

→ Velocity of transverse wave in a string:

v = \(\sqrt{\frac{T}{m}}=\sqrt{\frac{T}{\pi r^{2} \rho}}\), where

ρ = density of the material of string

r = radius of string

T = tension Applied on the string

m = mass per unit length of the string

→ Newton’s form ula for velocity of sound in air:

v = \(\sqrt{\frac{P}{\rho}}\)

P = air pressure

ρ = density of air

→ Velocity of elastic waves or longitudinal waves in a medium is:

v = \(\sqrt{\frac{E}{\rho}}\)

E = coefficient of elasticity of the medium

ρ = density of the medium

→ Leplace’s formula for velocity of sound is air/gases:

v = \(\sqrt{\frac{\gamma \mathrm{P}}{\rho}}\) where

E = γP = adiabatic elasticity of air/gas

ρ = density of air/gas

γ = C_{P}/C_{V}.

→ Velocity of wave in gas/liquid medium (Longitudinal wave):

V = \(\sqrt{\frac{Y}{\rho}}\), where

Y = Young’s modulus

ρ = coefficient of rigidity

→ Velocity as a function of:

1. temperature, \(\frac{v_{1}}{v_{2}}=\sqrt{\frac{T_{1}}{T_{2}}}\)

2. density, \(\frac{v_{1}}{v_{2}}=\sqrt{\frac{\rho_{1}}{\rho_{2}}}\)

→ The equation of a plane simple harmonic wave (progressive wave) travelling from left to right is:

y = A sin 2π(\(\frac{\mathrm{t}}{\mathrm{T}}-\frac{\mathrm{x}}{\lambda}\))

= A sin \(\frac{2 \pi}{\lambda}\)(vt – x)

= A sin (ωt – kx)

and from right to left i.e. along – X axis is obtained by replacing

x = -x, i.e. y = A sin \(\frac{2 \pi}{\lambda}\)(vt – x)

→ Phase difference = \(\frac{2 \pi}{\lambda}\) × path difference

or

ΔΦ = \(\frac{2 \pi}{\lambda}\) × Δx

→ Total energy transmitted per Unit volume in waves is given by

E = 2π^{2} ρ v^{2} A^{2}

= \(\frac{2 \pi^{2} \rho v^{2} A^{2}}{\lambda^{2}}\)

→ Intensity of wave = \(\frac{2 \pi^{2} \rho v^{2} A^{2}}{\text { area } \times \text { time }}\)

→ I_{max} = (A_{1} + A_{2})^{2}.

→ I_{min} = (A_{1} – A_{2})^{2}.

→ Apparent frequency of sound when:

1. Source moves towards listener at rest is

ν’ = \(\frac{v}{v-v_{s}}\)ν

2. When source moves away from listener at rest is

ν’ = \(\frac{v}{v+v_{s}}\)ν

3. When listener moves towards source at rest is

ν’ = \(\frac{\mathbf{v}+\mathbf{v}_{0}}{\mathbf{v}}\)ν

4. When listener moves away from source at rest is

ν’ = \(\frac{\mathbf{v}-\mathbf{v}_{0}}{\mathbf{v}}\)ν

5. When both source and listener move towards each other

ν” = \(\frac{v-v_{0}}{v+v_{s}}\)ν

6. If both move away from each other, then

ν” = \(\frac{v-v_{0}}{v+v_{s}}\)ν

→ Sabine’s formula for reverberation time is

t = \(\frac{0.166 \mathrm{~V}}{\sum \alpha \mathrm{s}}\), Where

k = constant

V = volume of the hall

α = coefficient of absorption

s = area exposed to sound

→ Particle velocity at any instant in a progressive wave is

v = v_{o} cos 2π (\(\frac{t}{T}-\frac{x}{\lambda}\))

Where v_{o} = \(\frac{2 \pi}{\lambda}\) A = 2πAv

= velocity amplitude.

→ Particle acceleration at any instant of time in a progressive wave is

where a_{o} = a_{o} sin 2π (\(\frac{t}{T}-\frac{x}{\lambda}\))

where a_{o} = 4π^{2} v^{2} = ω^{2}

= acceleration amplitude.