WAVES
WAVE MOTION
Wave motion is a type of motion in which the disturbance travels from one point of the medium to another but the particles of the medium do not travel from one point to another.
For the propagation of waves, the medium must have inertia and elasticity. These two properties of the medium decide the speed of the wave.
There are two types of waves
Mechanical waves : These waves require a material medium for their propagation.
For example : sound waves, waves in stretched strings etc.
Non-mechanical waves or electromagnetic waves : These waves do not require any material medium for their propagation. For example : light waves, x-rays etc.
There are two types of mechanical waves
Transverse waves : In the transverse wave, the particles of medium oscillate in a direction perpendicular to the direction of wave propagation. Waves in stretched string, waves on the water surface are transverse in nature.
Transverse waves can travel only in solids and the surface of liquids.
Transverse waves propagate in the form of crests and troughs.
All electromagnetic waves are transverse in nature.
Longitudinal waves : In longitudinal waves particles of medium oscillate about their mean position along the direction of wave propagation.
Sound waves in air are longitudinal. These waves can travel in solids, liquids and gases.
Longitudinal waves propagate through the medium with the help of compressions and rarefactions.
EQUATION OF A HARMONIC WAVE
Harmonic waves are generated by sources that execute simple harmonic motion.
A harmonic wave travelling along the positive direction of x-axis is represented by
where,
y = displacement of the particle of the medium at a location x at time t
A = amplitude of the wave
λ = wavelength
T = time period
v = wave velocity in the medium
ω = angular frequency
K = angular wave number or propagation constant.
If the wave is travelling along the negative direction of x-axis then
DIFFERENTIAL EQUATION OF WAVE MOTION
RELATION BETWEEN WAVE VELOCITY AND PARTICLE VELOCITY
The equation of a plane progressive wave is
... (i)
The particle velocity
... (ii)
Slope of displacement curve or strain
... (iii)
Dividing equation (ii) by (iii), we get
i.e., Particle velocity = – wave velocity × strain.
Particle velocity changes with the time but the wave velocity is constant in a medium.
RELATION BETWEEN PHASE DIFFERENCE, PATH DIFFERENCE AND TIME DIFFERENCE
Phase difference of 2π radian is equivalent to a path difference λ and a time difference of period T.
Phase difference = × path difference
Phase difference = × time difference
Time difference = × path difference
SPEED OF TRANSVERSE WAVES
The speed of transverse waves in solid is given by
where η is the modulus of rigidity of the solid and ρ is the density of material.
The speed of transverse waves on stretched string is given by
where T is the tension in the string and μ is the mass per unit length of the string.
SPEED OF LONGITUDINAL WAVES
The speed of longitudinal waves in a medium of elasticity E and density ρ is given by
For solids, E is replaced by Young's modulus (Y)
For liquids and gases, E is replaced by bulk modulus of elasticity (B)
The density of a solid is much larger than that of a gas but the elasticity is larger by a greater factor.
vsolid > vliquid > vgas
SOUND WAVES
Sound waves are captivating phenomena that permeate our daily lives, shaping our auditory experiences and connecting us to the world of sound. From the rhythmic beats of music to the soothing whispers of nature, sound waves surround us, conveying information and evoking emotions.
Sound waves are mechanical waves, requiring a medium—such as air, water, or solids—to propagate. As objects vibrate, they create waves that travel through the medium, carrying energy and compressions and rarefactions that our ears perceive as sound.
Understanding the behaviour of sound waves enables us to comprehend a diverse range of auditory phenomena, including pitch, volume, timbre, and resonance.
Pitch is determined by the frequency of sound waves, with higher frequencies corresponding to higher pitches and lower frequencies to lower pitches.
Volume, on the other hand, is related to the amplitude or intensity of sound waves, with greater amplitudes resulting in louder sounds. Timbre refers to the quality or character of a sound, influenced by factors such as harmonics and overtones.
Beyond our auditory experiences, sound waves find practical applications in numerous fields. Industries such as music, film, and telecommunications rely on the principles of sound waves to create immersive audio experiences and facilitate communication. Medical professionals employ ultrasound imaging, which utilises sound waves, to visualise internal structures and aid in diagnostic procedures.
SPEED OF SOUND IN A GAS
NEWTON'S FORMULA
where P is the atmospheric pressure and ρ is the density of air at STP.
LAPLACE'S CORRECTION
where γ is the ratio of two specific heats Cp and Cv
POWER AND INTENSITY OF WAVE MOTION
If a wave is travelling in a stretched string, energy is transmitted along the string.
Power of the wave is given by
where μ is mass per unit length.
Intensity is flow of energy per unit area of cross section of the string per unit time.
PRINCIPLE OF SUPERPOSITION OF WAVES
If two or more waves arrive at a point simultaneously then the net displacement at that point is the algebraic sum of the displacement due to individual waves.
y = y1 + y2 + ............... + yn.
where y1, y2 .......... yn are the displacement due to individual waves and y is the resultant displacement.
INTERFERENCE OF WAVES
When two waves of equal frequency and nearly equal amplitude travelling in the same direction having the same state of polarisation in the medium superimpose, then intensity is different at different points. At some points intensity is large, whereas at other points it is nearly zero.
Consider two waves
y1 = A1sin (ωt – kx) and y2 = A2 sin (ωt – kx + φ)
By principle of superposition
y = y1 + y2 = A sin (ωt – kx + δ)
where, A2 = A12 + A22 + 2A1A2 cos φ,
and
As intensity I ∝ A2
So, resultant intensity I = I1 + I2 +
For constructive interference (maximum intensity) :
Phase difference, φ = 2nπ and path difference = nλ where n = 0, 1, 2, 3, ...
⇒ Amax = A1 + A2 and Imax = I1 + I2 +
For destructive interference (minimum intensity) :
Phase difference, φ = (2n + 1)π,
and path difference = ; where n = 0, 1, 2, 3, ...
⇒ Amin = A1 – A2 and Imin = I1 + I2 –
RESULTS
The ratio of maximum and minimum intensities in any interference wave form.
Average intensity of interference in wave form :
Put the value of Imax and Imin
or Iav = I1 + I2
If A = A1 = A2 and I1 = I2 = I
then Imax = 4I, Imin = 0 and Iav = 2I
Condition of maximum contrast in interference wave form
A1 = A2 and I1 = I2
then Imax = 4I and Imin = 0
For perfect destructive interference we have a maximum contrast in interference wave form.
REFLECTION OF WAVES
A mechanical wave is reflected and refracted at a boundary separating two media according to the usual laws of reflection and refraction.
When a sound wave is reflected from a rigid boundary or denser medium, the wave suffers a phase reversal of π but the nature does not change i.e., on reflection the compression is reflected back as compression and rarefaction as rarefaction.
When a sound wave is reflected from an open boundary or rarer medium, there is no phase change but the nature of the wave is changed i.e., on reflection, the compression is reflected back as rarefaction and rarefaction as compression.
POINTS TO REMEMBER
For a wave, v = f λ
The wave velocity of sound in air
Particle velocity is given by. It changes with time. The wave velocity is the velocity with which disturbances travel in the medium and is given by .
When a wave reflects from a denser medium the phase change is π and when the wave reflects from a rarer medium, the phase change is zero.
In a tuning fork, the waves produced in the prongs are transverse whereas in the stem is longitudinal.
A medium in which the speed of a wave is independent of the frequency of the waves is called non-dispersive. For example air is a non-dispersive medium for the sound waves.
Transverse waves can propagate in medium with shear modulus of elasticity e.g., solid whereas longitudinal waves need bulk modulus of elasticity hence can propagate in all media solid, liquid and gas.
ENERGY TRANSPORTED BY A HARMONIC WAVE ALONG A STRING
Kinetic energy of a small element of length dx is
where μ = mass per unit length
and potential energy stored
BEATS
When two wave trains slightly differing in frequencies travel along the same straight line in the same direction, then the resultant amplitude is alternately maximum and minimum at a point in the medium. This phenomenon of waxing and waning of sound is called beats.
Let two sound waves of frequencies n1 and n2 are propagating simultaneously and in the same direction. Then at x=0
y1 = A sin 2π n1t, and y2 = A sin 2π n2t,
For simplicity we take the amplitude of both waves to be the same.
By principle of superposition, the resultant displacement at any instant is
y = y1 + 2 = 2A cos 2π nAt sin 2π navt
where ,
⇒ y = Abeat Sin 2π navt ..................(i)
It is clear from the above expression (i) that
Abeat = 2A cos 2πnAt, amplitude of resultant wave varies periodically as frequency
A is maximum when
A is minimum when
Since intensity is proportional to amplitude i.e.,
For Imax cos 2π nAt = ± 1 For Imin
i.e., 2π nAt = 0,π, 2π 2π nAt = π/2, 3π/2
i.e., t = 0, 1/2nA, 2/2nA t = 1/4nA, 3/4nA.......
So time interval between two consecutive beat is
Number of beats per sec is given by
So beat frequency is equal to the difference of frequency of two interfering waves.
To hear beats, the number of beats per second should not be more than 10. (due to hearing capabilities of human beings)
FILING/LOADING A TUNING FORK
On filing the prongs of the tuning fork, it raises its frequency and on loading it decreases the frequency.
When a tuning fork of frequency ν produces Δν beats per second with a standard tuning fork of frequency ν0, then
If the beat frequency decreases or reduces to zero or remains the same on filling the unknown fork, then
If the beat frequency decreases or reduces to zero or remains the same on loading the unknown fork with a little wax, then
If the beat frequency increases on loading, then
DOPPLER EFFECT
When a source of sound and an observer or both are in motion relative to each other there is an apparent change in frequency of sound as heard by the observer. This phenomenon is called the Doppler's effect.
Apparent change in frequency
When source is in motion and observer at rest
when source moving towards observer
when source moving away from observer
Here V = velocity of sound
VS = velocity of source
ν0 = source frequency.
When source is at rest and observer in motion
when observer moving towards source
when observer moving away from source and
V0 = velocity of observer.
When source and observer both are in motion
If source and observer both move away from each other.
If source and observer both move towards each other.
When the wind blows in the direction of sound, then in all above formulae V is replaced by (V + W) where W is the velocity of wind. If the wind blows in the opposite direction to sound then V is replaced by (V – W).
POINTS TO REMEMBER
The motion of the listener causes a change in the number of waves received by the listener and this produces an apparent change in frequency.
The motion of the source of sound causes change in wavelength of the sound waves, which produces apparent change in frequency.
If a star goes away from the earth with velocity v, then the frequency of the light emitted from it changes from ν to ν'.
ν' = ν (1–v/c), where c is the velocity of light and where is called Doppler’s shift.
If the wavelength of the observed waves decreases then the object from which the waves are coming is moving towards the listener and vice versa.
STATIONARY OR STANDING WAVES
When two progressive waves having the same amplitude, velocity and time period but travelling in opposite directions superimpose, then a stationary wave is produced.
Let two waves of same amplitude and frequency travel in opposite direction at same speed, then
y1 = A sin (ωt –kx) and y2 = A sin (ωt + kx)
By principle of superposition
y = y1 + y2 = (2A cos kx) sin ωt ...(i)
y = AS sinωt
It is clear that amplitude of stationary wave As vary with position
As = 0, when cos kx = 0 i.e., kx = π/2, 3π/2............
i.e., x = λ/4, 3λ/4...................[as k = 2π/λ]
These points are called nodes and spacing between two nodes is λ/2.
As is maximum, when cos kx is max
i.e., kx = 0, π , 2π, 3π i.e., x = 0, λl/2, 2λ/2....
It is clear that antinodes (where As is maximum) are also equally spaced with spacing λ/2.
The distance between node and antinode is λ/4 (see figure)
POINTS TO REMEMBER
When a string vibrates in one segment, the sound produced is called fundamental note. The string is said to vibrate in fundamental mode.
The fundamental note is called first harmonic, and is given by, where v = speed of wave.
If the fundamental frequency be then , , ... are respectively called second, third, fourth ... harmonics respectively.
If an instrument produces notes of frequencies .... where ....., then is called first overtone, is called second overtone, is called the third overtone ... so on.
Harmonics are the integral multiples of the fundamental frequency. If ν0 is the fundamental frequency, then nν0 is the frequency of the n-th harmonic.
Overtones are the notes of frequency higher than the fundamental frequency actually produced by the instrument.
In the strings all harmonics are produced.
STATIONARY WAVES IN AN ORGAN PIPE
In the open organ pipe all the harmonics are produced.
In an open organ pipe, the fundamental frequency or first harmonic is , where v is velocity of sound and l is the length of the air column [see fig. (a)]
(a)
,
(b)
,
(c)
,
Similarly the frequency of second harmonic or first overtone is [see fig (b)],
Similarly the frequency of third harmonic and second overtone is [(see fig. (c)]
Similarly ....
In the closed organ pipe only the odd harmonics are produced. In a closed organ pipe, the fundamental frequency (or first harmonic) is (see fig. a)
(a) (b) (c)
Similarly the frequency of third harmonic or first overtone (IInd harmonic absent) is (see fig. b)
Similarly ........
End Correction
It is observed that the antinode actually occurs a little above the open end. A correction is applied for this which is known as end correction and is denoted by e.
For closed organ pipe : l is replaced by l + e where e = 0.3D, D is the diameter of the tube.
For open organ pipe : l is replaced by l + 2e where e = 0.3D
In resonance tube, the velocity of sound in air given by v = 2v (l2-l1)
where ν = frequency of tuning fork, ll = 1st resonating length, l2 = 2nd resonating length.
RESONANCE TUBE
It is used to determine velocity of sound in air with the help of a tuning fork of known frequency.
Let l1 and l2 are lengths of first and second resonances then
and
Speed of sound in air is
where υ is the frequency
For vibrating strings/open organ pipe
For closed organ pipe
COMPARISON OF PROGRESSIVE (OR TRAVELLING) AND STATIONARY (OR STANDING) WAVE
COMPARATIVE STUDY OF INTERFERENCE, BEATS AND STATIONARY WAVE
Characteristics of Sound:
Frequency: The frequency of sound refers to the number of vibrations or cycles per second and is measured in hertz (Hz). It determines the pitch of the sound, with higher frequencies corresponding to higher-pitched sounds and lower frequencies corresponding to lower-pitched sounds.
Amplitude: Amplitude represents the magnitude or intensity of a sound wave. It is measured as the maximum displacement of particles in the medium from their rest position. Amplitude determines the loudness or volume of the sound, with larger amplitudes producing louder sounds and smaller amplitudes resulting in softer sounds.
Wavelength: The wavelength of a sound wave is the distance between two consecutive points in a wave that are in phase, such as two crests or two troughs. It is inversely related to the frequency of the sound, meaning that higher-frequency sounds have shorter wavelengths and lower-frequency sounds have longer wavelengths.
Speed: The speed of sound refers to how fast sound waves travel through a medium. In a given medium, the speed of sound is determined by the properties of the medium, such as its density and elasticity. In general, sound travels faster in solids, slower in liquids, and even slower in gases.
Timbre: Timbre refers to the quality or character of a sound that distinguishes it from others with the same pitch and loudness. It is influenced by the unique combination of harmonics and overtones present in a sound. Timbre allows us to differentiate between different musical instruments or recognize familiar voices.
Reflection: Sound waves can undergo reflection when they encounter a surface that is capable of reflecting sound. The reflection of sound waves enables phenomena like echoes, where the original sound wave is reflected and heard after a short delay.
Diffraction: Diffraction occurs when sound waves encounter an obstacle or a narrow opening, causing them to bend and spread out around the obstacle or through the opening. This phenomenon allows sound to travel around corners and enables us to hear sounds even when the source is not directly in our line of sight.
Interference: Interference occurs when two or more sound waves interact with each other. Depending on the phase relationship between the waves, interference can result in constructive interference, where the waves reinforce each other and produce a louder sound, or destructive interference, where the waves cancel each other out and produce a quieter sound.
NATURE OF SOUND
Musical sound - It consists of quick, regular and periodic succession of compressions and rarefactions without a sudden change in amplitude.
Noise - It consists of slow, irregular and a periodic succession of compressions and rarefactions that may have sudden changes in amplitude.
Pitch, loudness and quality are the important characteristics of musical sound.
Pitch depends on frequency
loudness depends on intensity
quality depends on the number and intensity of overtones.
The detailed characteristics are as follows:
Characteristics of Musical Sound:
Pitch: Pitch refers to the perceived highness or lowness of a musical sound and is determined by the frequency of the sound wave. Higher frequencies result in higher-pitched sounds, while lower frequencies produce lower-pitched sounds. Pitch allows us to distinguish between different musical notes and melodies.
Melody: Melody is a sequence of pitches played or sung in a specific order, creating a musical line or tune. It is the horizontal aspect of music, characterised by the arrangement and organisation of different pitches and intervals.
Harmony: Harmony is the vertical aspect of music, involving the simultaneous sounding of two or more pitches. It refers to the combination and relationship of different musical notes played together, creating chords, progressions, and tonal structures. Harmony adds richness and depth to musical compositions.
Timbre: Timbre, also known as tone colour or quality, refers to the unique characteristic of a musical sound that distinguishes it from other sounds with the same pitch and loudness. It is influenced by various factors, including the instrument or voice producing the sound, the presence of harmonics and overtones, and the way the sound is shaped and modified.
Dynamics: Dynamics refers to the variations in loudness or volume of a musical sound. It allows musicians to express emotions and convey musical intentions through changes in intensity. Musical notations such as piano (soft) and forte (loud) indicate the desired dynamics of a composition.
Rhythm: Rhythm is the pattern of duration and timing of musical sounds and silences. It encompasses elements such as beat, tempo, metre, and rhythmic patterns. Rhythm provides the framework and pulse that drive a musical piece, creating a sense of movement and groove.
Texture: Texture refers to the relationship between different musical parts or voices within a composition. It can be characterised as monophonic (single melodic line), homophonic (melody with accompanying chords), or polyphonic (multiple independent melodies). Texture adds richness and complexity to musical compositions.
Expression and Articulation: Expression and articulation involve the manipulation and shaping of musical sounds to convey specific emotions, moods, or stylistic elements. This includes techniques such as legato (smooth and connected), staccato (short and detached), vibrato (trembling effect), and dynamics variations.
Interval - The ratio of the frequencies of the two notes is called the interval between them.
For example, the interval between two notes of frequencies 512 Hz and 1024 Hz is 1 : 2 (or 1/2).
Two notes 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 sound are the following:
Octave (1 : 2), majortone (8 : 9), minortone (9 : 10) and semitone (15 : 16)
Major diatonic scale - It consists of eight notes. The consecutive notes have either of the following three intervals. They are 8 : 9 ; 9 : 10 and 15 : 16.
ACOUSTICS
The branch of physics that deals with the process of generation, reception and propagation of sound is called acoustics.
Acoustics may be studied under the following three subtitles.
Electro acoustics. This branch deals with electrical sound production with music.
Musical acoustics. This branch deals with the relationship of sound with music.
Architectural acoustics. This branch deals with the design and construction of buildings.
REVERBERATION
Multiple reflections which are responsible for a series of waves falling on listener’s ears, giving the impression of a persistence or prolongation of the sound are called C
The time gap between the initial direct note and the reflected note up to the minimum audibility level is called reverberation time.
Sabine Reverberation Formula for Time
Sabine established that the standard period of reverberation viz., the time that the sound takes to fall in intensity by 60 decibels or to one millionth of its original intensity after it was stopped, is given by
where V = volume of room, = α1 S1 + α2 S2 + ....
S1, S2 .... are different kinds of surfaces of room and
α1 , α2 .... are their respective absorption coefficients.
The above formula was derived by Prof C. Sabine.
SHOCK WAVES
The waves produced by a body moving with a speed greater than the speed of sound are called shock waves. These waves carry a huge amount of energy. It is due to the shock wave that we have a sudden violent sound called sonic boom when a supersonic plane passes by.
The rate of speed of the source to that of the speed of sound is called mach number.
INTENSITY OF SOUND
The sound intensities that we can hear range from 10–12 Wm–2 to 103 Wm–2.
The intensity level β, measured in terms of decibel (dB) is defined as
where I = measured intensity, I0 = 10–12 Wm–1
At the threshold β = 0
At the maximum,
LISSAJOUS FIGURES
When two simple harmonic waves having vibrations in mutually perpendicular directions superimpose on each other, then the resultant motion of the particle is along a closed path, called the Lissajous figures. These figures can be of many shapes depending on
ratio of frequencies or time periods of two waves
ratio of amplitude of two waves
phase difference between two waves.
