Class 11 Physics · NCERT chapter 15 · 9 topics · free to read
Wave Motion Basics
A wave is a disturbance that transfers energy (not matter) from one point to another through a medium or space. Waves are classified as transverse or longitudinal based on the direction of particle vibration relative to wave propagation, and as mechanical or electromagnetic based on whether they need a medium. NEET frequently tests classification of waves and basic wave properties.
v = f = T
What is a Wave?
A wave is a disturbance that propagates through space and time, transferring energy from one point to another without the physical transfer of matter.
When you throw a stone into a pond, ripples spread outward — but the water itself doesn't travel outward. The disturbance moves, not the medium. This is the essence of wave motion: energy propagates, matter does not.
A wave is a disturbance that propagates through a medium or space
Waves transfer energy and momentum — NOT matter
The particles of the medium oscillate about their mean positions — they don't travel with the wave
A wave requires a source of disturbance, a medium (for mechanical waves), and some physical connection between particles (elasticity)
Transverse Waves
In transverse waves, particles of the medium vibrate perpendicular (at right angles) to the direction of wave propagation. Crests and troughs are the characteristic features.
In a transverse wave, the particles of the medium vibrate perpendicular (at 90°) to the direction in which the wave travels. The highest point is called a crest and the lowest point is called a trough.
Particle displacement is perpendicular to wave velocity
Features: crests (peaks) and troughs (valleys)
Transverse waves can travel through solids and on surfaces of liquids — NOT through the bulk of liquids or gases
Electromagnetic waves (light, radio) are transverse — and they don't need a medium at all
One wavelength = crest-to-crest (or trough-to-trough) distance
Longitudinal Waves
In longitudinal waves, particles vibrate along (parallel to) the direction of wave propagation, creating alternating compressions and rarefactions.
In a longitudinal wave, particles vibrate parallel to (along) the direction of wave propagation. This creates regions of high density called compressions and regions of low density called rarefactions.
Particle displacement is parallel to wave velocity
Features: compressions (high pressure, high density) and rarefactions (low pressure, low density)
Longitudinal waves can travel through solids, liquids, AND gases — all three states of matter
Sound waves in air are the most important example of longitudinal waves
One wavelength = compression-to-compression (or rarefaction-to-rarefaction) distance
Vibration ⊥ propagation
Vibration ∥ propagation
Crests and troughs
Compressions and rarefactions
Solids + surface of liquids
Solids + liquids + gases
Needs rigidity (shear modulus)
Needs elasticity (bulk modulus)
Light, rope waves, S-waves
Sound, P-waves, slinky push-pull
Mechanical vs Electromagnetic Waves
Mechanical waves require a material medium for propagation; electromagnetic waves can travel through vacuum. This is the most fundamental classification of waves.
Mechanical waves need a material medium — the particles of the medium carry the disturbance forward. Remove the medium, and the wave can't propagate. Electromagnetic (EM) waves are self-sustaining oscillations of electric and magnetic fields — they don't need any medium and can travel through vacuum.
Mechanical waves: sound, water waves, seismic waves, waves on strings — all need a medium
Electromagnetic waves: light, radio, X-rays, microwaves — travel through vacuum at c = 3 10^8 m/s
A third category: matter waves (de Broglie waves) — associated with moving particles (quantum mechanics)
Speed of mechanical waves depends on elasticity and inertia of the medium
Speed of EM waves in vacuum is universal constant c , independent of frequency
Quick recall
Why can light from the Sun reach Earth but sound from the Sun cannot?
Light is an electromagnetic wave — it can travel through the vacuum of space. Sound is a mechanical wave — it requires a material medium (like air), and there's no air between the Sun and Earth.
Progressive (Travelling) Wave
A progressive wave continuously transfers energy from source to distant points. Key properties include amplitude, wavelength, frequency, time period, and wave speed.
A progressive wave (also called a travelling wave) is a wave that moves continuously through a medium, transferring energy from one point to another. Unlike standing waves (which we'll study later), every point in a progressive wave oscillates with the same amplitude.
Amplitude ( A ): Maximum displacement from mean position. Unit: metre (m)
Wavelength ( ): Distance between two consecutive points in the same phase (e.g., crest to crest). Unit: m
Time period ( T ): Time for one complete oscillation. Unit: second (s)
Frequency ( f or ): Number of oscillations per second. f = 1/T . Unit: Hz
Wave speed ( v ): Distance travelled by the wave per unit time. v = f = /T
v = f = T
In a progressive wave, all particles oscillate with the same amplitude but with a phase difference that depends on their separation. The phase changes continuously along the direction of propagation.
Wave Equation
The displacement of a progressive wave is described by y = A (kx - t + ) , where k = 2 / is the wave number and = 2 /T is the angular frequency. The sign of the kx and t terms determines the direction of propagation. NEET frequently tests extraction of wave parameters from the equation.
y = A (kx - t)
The Travelling Wave Equation
The mathematical description of a sinusoidal progressive wave, giving the displacement of any particle at any position x and any time t.
y(x, t) = A (kx - t + _0)
This single equation tells you the displacement y of any particle at position x at any time t . It's the complete mathematical description of a sinusoidal travelling wave moving in the +x direction.
A = amplitude — maximum displacement (always positive)
k = wave number (also called angular wave number or propagation constant)
= angular frequency — how fast the phase changes with time
_0 = initial phase — phase at x = 0, t = 0
(kx - t + _0) = total phase of the wave at point x at time t
Wave Number & Angular Frequency
Wave number k = 2π/λ measures how many radians of phase change per metre, and angular frequency ω = 2π/T measures radians of phase change per second.
k = 2 (unit: rad/m)
= 2 T = 2 f (unit: rad/s)
Wave number k tells you how many radians of phase fit per metre of wave. Higher k → shorter wavelength → more tightly packed waves. Angular frequency tells you how many radians of phase pass per second. Higher → higher frequency → faster oscillation.
k = 2 / — spatial analogue of . Think: k is to space what is to time
= 2 f = 2 /T — converts frequency from cycles/s to radians/s
Wave speed: v = /k = f — a crucial relation connecting spatial and temporal parameters
Phase difference between two points separated by x : = k x = 2 x
Quick recall
A wave is given by y = 3 (36t - 0.018x) cm. Find amplitude, frequency, wavelength, and speed.
A = 3 cm. = 36 rad/s → f = 36/2 5.73 Hz. k = 0.018 rad/cm → = 2 /0.018 349 cm. v = /k = 36/0.018 = 2000 cm/s = 20 m/s.
Direction of Wave Propagation
The sign between kx and ωt in the wave equation determines the direction of propagation. Negative sign (kx - ωt) means +x direction; positive sign (kx + ωt) means -x direction.
The direction of wave propagation is hidden in the sign between kx and t :
y = A (kx - t) → wave moves in +x direction (signs of kx and t are opposite)
y = A (kx + t) → wave moves in −x direction (signs of kx and t are same)
Equivalent forms: y = A ( t - kx) also moves in +x direction (it's just -A (kx - t) )
Quick recall
In which direction does the wave y = 5 (4x + 8t) travel? What is its speed?
Both kx and t terms have the same sign (positive) → wave travels in −x direction. Speed v = /k = 8/4 = 2 m/s.
Phase & Phase Difference
Phase tells you the exact state (displacement + direction of motion) of an oscillating particle. Phase difference between two points determines whether they are in step, out of step, or somewhere in between.
The phase of a wave at position x at time t is the argument of the sine function: = kx - t + _0 . It tells you the exact state of oscillation — not just where the particle is, but which way it's heading.
= k x = 2 x (phase diff. between two points at same time)
= t = 2 T t (phase diff. at same point, different times)
Path difference x = corresponds to phase difference = 2 (one full cycle)
Path difference x = /2 → = → particles are in opposite phase (antiphase)
Relation: = 2 x — this connects path difference to phase difference
Two particles in phase ( = 0, 2 , 4 , ... ) oscillate identically
Two particles in antiphase ( = , 3 , 5 , ... ) oscillate in exactly opposite manner
Speed of Waves
Wave speed depends on the properties of the medium, not the source. For transverse waves on a string, v = T/ . For sound in a gas, v = P/ (Newton-Laplace formula). NEET tests factors affecting speed of sound — temperature, pressure, humidity, and the common misconception about amplitude/frequency dependence.
v = T (string) v = P (gas)
Speed on a Stretched String
The speed of a transverse wave on a string depends on the tension in the string and its linear mass density (mass per unit length).
v = T
Here T is the tension in the string (in newtons) and is the linear mass density = mass per unit length (in kg/m). Higher tension → faster wave (tighter string = quicker response). Higher → slower wave (heavier string = more inertia).
T = tension in the string (NOT temperature). Unit: newton (N)
= m/L = linear mass density. Unit: kg/m
If cross-sectional area is A and material density is , then = A
Speed does NOT depend on frequency or amplitude — only on medium properties ( T and )
Quick recall
A string of length 2 m and mass 10 g is stretched with a tension of 200 N. Find the wave speed.
= m/L = 0.01/2 = 0.005 kg/m. v = T/ = 200/0.005 = 40000 = 200 m/s .
Speed of Sound in Gas
Newton assumed isothermal propagation (v = √(P/ρ)) but got the wrong answer. Laplace corrected it by recognizing sound propagation is adiabatic: v = √(γP/ρ).
Newton's formula assumed sound propagation is isothermal (slow, temperature has time to equalize): v = P/ . This gave v 280 m/s for air — about 15% too low compared to the experimental value (~332 m/s at 0°C).
Laplace's correction: Sound compressions and rarefactions happen so fast that heat has no time to escape — the process is adiabatic, not isothermal. Replace P with P :
v = P (Newton-Laplace formula)
For air: = 1.4 (diatomic), so v = 1.4 P/ 332 m/s at 0°C. This matches experiment perfectly.
Newton's (wrong): v = P/ — assumed isothermal — gave ~280 m/s
Laplace's (correct): v = P/ — assumed adiabatic — gives ~332 m/s
= C_p/C_v = ratio of specific heats (1.4 for diatomic gases like air)
Laplace's correction multiplied Newton's value by : v_ Laplace = v_ Newton
Factors Affecting Speed of Sound
Speed of sound increases with temperature (∝ √T), is independent of pressure (at constant temperature), and increases with humidity. These dependencies are NEET favourites.
Using the ideal gas law PV = nRT and = PM/(RT) (where M is molar mass), we can rewrite the Newton-Laplace formula as:
v = R T M
This reveals the true dependencies: speed depends on , T (temperature), and M (molar mass) — but NOT on pressure!
Temperature: v T (in Kelvin). Speed increases ~0.6 m/s per °C rise near room temperature
Pressure: NO effect (at constant temperature). Increasing P increases proportionally, so P/ stays constant
Humidity: Moist air has lower M (water vapour M = 18 replaces N_2 with M = 28 ) → speed is faster in humid air
Molar mass: v 1/ M . Sound is faster in lighter gases (He > air > CO _2 )
Density alone: v 1/ only if comparing at same pressure. This is often misleading — use the T and M form instead
v_1 v_2 = T_1 T_2 (same gas, different temperatures)
Superposition Principle
When two or more waves overlap, the resultant displacement at any point is the algebraic sum of the individual displacements. This leads to constructive interference (waves add up) and destructive interference (waves cancel). The path difference–phase difference relation governs all interference phenomena.
y = y_1 + y_2 (algebraic sum)
Principle of Superposition
When two or more waves pass through the same region, each wave travels independently. The net displacement at any point is the vector (algebraic) sum of individual displacements.
The principle of superposition states: when two or more waves overlap in a region, the resultant displacement at any point at any instant is the algebraic sum of the displacements due to each individual wave at that point.
y(x, t) = y_1(x, t) + y_2(x, t) + y_3(x, t) +
Each wave behaves as if the other waves don't exist — waves pass through each other unchanged
The sum is algebraic: if y_1 = +3 cm and y_2 = -2 cm at the same point, then y = +1 cm
After crossing, each wave continues with its original shape, speed, and direction — unaltered
This principle works because the wave equation is linear — solutions can be added
Constructive & Destructive Interference
Two waves meeting in phase (crest meets crest) produce constructive interference (larger amplitude). Waves meeting in antiphase (crest meets trough) produce destructive interference (reduced/zero amplitude).
When two waves of the same frequency overlap, the result depends on their phase difference:
Two waves of amplitudes 5 cm and 3 cm interfere. What are the maximum and minimum possible resultant amplitudes?
Maximum (constructive): A_ = 5 + 3 = 8 cm . Minimum (destructive): A_ = 5 - 3 = 2 cm . The amplitude can never exceed 8 cm or go below 2 cm.
Path Difference & Phase Difference
Path difference (extra distance one wave travels) and phase difference are related by Δφ = (2π/λ)Δx. Constructive interference when path difference = nλ; destructive when path difference = (n + ½)λ.
= 2 x
Path difference ( x ) is the extra distance one wave travels compared to the other. It directly determines the phase difference and hence whether interference is constructive or destructive.
Constructive: x = n where n = 0, 1, 2, ... → = 2n
Destructive: x = (2n+1) 2 where n = 0, 1, 2, ... → = (2n+1)
x = → = 2 → one full cycle → constructive
x = /2 → = → half cycle → destructive
This relation is the foundation for Young's double slit, thin film interference, and all wave optics
Path diff = n
Path diff = (2n+1) /2
Phase diff = 2n
Phase diff = (2n+1)
Crest meets crest
Crest meets trough
A_R = A_1 + A_2 (max)
A_R = |A_1 - A_2| (min)
I_R = ( I_1 + I_2 )^2
I_R = ( I_1 - I_2 )^2
Standing Waves
When two identical waves travel in opposite directions, they superpose to form standing (stationary) waves. These waves have fixed points of zero displacement (nodes) and maximum displacement (antinodes). Unlike progressive waves, standing waves do NOT transfer energy. NEET heavily tests node–antinode patterns, the equation y = 2A (kx) ( t) , and differences between standing and progressive waves.
y = 2A (kx) ( t)
Formation of Standing Waves
When an incident wave and its reflection (same amplitude, same frequency, opposite direction) superpose, the result is a standing wave — a wave that oscillates in place without travelling.
Consider two waves of identical amplitude A , frequency f , and wavelength , travelling in opposite directions:
y_1 = A (kx - t) and y_2 = A (kx + t)
By superposition and using the identity C + D = 2 C+D 2 C-D 2 :
y = y_1 + y_2 = 2A (kx) ( t)
Standing waves form when two identical waves travel in opposite directions and superpose
Most common scenario: a wave reflects off a boundary and interferes with the incoming wave
The resultant wave does not travel — it oscillates in place
Amplitude varies with position: 2A (kx) — some points always have zero amplitude (nodes)
Nodes & Antinodes
Nodes are points of permanent zero displacement; antinodes are points of maximum displacement. They alternate at regular intervals of λ/4 apart.
In the standing wave y = 2A (kx) ( t) , the amplitude at position x is 2A| (kx)| .
Nodes: Points where (kx) = 0 → amplitude is permanently zero → kx = 0, , 2 , ... → x = 0, /2, , 3 /2, ...
Antinodes: Points where | (kx)| = 1 → amplitude is maximum ( 2A ) → kx = /2, 3 /2, ... → x = /4, 3 /4, 5 /4, ...
Distance between consecutive nodes = /2
Distance between consecutive antinodes = /2
Distance between a node and the nearest antinode = /4
Nodes and antinodes alternate — a node always has an antinode on each side
At a node, the two constituent waves always cancel (destructive interference). At an antinode, they always reinforce (constructive interference). Between nodes, all particles oscillate in phase with each other but with varying amplitudes.
Quick recall
In a standing wave, the distance between the 3rd node and the 7th node is 80 cm. Find the wavelength.
Between 3rd and 7th node, there are 7 - 3 = 4 gaps, each of length /2 . So 4 /2 = 80 cm → 2 = 80 cm → = 40 cm .
Standing vs Progressive Waves
Standing waves have nodes and antinodes, don't transfer energy, and have position-dependent amplitudes. Progressive waves have uniform amplitude and continuously transfer energy.
Wave pattern moves forward
Wave pattern oscillates in place
All particles have the same amplitude
Amplitude varies: zero at nodes, max at antinodes
Energy is continuously transferred
No net energy transfer (energy oscillates between KE and PE)
Every particle has a different phase
Particles between two nodes are in phase; across a node, they're in antiphase
Equation: y = A (kx - t)
Equation: y = 2A (kx) ( t)
No fixed points of zero displacement
Nodes: permanently at rest
All points reach zero displacement at different times
All points (between nodes) reach zero simultaneously
Progressive waves: x and t appear together in the argument: (kx - t)
Standing waves: x and t appear separately: (kx) ( t)
In standing waves, wavelength is still defined as twice the node-to-node distance
Standing waves store energy; progressive waves transport it
Vibrations of Strings
A string fixed at both ends vibrates in standing wave patterns called harmonics. The fundamental frequency is f_1 = 1 2L T/ , and all harmonics (integer multiples) are present. The three laws of transverse vibrations relate frequency to length, tension, and linear mass density. NEET loves numerical problems on harmonics and sonometer.
f_n = n 2L T (n = 1, 2, 3, ...)
Boundary Conditions
A string clamped at both ends must have nodes at both endpoints. This constrains which standing wave patterns can exist, giving rise to discrete frequencies (harmonics).
When a string is fixed at both ends (like a guitar string), the endpoints cannot move — they must be nodes. This boundary condition restricts the possible standing wave patterns: only those wavelengths that fit a whole number of half-wavelengths in the string length L are allowed.
L = n _n 2 _n = 2L n (n = 1, 2, 3, ...)
n = 1 : Fundamental mode — one loop (half-wavelength fits in L ), _1 = 2L
n = 2 : 2nd harmonic (1st overtone) — two loops, _2 = L , one node in the middle
n = 3 : 3rd harmonic (2nd overtone) — three loops, _3 = 2L/3 , two internal nodes
General: n th harmonic has n loops, (n-1) internal nodes, and (n+1) total nodes (including endpoints)
A string fixed at both ends supports ALL harmonics — n = 1, 2, 3, 4, ...
Frequency of Harmonics
The frequency of the nth harmonic of a string fixed at both ends: f_n = (n/2L)√(T/μ). All harmonics are present.
Since v = f and _n = 2L/n , the frequency of the n th harmonic is:
f_n = v _n = nv 2L = n 2L T
Fundamental frequency: f_1 = 1 2L T (this is the lowest possible frequency)
n th harmonic: f_n = nf_1 — all harmonics are integer multiples of the fundamental
Ratio of frequencies: f_1 : f_2 : f_3 : ... = 1 : 2 : 3 : ... (harmonic series)
The quality (timbre) of a musical note depends on which harmonics are present and their relative amplitudes
Quick recall
A string of length 1 m and mass 5 g is under 80 N tension. Find the fundamental frequency and the frequency of the 3rd harmonic.
Three laws govern the fundamental frequency of a vibrating string: inversely proportional to length, proportional to √tension, and inversely proportional to √(linear mass density). These laws form the basis of the sonometer experiment.
From f_1 = 1 2L T/ , we can extract three independent laws by varying one quantity at a time:
Law of Length: f 1/L (keeping T and constant). Halve the length → double the frequency
Law of Tension: f T (keeping L and constant). Quadruple the tension → double the frequency
Law of Linear Mass Density: f 1/ (keeping L and T constant). Use a string 4× heavier → halve the frequency
f_1 f_2 = L_2 L_1 T_1 T_2 _2 _1
Quick recall
If the tension in a string is increased by 44%, by what percentage does the fundamental frequency change?
f T . New T' = 1.44T . f'/f = 1.44 = 1.2 . Frequency increases by 20%.
Organ Pipes
An open pipe supports all harmonics ( f_n = nv/2L ), while a closed pipe supports only odd harmonics ( f_n = nv/4L where n = 1, 3, 5, ... ). The open end is always an antinode; the closed end is always a node. End correction ( e 0.6r ) accounts for the antinode forming slightly beyond the open end. NEET heavily tests open vs closed pipe frequency ratios.
A pipe open at both ends has pressure nodes (displacement antinodes) at both openings. All harmonics are present. The fundamental frequency is v/2L.
In an open pipe (open at both ends), air is free to vibrate at both openings. So both ends are displacement antinodes (and pressure nodes). The standing wave pattern must fit antinodes at both ends.
L = n _n 2 f_n = nv 2L (n = 1, 2, 3, ...)
Fundamental ( n = 1 ): One node in the middle, antinodes at both ends. _1 = 2L , f_1 = v/2L
2nd harmonic ( n = 2 ): Two nodes inside. _2 = L , f_2 = 2f_1
An open pipe sounds richer than a closed pipe of same length because all harmonics are present
Open pipe frequency formula is identical to that of a string fixed at both ends
Closed Pipe
A pipe closed at one end has a displacement node at the closed end and an antinode at the open end. Only odd harmonics (1st, 3rd, 5th, ...) are present. Fundamental frequency is v/4L.
In a closed pipe (one end closed, one end open), the closed end is a displacement node (air can't move there) and the open end is a displacement antinode. This asymmetric boundary condition means only odd harmonics survive.
L = n _n 4 f_n = nv 4L (n = 1, 3, 5, 7, ...)
Fundamental ( n = 1 ): Only a quarter-wavelength fits. _1 = 4L , f_1 = v/4L
1st overtone ( n = 3 ): 3rd harmonic. f_3 = 3f_1 — NOT 2f_1 !
2nd overtone ( n = 5 ): 5th harmonic. f_5 = 5f_1
Only ODD harmonics: f_1 : f_3 : f_5 = 1 : 3 : 5 . Even harmonics ( 2f_1, 4f_1, ... ) are missing
The fundamental of a closed pipe ( v/4L ) is half that of an open pipe of same length ( v/2L )
Both ends: antinodes
Open end: antinode; closed end: node
f_1 = v/2L
f_1 = v/4L (half of open pipe)
ALL harmonics: 1, 2, 3, 4, ...
Only ODD harmonics: 1, 3, 5, 7, ...
Richer sound (more overtones)
Hollow/nasal sound (missing even harmonics)
Flute, organ pipe (open)
Bottle (blow across), clarinet
Quick recall
A closed pipe of length 85 cm resonates in its 3rd harmonic. If the speed of sound is 340 m/s, find the frequency.
Closed pipe: f_n = nv/4L with n odd. 3rd harmonic: n = 3 . f_3 = 3 340/(4 0.85) = 1020/3.4 = 300 Hz .
End Correction
The displacement antinode at an open end forms slightly beyond the physical end of the pipe. This extra distance e ≈ 0.6r (where r is the pipe radius) must be added to the effective length.
In practice, the displacement antinode at an open end forms slightly beyond the open end of the pipe — not exactly at the edge. This extra distance is called the end correction e .
e 0.6r (where r = inner radius of the pipe)
Open pipe: Both ends need correction. Effective length = L + 2e . So f_1 = v/(2(L + 2e))
Closed pipe: Only the open end needs correction. Effective length = L + e . So f_1 = v/(4(L + e))
End correction is typically small compared to L — but it matters in resonance tube experiments
The resonance tube experiment uses end correction to find the speed of sound accurately
v = 2f(L_2 - L_1) (resonance tube experiment)
Quick recall
In a resonance tube experiment, the first and second resonances are at lengths 16 cm and 50 cm. The tuning fork frequency is 512 Hz. Find the speed of sound and end correction.
= 2(L_2 - L_1) = 2(50 - 16) = 68 cm = 0.68 m. v = f = 512 0.68 = 348.2 m/s . End correction: e = (L_2 - 3L_1)/2 = (50 - 48)/2 = 1 cm .
Beats
When two sound waves of slightly different frequencies superpose, the resultant intensity rises and falls periodically — this phenomenon is called beats. The beat frequency equals the absolute difference of the two frequencies: f_ beat = |f_1 - f_2| . Beats are used to tune musical instruments. NEET tests beat frequency calculations and tuning fork problems.
f_ beat = |f_1 - f_2|
What are Beats?
Beats occur when two sound waves with nearly equal frequencies arrive at the same point. The resultant sound alternates between loud (constructive) and soft (destructive) at a rate equal to the difference of the frequencies.
When two tuning forks of slightly different frequencies (say 256 Hz and 260 Hz) are sounded together, you hear a single tone that periodically gets louder and softer. This periodic rise and fall of sound intensity is called beats.
Mathematically, if y_1 = A (2 f_1 t) and y_2 = A (2 f_2 t) , the superposition gives:
y = 2A (2 f_1 - f_2 2 t ) (2 f_1 + f_2 2 t )
The resultant sounds like a tone of frequency (f_1 + f_2)/2 (average frequency) — this is the pitch you hear
The amplitude varies as 2A ( (f_1 - f_2)t ) — this is the 'envelope' that waxes and wanes
Beat frequency = |f_1 - f_2| — the number of loud-soft cycles per second
Beat period = 1/|f_1 - f_2| — time between consecutive loud sounds
Beat Frequency Calculation
Beat frequency equals the absolute difference of the two source frequencies. Problems often involve tuning forks, where one fork's frequency is altered by loading (wax) or filing.
f_ beat = |f_1 - f_2|
Beat frequency is always positive (absolute value)
If 5 beats are heard in 2 seconds, beat frequency = 5/2 = 2.5 Hz
Loading a tuning fork (adding wax) → decreases its frequency
Filing a tuning fork (removing material) → increases its frequency
If beats increase after loading → the loaded fork was originally the higher-frequency one (moved away from the other fork)
If beats decrease after loading → the loaded fork was originally the higher-frequency one (moved closer to the other)
Quick recall
A tuning fork of unknown frequency produces 4 beats/s with a 256 Hz fork. When wax is added to the unknown fork, beats become 6/s. Find the unknown frequency.
Unknown fork: f = 256 4 → f = 252 or 260 Hz. Adding wax decreases f . If f = 260 , wax makes it closer to 256 → beats should decrease. But beats increased to 6 → f must be 252 Hz. Wax makes 252 even lower → further from 256 → beats increase to 6. So f = 252 Hz .
Applications of Beats
Beats are used to tune musical instruments (tune until beats disappear = frequencies matched), detect small frequency differences, and in radar/sonar technologies.
The most important practical application of beats is tuning musical instruments. A musician strikes a tuning fork (known frequency) and plays a note on their instrument simultaneously. If they hear beats, the instrument is slightly out of tune. They adjust the string tension until the beats disappear — meaning the frequencies are now identical.
Tuning instruments: Adjust until beat frequency → 0 (frequencies match perfectly)
Measuring unknown frequency: Use beats with a known fork to find the unknown fork's frequency
Detecting tiny frequency differences: Beats make even 1 Hz differences audible as distinct pulsations
Piano tuners rely on beats between strings and tuning forks to achieve perfect pitch
Doppler Effect
When a source and/or observer are in relative motion, the observed frequency differs from the emitted frequency. The general formula is f' = f v v_o v v_s . The sign convention depends on the direction of motion relative to the source–observer line. NEET frequently tests the formula application, sign conventions, and special cases.
f' = f ( v + v_o v - v_s ) (approaching)
The Doppler Effect
The apparent change in frequency of a wave due to relative motion between the source and observer. Frequency increases when they approach each other and decreases when they move apart.
The Doppler effect is the change in observed frequency of a wave when the source and/or observer are in motion relative to the medium. Named after Austrian physicist Christian Doppler (1842).
Source and observer approaching: observed frequency increases ( f' > f )
Source and observer receding: observed frequency decreases ( f' < f )
The Doppler effect applies to ALL waves — sound, light, water waves
For sound: the effect depends on the velocities of source AND observer relative to the medium (air)
For light: only the relative velocity between source and observer matters (no medium needed)
NCERT derives the Doppler formula for sound waves only. For light, the relativistic Doppler effect is different (not in NEET syllabus). But NEET may ask conceptual questions about Doppler effect in light (redshift/blueshift).
Doppler Effect Formula
The general Doppler formula f' = f(v ± v_o)/(v ∓ v_s) with the NCERT sign convention: upper signs for approaching, lower signs for receding.
f' = f ( v v_o v v_s )
Where v = speed of sound in the medium, v_o = speed of observer, v_s = speed of source, f = actual frequency emitted, f' = apparent frequency observed.
Numerator ( v v_o ): Use + when observer moves TOWARD source, − when moving AWAY
Denominator ( v v_s ): Use − when source moves TOWARD observer, + when moving AWAY
Both approaching → f' = f(v + v_o)/(v - v_s) — maximum increase in frequency
Both receding → f' = f(v - v_o)/(v + v_s) — maximum decrease in frequency
Speed of medium wind? Add/subtract wind speed to v depending on direction
Special Cases
The most commonly tested scenarios: only source moving, only observer moving, both moving, and the case of zero apparent change.
Source stationary, observer approaching: f' = f ( v + v_o v ) = f (1 + v_o v )
Source stationary, observer receding: f' = f ( v - v_o v ) = f (1 - v_o v )
Observer stationary, source approaching: f' = f ( v v - v_s )
Observer stationary, source receding: f' = f ( v v + v_s )
Source and observer moving with the same velocity in the same direction: f' = f (no change — relative motion with respect to medium is what matters, but net effect cancels only if v_s = v_o )
Quick recall
A car moving at 36 km/h approaches a stationary observer while honking at 400 Hz. Speed of sound = 340 m/s. Find the apparent frequency.
A train moves away from a station at 30 m/s. Its whistle frequency is 500 Hz. Speed of sound = 330 m/s. Find the frequency heard by a person on the platform.
The Doppler effect is used in SONAR, radar speed guns, medical ultrasound, and astronomy (redshift of galaxies). NEET may ask conceptual questions on these applications.
SONAR: Ships emit ultrasound pulses. The reflected wave from a submarine/fish has a Doppler-shifted frequency → gives speed and direction of the target
Radar speed guns: Police radar sends microwaves at a car. The reflected wave is Doppler-shifted → gives the car's speed
Medical ultrasound (Doppler echocardiography): Ultrasound reflects off blood cells. The frequency shift reveals blood flow speed — used to detect heart valve defects
Redshift of galaxies: Light from distant galaxies is shifted toward red (lower frequency) → galaxies are moving away → universe is expanding (Hubble's observation)
Blueshift: Light from an approaching star shifts toward blue (higher frequency) — rare, but seen in some nearby galaxies