Waves — Wave Motion, Sound & Doppler Effect

Introduction to Waves

A wave is a disturbance that transfers energy from one point to another without the net transfer of matter. Waves are classified as mechanical (require a medium — sound, water, string waves) and electromagnetic (do not require a medium — light, radio waves). In NEET physics, mechanical waves, particularly sound waves and waves on strings, are the primary focus.

Key parameters of a wave: Wavelength (λ) — distance between two consecutive identical points; Frequency (f) — number of oscillations per second (Hz); Time period (T) — time for one complete oscillation (T = 1/f); Amplitude (A) — maximum displacement from equilibrium; Wave speed (v) — speed of propagation (v = fλ).

Types of Waves

Transverse Waves

In transverse waves, particles oscillate perpendicular to the direction of wave propagation. Examples: waves on a string, light waves, ripples on water surface. The wave profile at any instant is a sine or cosine function. Crests are the highest points, troughs are the lowest points. Distance between successive crests or troughs equals one wavelength.

Longitudinal Waves

In longitudinal waves, particles oscillate parallel to the direction of wave propagation. Examples: sound waves in air, seismic P-waves. The wave consists of alternate compressions (high pressure/density regions) and rarefactions (low pressure/density regions). Distance between successive compressions or rarefactions equals one wavelength. Sound waves in fluids are always longitudinal; in solids, they can be both longitudinal and transverse.

Progressive Waves

A progressive (travelling) wave transfers energy continuously. The wave function for a sinusoidal progressive wave travelling in the positive x-direction: y(x,t) = A sin(ωt − kx + φ) where ω = 2πf (angular frequency), k = 2π/λ (wave number), and φ is the initial phase. For a wave travelling in the negative x-direction: y(x,t) = A sin(ωt + kx + φ).

Wave Equation & Speed

Speed of Transverse Waves on a String

The speed of a transverse wave on a stretched string depends on tension T and linear mass density μ (mass per unit length): v = √(T/μ). This is derived from Newton's second law applied to a small element of the string. The speed is independent of frequency and amplitude — only the medium properties matter.

Speed of Sound Waves

The speed of longitudinal waves (sound) in a medium depends on the elastic properties and density. In a gas: v = √(γP/ρ) = √(γRT/M) where γ = C_p/C_v (adiabatic index), P is pressure, ρ is density, M is molar mass, and T is temperature. In air at NTP: v ≈ 332 m/s. The speed of sound increases with temperature: v_t = v₀√(1 + t/273) ≈ v₀ + 0.6t m/s.

In solids: v = √(Y/ρ) for longitudinal waves, v = √(G/ρ) for transverse waves. In liquids: v = √(B/ρ) where B is bulk modulus.

Wave Equation — Detailed Derivation

The wave function for a sinusoidal progressive wave travelling in the positive x-direction is derived from the fact that the disturbance (displacement) at any point x and time t is a function of the form y = f(t − x/v). For a periodic wave, the source executes simple harmonic motion: y(0,t) = A sin(ωt). If the wave travels with speed v, the disturbance at a point x is delayed by time x/v. Therefore:

y(x,t) = A sin[ω(t − x/v)] = A sin(ωt − ωx/v)

Since ω/v = 2πf/v = 2π/λ = k (wave number), we get the standard form: y(x,t) = A sin(ωt − kx + φ) where φ is the initial phase. This equation satisfies the one-dimensional wave equation: ∂²y/∂t² = v² ∂²y/∂x².

Phase Difference & Path Difference

Phase difference (Δφ) and path difference (Δx) are related by: Δφ = (2π/λ) Δx. Similarly, time difference Δt is related as: Δφ = ωΔt = 2πfΔt. Points separated by a path difference of λ have a phase difference of 2π (they are in phase). Points separated by λ/2 have a phase difference of π (they are out of phase).

For two waves of the same frequency travelling in the same direction, the resultant amplitude depends on their phase difference: if Δφ = 0, 2π, 4π,... → constructive interference (A_R = A₁ + A₂). If Δφ = π, 3π, 5π,... → destructive interference (A_R = |A₁ − A₂|).

Phase Velocity

The phase velocity (wave speed) is the speed at which a point of constant phase travels: v_p = ω/k = fλ. For non-dispersive media (e.g., sound in air, waves on an ideal string), the phase velocity is independent of frequency. In dispersive media, different frequencies travel at different speeds, causing a wave pulse to spread out over time.

Sound Waves

Characteristics of Sound

Sound is a longitudinal mechanical wave. Human ear can detect frequencies from 20 Hz to 20,000 Hz. Frequencies below 20 Hz are infrasonic (earthquakes, whales), above 20 kHz are ultrasonic (bats, medical imaging). The loudness of sound depends on intensity (power per unit area): I = ½ρvω²A². Sound level in decibels: β = 10 log(I/I₀) where I₀ = 10⁻¹² W/m² (threshold of hearing). Threshold of pain: 120 dB (I = 1 W/m²).

Intensity & Inverse Square Law

The intensity of sound from a point source decreases with distance: I ∝ 1/r². The amplitude decreases as 1/r. The sound level in dB decreases by approximately 6 dB for each doubling of distance from a point source.

Reflection & Echo

Sound waves obey the laws of reflection (angle of incidence = angle of reflection). An echo is heard when the reflected sound reaches the ear with a delay ≥ 0.1 s (requires reflecting surface at least 17 m away). Reverberation is the persistence of sound due to multiple reflections. Acoustics of halls is designed to optimise reverberation time.

Reflection of Waves — Phase Change

When a wave is reflected from a boundary, the phase change depends on the nature of the boundary:

Rigid boundary (fixed end): The reflected wave undergoes a phase change of π (180°). This is equivalent to a path difference of λ/2. For a transverse wave on a string fixed at the end, the displacement of the fixed point must always be zero, so the reflected wave is inverted relative to the incident wave. The incident and reflected waves interfere to produce a node at the rigid boundary.

Free boundary (free end): The reflected wave undergoes no phase change (0°). The end is free to move, and the reflected wave is not inverted. An antinode is formed at the free boundary.

Sound waves: When sound reflects from a rigid wall (closed end of a pipe), a displacement node (pressure antinode) is formed. When sound reflects from an open end, a displacement antinode (pressure node) is formed. This explains why closed pipes have a node at the closed end and an antinode at the open end.

Phase Change on Reflection — Comparison

Table 3: Phase Change on Reflection

Boundary Type	Phase Change	Path Difference Equivalent	Result at Boundary	Example
Rigid (Fixed)	π (180°)	λ/2	Node	String fixed end, closed pipe
Free (Open)	0°	0	Antinode	String free end, open pipe
Rarer to denser	π (180°)	λ/2	Node-like	Sound: air to water
Denser to rarer	0°	0	Antinode-like	Sound: water to air

Intensity & Amplitude Relationship

The intensity of a wave is proportional to the square of its amplitude: I ∝ A². For a sound wave: I = ½ρvω²A² where ρ is the density of the medium, v is the wave speed, and ω is the angular frequency. This means that doubling the amplitude quadruples the intensity, and increasing the frequency (keeping amplitude constant) also increases the intensity proportionally to f².

In interference of two waves of equal amplitude A and phase difference Δφ, the resultant amplitude: A_R = 2A cos(Δφ/2). The resultant intensity: I_R = I₁ + I₂ + 2√(I₁I₂) cos(Δφ). For equal intensities I₀: I_R = 4I₀ cos²(Δφ/2). At constructive interference (Δφ = 0): I_R = 4I₀. At destructive interference (Δφ = π): I_R = 0.

The loudness of sound is logarithmic: β = 10 log(I/I₀) dB. A 10× increase in intensity corresponds to a 10 dB increase in sound level. A 100× increase corresponds to 20 dB.

Interference of Sound Waves

When two sound waves of the same frequency travel in the same medium and meet at a point, they interfere. The resultant displacement is the algebraic sum of the individual displacements (superposition principle). The nature of interference depends on the phase difference between the waves arriving at that point.

Constructive interference occurs when the waves arrive in phase (path difference = nλ). The resultant amplitude is maximum (A₁ + A₂). For sound, this produces a loud sound. Destructive interference occurs when the waves arrive out of phase (path difference = (2n+1)λ/2). The resultant amplitude is minimum (|A₁ − A₂|), producing a soft or silent region.

For two sources emitting sound in phase, the condition for constructive interference at a point is: path difference = nλ, and for destructive interference: path difference = (2n+1)λ/2. If the sources are out of phase by π, these conditions are reversed.

In Young's double-slit experiment with sound (using two loudspeakers driven by the same oscillator), alternating loud and soft regions are observed as the listener moves across the room. The angular position of maxima: sin θ = nλ/d where d is the separation between sources.

Characteristics of Musical Sound

Musical sound has three distinguishing characteristics: Pitch (determined by frequency — higher frequency = higher pitch), Loudness (determined by intensity/amplitude — greater amplitude = louder sound, measured in dB), and Quality or Timbre (determined by the waveform shape / presence of harmonics — distinguishes the same note played on different instruments).

A musical note consists of a fundamental frequency (first harmonic) along with several overtones (higher harmonics). The relative amplitudes of these harmonics determine the timbre. A pure tone has only the fundamental frequency (sine wave). Most musical instruments produce complex waves with many harmonics. The human ear can distinguish these differences, allowing us to identify different instruments playing the same note.

Consonance and Dissonance: Certain frequency combinations sound pleasant (consonance — e.g., octave 2:1, fifth 3:2, fourth 4:3). Others sound harsh (dissonance — e.g., semitone 16:15). The concept of beats explains this — consonant intervals have simple frequency ratios and produce few or no audible beats.

Standing (Stationary) Waves

Standing waves are formed by the superposition of two identical waves travelling in opposite directions. The wave function: y(x,t) = 2A sin(kx) cos(ωt). In standing waves, all particles oscillate with the same frequency but different amplitudes. Points of zero displacement are called nodes (permanent rest). Points of maximum displacement are called antinodes. Distance between successive nodes = λ/2. Distance between successive antinodes = λ/2. Distance between a node and adjacent antinode = λ/4.

Energy is not transported in standing waves — it is trapped between nodes. The medium particles oscillate between extreme positions, converting potential to kinetic energy and back. Standing waves represent the normal modes of vibration of a bounded system — only specific frequencies (eigenfrequencies) produce sustained standing waves.

Formation of Standing Waves

When a travelling wave reflects from a rigid boundary, it undergoes a phase change of π (equivalent to λ/2 path difference). From a free boundary, no phase change occurs. Standing waves are produced when a wave and its reflected counterpart interfere. In strings fixed at both ends, nodes form at the ends; in open pipes, antinodes form at the open ends.

Vibrations in Strings & Pipes

Standing Waves in Stretched Strings

For a string fixed at both ends of length L, standing waves exist only at specific frequencies (natural frequencies or harmonics). The fundamental frequency (first harmonic): f₁ = v/2L. Higher harmonics: f_n = nv/2L for n = 1, 2, 3... The integer n denotes the number of loops (segments). The wavelength of the nth harmonic: λ_n = 2L/n.

Laws of vibrating strings: (i) f ∝ 1/L (law of length), (ii) f ∝ √T (law of tension), (iii) f ∝ 1/√μ (law of mass). These are collectively known as Sonometer laws.

Standing Waves in Air Columns (Pipes)

Closed pipe (closed at one end, open at the other): The closed end is always a node (displacement) / pressure antinode. The open end is always an antinode (displacement) / pressure node. Fundamental frequency: f₁ = v/4L. Only odd harmonics exist: f_n = (2n−1)v/4L for n = 1, 2, 3,... Wavelength: λ_n = 4L/(2n−1). The first three resonances correspond to n = 1 (fundamental, f₁), n = 2 (third harmonic, 3f₁), n = 3 (fifth harmonic, 5f₁).

Open pipe (open at both ends): Both ends are displacement antinodes (pressure nodes). Fundamental frequency: f₁ = v/2L. All harmonics exist: f_n = nv/2L. Wavelength: λ_n = 2L/n. The first three resonances are the fundamental (f₁), second harmonic (2f₁), and third harmonic (3f₁).

End correction: For real pipes, the effective length is slightly more than the geometric length because the antinode lies slightly beyond the open end: L_eff = L + e where e = 0.6r (for a circular pipe of radius r) or approximately 0.3d where d is the diameter. The end correction is important in accurate determination of the speed of sound using resonance column experiments. For a closed pipe: f₁ = v/4(L + e). For an open pipe: f₁ = v/2(L + 2e) (both ends have end correction).

String Instruments — Laws of Vibration

The frequency of vibration of a stretched string depends on three factors:

(i) Law of length: f ∝ 1/L (for constant T and μ). Doubling the length halves the frequency. This is why bass strings in musical instruments are longer than treble strings.

(ii) Law of tension: f ∝ √T (for constant L and μ). Increasing tension increases frequency. This is how string instruments are tuned — tightening increases pitch.

(iii) Law of mass: f ∝ 1/√μ (for constant L and T). Thicker strings (higher μ) produce lower frequencies. This is why the bass strings of a guitar are thicker than the treble strings.

Combining all three: f = (1/2L) √(T/μ) which is exactly f₁ = v/2L since v = √(T/μ).

Sonometer Experiment

A sonometer (or monochord) is used to verify the laws of vibrating strings. It consists of a hollow wooden box with a long wire stretched over two bridges. One end of the wire is fixed, the other passes over a pulley carrying a load to provide tension. The length of the vibrating segment can be adjusted by moving a movable bridge. A tuning fork of known frequency is struck and held near the wire, and the length is adjusted until the wire vibrates in resonance with the fork (maximum loudness). Using the sonometer, one can verify:

(a) f ∝ 1/L: For constant T and μ, measure L for different tuning forks. fL = constant.

(b) f ∝ √T: For constant L, vary tension and find the resonant frequency using a set of tuning forks.

(c) f ∝ 1/√μ: Use wires of different materials/thicknesses (different μ) keeping L and T constant.

The sonometer is also used to determine the unknown frequency of a tuning fork and to find the AC mains frequency using a magnet and solenoid arrangement.

Echo, Reverberation & Acoustics of Buildings

Echo: An echo is the reflection of sound that reaches the listener after a delay of at least 0.1 s (the persistence of hearing). Since the speed of sound in air is approximately 340 m/s, the minimum distance for a distinct echo is: d = vt/2 = 340 × 0.1/2 = 17 m. If the reflecting surface is closer than 17 m, the reflected sound overlaps with the original sound and is perceived as reverberation rather than a distinct echo.

Multiple Echoes: When sound reflects multiple times between two or more surfaces, multiple echoes are heard. This is commonly observed in hills, canyons, large empty halls, and stairwells. The time interval between successive echoes depends on the distances between reflecting surfaces.

Reverberation: Reverberation is the persistence of sound in an enclosed space after the original sound source has stopped. It is caused by multiple reflections of sound from walls, ceiling, and floor. The reverberation time (RT₆₀) is the time required for the sound intensity to decay by 60 dB (i.e., to one-millionth of its original intensity) after the source stops. Sabine's formula for reverberation time is: RT = 0.161V/A, where V is the volume of the room (in m³) and A is the total absorption (in sabins). An optimal reverberation time is crucial for different applications: concert halls (1.8–2.2 s), theatres (1.0–1.5 s), classrooms (0.5–0.8 s), recording studios (0.2–0.4 s).

Acoustic Treatment: Sound absorption can be increased using porous materials (fibreglass, foam, carpets, curtains), resonant panels, or Helmholtz resonators. Diffusers scatter sound waves to prevent echoes and standing waves. Bass traps absorb low-frequency sound. The choice of materials and their placement determines the acoustic quality of a hall.

Sabine's Law: Wallace Sabine found empirically that RT = 0.161V/Σ(α_iS_i), where α_i is the absorption coefficient of surface i (ranging from 0 for perfect reflection to 1 for perfect absorption) and S_i is its surface area. Common absorption coefficients: brick wall 0.03, carpet 0.3–0.6, acoustic tiles 0.5–0.9, audience 0.8–0.9 per person.

Human Ear & Hearing

The human ear is a remarkable transducer that converts sound waves (pressure variations) into electrical signals that the brain interprets. The audible frequency range for young humans is typically 20 Hz to 20,000 Hz (20 kHz). With age, the upper limit gradually decreases to about 12–16 kHz by middle age — this is called presbycusis (age-related hearing loss).

Structure of the Ear: The outer ear (pinna and ear canal) collects and directs sound waves to the eardrum. The middle ear (three small bones: malleus, incus, stapes) amplifies the vibrations and transmits them to the oval window of the inner ear. The inner ear (cochlea) is a fluid-filled, spiral-shaped organ lined with hair cells of different lengths, each tuned to a specific frequency. Different regions of the cochlea respond to different frequencies (tonotopy): the base responds to high frequencies, the apex to low frequencies.

Loudness Perception: The perceived loudness depends on both intensity and frequency. The threshold of hearing is about 10⁻¹² W/m² at 1 kHz. The threshold of pain is about 1 W/m². The ear is most sensitive in the 2–5 kHz range (conversational speech frequencies). Equal-loudness contours (Fletcher–Munson curves) show the sound pressure level required at different frequencies to produce the same perceived loudness.

Sound Localisation: Humans can localise the direction of sound using two cues: interaural time difference (ITD) — the time delay between sound reaching the two ears (effective for frequencies below 1.5 kHz when the wavelength is larger than the head); and interaural level difference (ILD) — the difference in sound intensity at the two ears (effective for frequencies above 1.5 kHz when the head acts as a barrier creating a sound shadow).

Resonance & Tuning

Resonance occurs when a system is driven at its natural frequency, resulting in a large amplitude of vibration. The driving force is in phase with the velocity of the oscillator, maximising energy transfer. In the absence of damping, the amplitude would theoretically become infinite at resonance.

Examples of resonance:

Tacoma Narrows Bridge (1940): Wind-induced vibrations matched the bridge's natural frequency, causing large-amplitude torsional oscillations that destroyed the bridge. This is a classic cautionary example of mechanical resonance.

Musical instruments: The air column in a flute/guitar body resonates at specific frequencies, amplifying certain harmonics. The soundboard of a piano or violin resonates to amplify the string vibrations. The resonance of the air column in an organ pipe determines the pitch produced.

Other examples: Tuning a radio circuit (electrical resonance), microwave oven (water molecules resonate at ~2.45 GHz), breaking a wine glass with sound (acoustic resonance), and soldiers breaking step while crossing a bridge to avoid resonant excitation.

In the context of sound, resonance is used to determine the speed of sound (resonance column experiment), to tune instruments, and in the design of musical instruments and auditoriums.

Resonance Column Experiment

This experiment uses a closed pipe with variable length (water column) to determine the speed of sound. The first resonance occurs at L₁ = λ/4 (approximately). The second resonance at L₂ = 3λ/4. Therefore, λ = 2(L₂ − L₁) and v = fλ. This is a classic NEET experiment-based question.

Beats

When two sound waves of nearly equal frequencies (f₁ and f₂) travel in the same direction and superimpose, the intensity at a point varies periodically with time. This phenomenon is called beats. The beat frequency: f_beat = |f₁ − f₂|. The number of beats per second equals the difference in frequencies. Beat period T_beat = 1/f_beat.

Beat Frequency — Mathematical Derivation

Consider two waves of equal amplitude A and slightly different angular frequencies ω₁ and ω₂: y₁ = A sin(ω₁t − k₁x) and y₂ = A sin(ω₂t − k₂x). At a fixed point x = 0 (for simplicity), the displacements are: y₁ = A sin(ω₁t) and y₂ = A sin(ω₂t). By the principle of superposition:

y = y₁ + y₂ = A[sin(ω₁t) + sin(ω₂t)]

Using the trigonometric identity sin α + sin β = 2 sin((α+β)/2) cos((α−β)/2):

y = 2A cos[(ω₁ − ω₂)t/2] × sin[(ω₁ + ω₂)t/2]

The resultant wave has an average angular frequency (ω₁ + ω₂)/2 (≈ either frequency) and its amplitude is modulated by the cos term with angular frequency (ω₁ − ω₂)/2. The intensity varies as cos²((ω₁ − ω₂)t/2). The intensity maxima occur twice per modulation cycle (when cos = ±1), so the beat frequency is twice the modulation frequency:

f_beat = 2 × [(ω₁ − ω₂)/4π] = |f₁ − f₂|

The human ear can distinguish beats up to about 10–15 beats per second. Beyond that, the beats merge into a continuous roughness or dissonance.

Graphical Representation of Beats

If we plot the displacement of the two individual waves and their resultant, we observe that the resultant amplitude varies sinusoidally between 2A (constructive interference) and 0 (destructive interference). The envelope of the resultant wave is given by 2A cos(π(f₁−f₂)t). The time interval between successive maxima (loud sounds) is: t_max = 1/|f₁ − f₂|. During one beat period, the resultant wave completes approximately (f₁ + f₂)/2f_beat oscillations.

Applications of beats include: tuning musical instruments (two instruments produce beats when slightly out of tune; adjusting to minimise beats brings them into unison), detecting small frequency differences in Doppler shift measurements, and in heterodyne detection used in radio receivers and ultrasonic imaging.

Beats in instrument tuning: When tuning a violin or guitar, the musician sounds a reference tuning fork alongside the string. If beats are heard, the string is slightly out of tune. The tension is adjusted until the beat frequency drops to zero (unison). Similarly, piano tuners use beat frequencies to tune intervals.

Beats with more than two waves: When multiple waves with closely spaced frequencies interfere, the beat pattern becomes more complex. For n waves with frequencies f, f+Δf, f+2Δf, ..., the resultant is a periodic pulse train — this is the principle behind mode-locked lasers and certain musical effects like chorus and phasing.

Detection of partials in musical instruments: Beats can also occur between the harmonics of a single note (e.g., between the 2nd and 3rd harmonics if they are not perfectly harmonic). This causes the sensation of "roughness" and is an important factor in the perception of musical consonance and dissonance.

Doppler Effect

The Doppler effect is the apparent change in frequency of a wave when the source, observer, or both are in relative motion. For sound waves, the formula depends on the velocities of source (v_s) and observer (v_o) relative to the medium:

f' = f × (v ± v_o) / (v ∓ v_s)

where v is the speed of sound in the medium. Sign convention: Numerator: + if observer moves towards source, − if away. Denominator: − if source moves towards observer, + if away. The apparent frequency increases when source and observer approach each other, and decreases when they recede.

Derivation of Doppler Effect Formulas

Case 1 — Source moving towards stationary observer: The source moves with velocity v_s towards the observer. In one second, the source emits f waves. The first wave travels a distance v, while the source itself moves v_s. The waves are compressed into a shorter distance (v − v_s). The apparent wavelength: λ' = (v − v_s)/f. The apparent frequency: f' = v/λ' = vf/(v − v_s).

Case 2 — Source moving away from stationary observer: The waves are stretched: λ' = (v + v_s)/f. Apparent frequency: f' = vf/(v + v_s).

Case 3 — Observer moving towards stationary source: The observer intercepts more waves per second. Relative speed of sound with respect to observer = v + v_o. The apparent frequency: f' = (v + v_o)/λ = (v + v_o)f/v.

Case 4 — Observer moving away from stationary source: Relative speed = v − v_o. Apparent frequency: f' = (v − v_o)f/v.

Case 5 — Both source and observer moving: Combine the effects: f' = f(v ± v_o)/(v ∓ v_s). The signs depend on directions of motion relative to each other.

Effect of wind: If wind is blowing with speed w from source to observer, the effective speed of sound changes to v_eff = v ± w (depending on direction). Replace v with v_eff in all formulas. Note that wind affects the wave speed, not the source or observer velocities relative to the ground.

Supersonic motion: When v_s > v, the source moves faster than the waves it produces. The wavefronts form a cone (Mach cone) with half-angle θ given by sin θ = v/v_s. The ratio v_s/v is called the Mach number. The Doppler formula breaks down when v_s ≥ v as the denominator becomes zero or negative.

Apparent Frequency Formulas — Quick Reference

Table 4: Doppler Effect — All Cases

Scenario	Source Motion	Observer Motion	Apparent Frequency f'	Effect
Source → stationary O	Towards O (vs)	Stationary	f × v/(v − vs)	Increases
Source ← stationary O	Away from O (vs)	Stationary	f × v/(v + vs)	Decreases
Stationary S → observer	Stationary	Towards S (vo)	f × (v + vo)/v	Increases
Stationary S ← observer	Stationary	Away from S (vo)	f × (v − vo)/v	Decreases
Both towards each other	Towards O (vs)	Towards S (vo)	f × (v + vo)/(v − vs)	Increases (strong)
Both away from each other	Away from O (vs)	Away from S (vo)	f × (v − vo)/(v + vs)	Decreases (strong)
Source → away, O → towards	Away from O (vs)	Towards S (vo)	f × (v + vo)/(v + vs)	Depends on magnitudes
Perpendicular motion	Perpendicular to S-O line	Any	f (unchanged)	No change

Special Cases of Doppler Effect

1. Source moving, observer stationary: f' = f × v/(v ∓ v_s).

2. Observer moving, source stationary: f' = f × (v ± v_o)/v.

3. Both source and observer moving: Use the general formula with appropriate signs.

4. Wind blowing from source to observer: Effective speed of sound changes. If wind speed is w in the direction from source to observer: v_eff = v + w for the forward direction.

5. Source moving perpendicular to the line joining source-observer: No Doppler shift (only radial component matters).

6. Source moving supersonically (v_s > v): The source outruns its own waves, producing a conical shock wave (Mach cone). The Mach angle: sin θ = v/v_s. This is the basis of sonic booms.

Doppler Effect in Reflection (Echo)

When sound reflects from a moving surface (e.g., a moving wall), the Doppler shift occurs twice — the wall acts first as an observer (moving), then as a source (moving at the same velocity). The final apparent frequency after reflection: f'' = f × (v + v_w)/(v − v_w) for a wall moving towards the source. This principle is used in radar speed guns and medical ultrasound (Doppler echocardiography).

Applications of Doppler Effect

Medical Ultrasound (Echocardiography): Ultrasonic waves (2–10 MHz) are directed at blood cells. The reflected waves undergo a Doppler shift proportional to the velocity of blood flow. By analysing the shift, doctors can measure blood flow velocity in the heart and blood vessels, detect blockages, and assess heart valve function. Colour Doppler imaging maps blood flow velocities onto a 2D image.

Radar Speed Guns: A radar gun emits microwaves that reflect off a moving vehicle. The frequency shift between the emitted and received waves is proportional to the vehicle's speed: Δf = 2f₀v/c, where c is the speed of light. The factor of 2 accounts for the double Doppler shift (out and back).

Astronomy (Redshift): The Doppler effect for light waves causes a shift in spectral lines. When a star or galaxy moves away from Earth, its spectral lines shift to longer wavelengths (redshift). The amount of redshift reveals the recessional velocity, which Edwin Hubble used to discover the expansion of the universe. Blueshift occurs when objects move towards Earth.

Flow Measurement (Laser Doppler Velocimetry): A laser beam is split and directed at a flowing fluid. Light scattered by particles in the fluid is Doppler-shifted. By measuring the shift, the flow velocity can be determined without disturbing the flow.

Weather Radar: Doppler radar is used in meteorology to detect the motion of raindrops and hail. The frequency shift indicates the speed and direction of wind and precipitation, helping forecasters track storms and tornadoes.

Military Sonar: Submarines use active sonar (sound navigation and ranging) to detect objects. The Doppler shift of the reflected sound reveals whether the target is moving towards or away from the submarine, and its speed.

Ultrasonic Waves & Applications

Ultrasonic waves are sound waves with frequencies above 20 kHz (the upper limit of human hearing). They have important applications due to their high frequency (short wavelength), which allows better resolution and directional beams. Common frequencies used in applications range from 20 kHz to several GHz.

Properties of ultrasonic waves: (i) They are highly directional and can be focused into narrow beams. (ii) They can penetrate solids and liquids. (iii) They produce cavitation in liquids (formation and collapse of tiny bubbles, releasing energy). (iv) They are reflected at interfaces between different media (used in echolocation). (v) They cause heating when absorbed by materials.

Applications: (1) Medical imaging (sonography) — ultrasonic waves are reflected from different tissue layers; the echoes are processed to create images of internal organs, foetus, etc. (2) Therapeutic ultrasound — used in physiotherapy for deep heating of tissues, and in surgery (high-intensity focused ultrasound, HIFU) to destroy tumours. (3) Sonar (Sound Navigation and Ranging) — ships and submarines use ultrasonic waves to detect underwater objects and measure depth. (4) Non-destructive testing (NDT) — ultrasonic waves are used to detect cracks and flaws in metal structures, welds, and pipelines. (5) Industrial cleaning — ultrasonic baths use cavitation to remove dirt from instruments, jewellery, and electronic components. (6) Insect and rodent repellers — devices that emit high-frequency sound to deter pests. (7) Welding plastics — ultrasonic vibrations generate heat at the interface of plastic parts, fusing them together.

Detection of ultrasonic waves: Since they cannot be heard by the human ear, special detectors are used: piezoelectric transducers (quartz or ceramic crystals that generate electric signals when subjected to ultrasonic pressure), magnetostriction oscillators, and microphones designed for high-frequency response. The same transducer often serves as both transmitter and receiver (pulse-echo principle).

In NEET, questions on ultrasonics typically focus on their properties (directional, high energy, penetrative) and the principle of echo-based distance measurement: distance = (v × t)/2, where t is the time between emission and reception of the reflected pulse.

Shock Waves & Sonic Boom

When a source moves through a medium at a speed greater than the wave speed (v_s > v), it creates a shock wave. The wavefronts form a cone (Mach cone) with the source at the apex. The half-angle of the cone (Mach angle) is given by: sin θ = v/v_s = 1/M, where M = v_s/v is the Mach number.

When an aircraft breaks the sound barrier (M > 1), the shock wave produces a sonic boom — a sudden, loud sound heard on the ground as the cone of compressed air passes by. The sonic boom is not a single event at the moment of breaking the barrier; rather, it is continuously generated as long as the aircraft travels supersonically. The boom is heard as two distinct thumps (from the nose and tail shock waves).

Key points for NEET: (1) Subsonic: v_s < v (M < 1). (2) Supersonic: v_s > v (M > 1). (3) Mach angle decreases as speed increases: sin θ = v/v_s. (4) The Doppler formula is not applicable for v_s ≥ v because the denominator becomes zero or negative — the source outruns its own waves. (5) A bullet fired from a rifle produces a mini sonic boom (crack sound) as it travels supersonically. (6) The crack of a whip is a mini sonic boom — the tip of the whip moves faster than sound.

Numerical Problem Solving — Waves

Solving numerical problems in waves requires a systematic approach. Here are step-by-step strategies for each type of problem commonly asked in NEET:

Strategy 1 — Wave Parameters from Equation

Given y = A sin(ωt ± kx + φ), identify: Amplitude A (coefficient of sin), ω (coefficient of t), k (coefficient of x). Then: f = ω/2π, T = 2π/ω, λ = 2π/k, v = ω/k = fλ. The sign determines direction: − for +x direction, + for −x direction.

Strategy 2 — Standing Wave Problems

Identify boundary conditions: (a) String fixed at both ends — nodes at ends, f_n = nv/2L. (b) Open pipe — antinodes at both ends, f_n = nv/2L. (c) Closed pipe — node at closed end, antinode at open end, f_n = (2n−1)v/4L. For string, n = number of loops. For pipes, n = 1,2,3,... For closed pipe, remember only odd harmonics exist.

Strategy 3 — Doppler Effect Problems

Step 1: Draw a diagram showing source, observer, and their velocities. Step 2: Determine direction of motion relative to each other. Step 3: Apply formula f' = f(v ± v_o)/(v ∓ v_s). Step 4: For reflection, treat the reflector as first an observer, then a source. Step 5: For wind, adjust the effective speed of sound. Remember: only the component of velocity along the line joining source and observer matters.

Strategy 4 — Beat Frequency Problems

Step 1: f_beat = |f₁ − f₂|. Step 2: For unknown fork, f = f_known ± f_beat. Step 3: To determine sign, use loading (decreases f) or filing (increases f). Step 4: If beat frequency increases after loading, unknown < known. If beat frequency decreases after loading, unknown > known.

Strategy 5 — Speed of Sound Problems

Use v = √(γRT/M) for gases. Remember v ∝ √T (at constant composition). v_t = v₀√(1 + t/273) ≈ v₀ + 0.6t. For resonance column: λ = 2(L₂ − L₁), v = fλ, end correction e = (L₂ − 3L₁)/2.

Common NEET Pitfalls to Avoid

(1) Confusing open and closed pipe harmonics — open: all harmonics, closed: only odd. (2) Forgetting the factor of 2 in round-trip problems (echo, sonar). (3) Using the wrong sign in Doppler effect — always check "towards = increases frequency". (4) Confusing ω (angular frequency) with f (frequency) — ω = 2πf. (5) Mixing up node and antinode positions — node: zero displacement, antinode: maximum displacement. (6) Forgetting that wave speed on a string depends on T and μ only, not on f. (7) Applying v = fλ incorrectly — this holds for all waves but the values of f and λ change according to the medium.

Quick Comparison Tables

Table 1: Transverse vs Longitudinal Waves

Property	Transverse Waves	Longitudinal Waves
Particle vibration	Perpendicular to propagation	Parallel to propagation
Requires medium?	Mechanical ones do; EM do not	Always requires a material medium
Can travel in solids?	Yes (and on surfaces)	Yes
Can travel in fluids?	No (except surface waves)	Yes
Can travel in vacuum?	EM waves can; mechanical cannot	No
Wave speed formula (solids)	v = √(G/ρ)	v = √(Y/ρ)
Wave speed formula (fluids)	N/A	v = √(B/ρ)
Waveform features	Crests & troughs	Compressions & rarefactions
Periodic shape	Can be polarized	Cannot be polarized
Examples	String waves, light, water ripples, seismic S-waves	Sound in air/water, seismic P-waves, spring slinky
Energy transfer	Transports energy perpendicular to vibration	Transports energy parallel to vibration

Table 2: Open vs Closed Pipe Comparison

Property	Open Pipe	Closed Pipe
End conditions	Both ends open (A at ends)	One closed (N), one open (A)
Displacement at ends	Antinode at both ends	Node at closed end, antinode at open end
Pressure at ends	Node at both ends	Antinode at closed end, node at open end
Fundamental frequency	f1 = v/2L	f1 = v/4L
Harmonics present	All harmonics (n = 1,2,3,...)	Only odd harmonics (n = 1,3,5,...)
General frequency	fn = nv/2L	fn = (2n−1)v/4L
Fundamental wavelength	λ1 = 2L	λ1 = 4L
Second resonance	2nd harmonic (2f1)	3rd harmonic (3f1)
Third resonance	3rd harmonic (3f1)	5th harmonic (5f1)
End correction	Leff = L + 2e (both ends)	Leff = L + e (one end)

Key NEET Formulas — Waves

⚠️ NEET Exam Tips & Common Mistakes

▶ Frequently Made Mistakes

❌ Frequency vs velocity: Frequency of a wave depends ONLY on the source, not on the medium. Wavelength and velocity change with medium, but frequency stays constant.

❌ Phase difference vs path difference: Δφ = (2π/λ)Δx. A path difference of λ corresponds to a phase difference of 2π (360°). Don't confuse the two.

❌ Wave equation sign convention: y = A sin(ωt − kx) → wave travelling in +x direction. y = A sin(ωt + kx) → wave travelling in −x direction. The sign before kx determines direction.

❌ Beats formula: Beat frequency = |f₁ − f₂|. The number of beats per second equals the difference in frequencies, NOT the sum.

❌ Doppler effect sign: When source moves TOWARDS observer: apparent f increases (higher pitch). When source moves AWAY: f decreases. Use the correct sign convention: f' = f(v±v₀)/(v∓v_s).

❌ Open vs closed pipe overtones: Open pipe: all harmonics present (f₀, 2f₀, 3f₀...). Closed pipe: only odd harmonics (f₀, 3f₀, 5f₀...). NEET frequently tests this difference.

❌ Node vs antinode: At nodes, displacement is always zero (destructive interference). At antinodes, displacement is maximum (constructive interference). Distance between adjacent nodes = λ/2.

❌ Transverse vs longitudinal: Light = transverse (particles vibrate ⟂ to wave direction). Sound = longitudinal (particles vibrate ∥ to wave direction). Remember: sound cannot travel in vacuum.

▶ Shortcut Formulas for NEET

⚡ Wave speed on a string: v = √(T/μ). T = tension, μ = mass per unit length.

⚡ Speed of sound in air: v = √(γP/ρ) = √(γRT/M). At STP, v ≈ 332 m/s. Increases by 0.6 m/s per °C rise.

⚡ Fundamental frequency of stretched string: f₀ = (1/2L)√(T/μ).

⚡ For open pipe: f₀ = v/2L, for closed pipe: f₀ = v/4L.

⚡ Beat frequency = |f₁ − f₂|. If you hear x beats/sec and one frequency is known, the unknown is f±x.

⚡ Doppler effect for sound: f' = f(v±v₀)/(v∓v_s). Numerator: + if observer moves toward source. Denominator: − if source moves toward observer.

⚡ In resonance tube experiment: first resonance = λ/4, second = 3λ/4. End correction e = (l₂ − 3l₁)/2.

Practice Questions