Waves

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Waves — Key Facts Wave equation: v = fλ; v = √(T/μ) for string (T=tension, μ=linear density) Transverse: displacement perpendicular to propagation (string, light); Longitudinal: displacement parallel (sound, spring) Doppler Effect: f’ = f × (v ± v₀)/(v ∓ v_s); source moving toward observer → higher pitch Stationary waves: nodes (zero amplitude) and antinodes (maximum amplitude); fixed end → node ⚡ Exam tip: Doppler applies to ALL waves (sound, light, water); formula differs slightly for light

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Waves — NEET/JEE Study Guide Wave types: mechanical (require medium) vs electromagnetic (no medium; c=3×10⁸ m/s) Doppler: moving source: denominator; moving observer: numerator; both: combine effects Beats: f_beat = |f₁ - f₂|; heard when two close frequencies interfere; used in tuning Standing waves on string fixed at both ends: L = n(λ/2), f_n = n(v/2L); fundamental n=1 Open pipe: L = n(λ/2); closed pipe: L = n(λ/4) for odd n only; overtones differ

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Waves — Comprehensive Notes

Wave Parameters and Classification

A wave is a disturbance that transfers energy through a medium without permanent displacement of the medium itself. The fundamental wave equation relates velocity (v), frequency (f), and wavelength (λ):

$$v = f\lambda = \frac{\lambda}{T}$$

where T is the time period. Wave speed in a stretched string is given by:

$$v = \sqrt{\frac{T}{\mu}}$$

where T is tension and μ = m/L is linear mass density (kg/m). This derivation comes from comparing wave equation with Newton’s second law applied to a small element of the string.

Transverse waves have particle displacement perpendicular to wave propagation direction. Examples include waves on strings, electromagnetic waves, and water waves. The wave profile moves horizontally while particles oscillate vertically.

Longitudinal waves have particle displacement parallel to propagation direction. Sound waves in air are the primary example. Compression and rarefaction regions travel through the medium. The speed of longitudinal waves in a fluid is:

$$v = \sqrt{frac{B}{ ho}}$$

where B is bulk modulus and ρ is density.

Wave Equation and Superposition

For a wave travelling in +x direction: y = A sin(kx - ωt + φ), where k = 2π/λ is wave number and ω = 2πf is angular frequency. The general solution to the wave equation ∂²y/∂x² = (1/v²)∂²y/∂t² is:

$$y(x,t) = f_1(vt - x) + f_2(vt + x)$$

representing waves travelling in +x and -x directions respectively. Superposition principle applies — when two or more waves occupy the same medium, the resultant displacement is the vector sum of individual displacements.

Phase velocity for deep water waves: v = √(gλ/2π). Note that deep water dispersion is v ∝ √λ, unlike strings where v is constant. This leads to wave broadening as waves travel across ocean basins.

Stationary (Standing) Waves

When two identical waves travel in opposite directions superpose, standing waves form:

$$y = 2A cos(omega t) sin(kx)$$

Nodes occur where sin(kx) = 0 → displacement always zero. Antinodes occur where |sin(kx)| = 1 → maximum amplitude 2A.

For a string fixed at both ends (both ends are nodes): $$L = frac{n\lambda}{2}, quad f_n = frac{nv}{2L}, quad n = 1, 2, 3, …$$

The fundamental frequency (n=1) is f₁ = v/2L. Harmonics are all integer multiples: f₂ = 2f₁, f₃ = 3f₁, etc.

For a pipe open at both ends (both ends are antinodes): $$L = frac{n\lambda}{2}, quad f_n = frac{nv}{2L}$$

For a pipe closed at one end (closed end = node, open end = antinode): $$L = frac{n\lambda}{4}, quad f_n = frac{nv}{4L}, quad n = 1, 3, 5, …$$

Only odd harmonics exist in closed pipes. The fundamental is f₁ = v/4L, first overtone is f₃ = 3v/4L = 3f₁.

Doppler Effect

When source or observer moves relative to the medium, observed frequency changes:

$$f’ = f left( frac{v pm v_o}{v mp v_s} right)$$

• Observer moving toward source: numerator uses +v₀
• Observer moving away: numerator uses -v₀
• Source moving toward observer: denominator uses -v_s
• Source moving away: denominator uses +v_s

JEE-specific tip: For light (electromagnetic waves), the Doppler formula differs because no medium exists. For a source approaching at speed βc (where β = v/c):

$$f’ = f sqrt{left( frac{1+eta}{1-eta} right)}$$

This comes from special relativity — time dilation affects the source frequency measurement.

Sonic boom occurs when source speed exceeds wave speed (v_s > v). The Mach cone angle satisfies sin θ = v/v_s = 1/M, where M is Mach number. The shock wave front is a cone extending behind the source.

Wave Energy and Intensity

For a travelling harmonic wave, total energy per unit length: $$E_{total} = frac{1}{2}mu omega^2 A^2$$

Kinetic energy and potential energy are both equal to (1/4)μω²A² at any point (they are in phase). The intensity (power per unit area) for spherical waves:

$$I = frac{P}{4pi r^2} = frac{1}{2} ho v omega^2 A^2$$

Note: I ∝ A² and I ∝ f² for sound waves. This is why doubling the amplitude quadruples the intensity — important for decibel calculations.

Beats occur when two waves of slightly different frequencies interfere: $$f_{beat} = |f_1 - f_2|$$

The beat frequency is the absolute difference. This principle is used in tuning musical instruments.

Wave Reflection and Refraction

When a wave meets a boundary:

• Fixed end reflection: phase inverted (wave inverts upon reflection)
• Free end reflection: no phase change

At interface between two media: $$frac{sin theta_1}{sin theta_2} = frac{v_1}{v_2} = frac{n_2}{n_1} = constant$$

This is Snell’s law. The frequency f remains constant across interface; wavelength changes because v changes.

⚡ Exam tips for JEE Advanced:

1. Wave speed on string v = √(T/μ) — remember T must be in newtons and μ in kg/m
2. In closed pipe problems, always check if n is odd only; n = 1, 3, 5…
3. Doppler in sound: moving observer uses numerator, moving source uses denominator
4. For light waves, use relativistic Doppler formula, not the classical one
5. Standing wave node/antinode positions can be asked with boundary conditions at 0, L/2, L etc.
6. Intensity I ∝ A² — useful for comparing loudness of sources

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