#wave physics

Doppler Effect

Why are standing waves formed in musical instruments like a guitar?

**Relative motion** changes effective wavelength encountered. For moving source approaching stationary observer, wavelength ahead λ' = (v - v_s)/f, so f' = v/λ' = f·v/(v - v_s) > f, basis for calculating apparent pitch shift in sound. Standing waves in a guitar arise from the interference of waves reflected at fixed ends (e.g., bridge and nut), producing discrete frequencies (harmonics) determined by string length. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Due to reflection at boundaries, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

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Doppler Effect

Which factor explains why a transverse wave’s speed is independent of its frequency in a uniform string?

**Doppler effect** describes apparent frequency shift due to relative motion between source and observer, f' = f·v/(v ∓ v_s) for source motion, f' = f·(v ± v_o)/v for observer motion, upper signs for approach increasing observed frequency. Motion towards observer compresses wavelength raising f'. The speed v = √((T/μ)) depends on tension and mass density, properties of the medium, not frequency, which is source-dependent. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Medium properties, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

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Superposition and Interference of Waves

What happens to the wavelength of a wave when it reflects off a free end without changing the medium?

**Superposition principle** states resultant displacement equals algebraic sum of individual waves, y = y₁ + y₂. For coherent waves with phase difference φ, resultant amplitude A = √(a₁² + a₂² + 2a₁a₂ cosφ), equal amplitudes give A = 2a cos(φ/2), constructive when φ = 2nπ, destructive when φ = (2n+1)π. Reflection at a free end does not alter the medium’s properties (tension, density), so the wave speed and frequency remain unchanged, keeping the wavelength constant ( λ = v/f ). Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields It remains unchanged, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Superposition and Interference of Waves

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Wave Speed, Energy and Power

A string of length 1.8 m and mass 0.045 kg has a fundamental frequency of 40 Hz. What is the tension in the string?

**Transverse wave velocity** depends on medium not frequency alone. For string under tension, v ∝ √(T/μ), calculation requires μ from mass and length, then square root evaluation, giving v in m/s, then f = v/λ for given wavelength. μ = (0.045/1.8) = 0.025 kg/m . v₁ = (v/2L) ⇒ 40 = (v/2 × 1.8) ⇒ v = 40 × 3.6 = 144 m/s . v = √((T/μ)) ⇒ 144 = √((T/0.025)) ⇒ 144² = (T/0.025) . T = 20736 × 0.025 = 518.4 N . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 518 N, illustrating

Ref: NCERT > Physics Book > Waves > Wave Speed, Energy and Power

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Beats Phenomenon

Two strings produce beats of 3 Hz. One has a frequency of 400 Hz. When the tension in the second string is slightly incr

**Superposition of nearly equal frequencies** creates resultant y = 2A cos(Δω·t/2) sin(ω_avg·t), envelope frequency Δf/2, beat frequency Δf. Total beats heard in Δt is f_beat·Δt, explaining counting over seconds. Let v₂ be the original frequency. |400 - v₂| = 3 ⇒ v₂ = 397 Hz or 403 Hz . Increasing tension increases frequency. If v₂ = 397 , new v₂’ > 397 , beat = 400 - v₂’ < 3 , becomes 2 Hz, consistent ( v₂’ = 398 ). If v₂ = 403 , beat increases, contradicts. So, v₂ = 397 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L)

Ref: NCERT > Physics Book > Waves > Beats Phenomenon

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Transverse and Longitudinal Waves

Which property of a medium allows longitudinal waves to propagate through it but not transverse waves?

**Nature of wave** determines energy transfer mechanism. Longitudinal nature of sound explains propagation through fluids with density variations carrying energy, unlike transverse waves needing rigidity for restoring force. Longitudinal waves require a medium with bulk modulus (resistance to compression), which fluids possess. Transverse waves need shear modulus (resistance to shearing), which fluids lack due to their inability to sustain shear stress. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Lack of shear modulus, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Transverse and Longitudinal Waves

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Beats Phenomenon

Two waves of frequencies 490 Hz and 495 Hz interfere. How many beats are heard in 8 seconds?

**Beat formation** is interference in time with time-varying amplitude. Frequencies close together generate slow modulation, count in given duration obtained by multiplying beat frequency by duration, e.g., 6 Hz × 5 s = 30 beats. Beat frequency: vbₑₐt = 495 - 490 = 5 Hz . Beats in 8 s: 5 × 8 = 40 . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 40, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Beats Phenomenon

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Vibrations of Air Columns - Open and Closed Pipes

A pipe open at both ends has a length of 0.8 m and resonates with a source of frequency 637.5 Hz. What is the harmonic n

**Organ pipe modes** illustrate boundary influence. Closed pipe odd harmonic series contrasts with open pipe full series, affecting timbre. Frequency scales inversely with length, explaining pitch variation with pipe length. v_n = (n v/2L) . 637.5 = (n × 340/2 × 0.8) = (n × 340/1.6) . 637.5 = n × 212.5 ⇒ n = (637.5/212.5) = 3 . Third harmonic. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 3, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Vibrations of Air Columns - Open and Closed Pipes

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Wave Equation and Displacement Relation

What is the primary physical quantity that remains constant for a wave reflecting off a boundary within the same medium?

**Wave equation** y(x,t) = A sin(kx - ωt + φ) describes displacement of progressive harmonic wave, where k = 2π/λ wave number (rad/m), ω = 2πf angular frequency (rad/s), v = ω/k wave speed (m/s). Sign of ωt indicates direction, amplitude A is maximum displacement. Reflection within the same medium preserves the wave’s frequency, as it depends on the source, while speed, wavelength, and amplitude may adjust based on boundary conditions. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Frequency, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Wave Equation and Displacement Relation

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Wave Speed, Energy and Power

A transverse wave on a string has a tension of 200 N and a linear mass density of 0.04 kg/m. What is the wavelength if t

**Energy transport** in waves scales with amplitude squared A² and frequency squared ω². Wave speed determines propagation rate, and understanding T and μ allows quantitative prediction of v and associated frequencies. Speed: v = √((T/μ)) = √((200/0.04)) = √(5000) ≈ 70.71 m/s . Wavelength: λ = (v/v) = (70.71/25) ≈ 2.83 m . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 2.83 m, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Wave Speed, Energy and Power

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Wave Equation and Displacement Relation

Which characteristic of a wave remains constant when it travels from one medium to another with different properties?

**Displacement relation** encodes λ = 2π/k and f = ω/2π. Comparing given equation y = a sin(kx - ωt) with standard form yields k and ω, hence λ = 2π/k and v = ω/k, essential for identifying propagation characteristics and phase. When a wave crosses a boundary between media, its speed and wavelength change due to differing medium properties, but frequency remains constant as it is determined by the source. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Frequency, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Wave Equation and Displacement Relation

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Transverse and Longitudinal Waves

What is the phase change when a longitudinal wave reflects off a rigid boundary?

**Wave classification** depends on particle vibration relative to propagation. Longitudinal waves have particle oscillation parallel to propagation, creating compressions and rarefactions as in sound in air; transverse have perpendicular oscillation. Tuning fork generates longitudinal sound because air cannot sustain shear. For a longitudinal wave, a rigid boundary (fixed end) causes a compression to reflect as a compression, resulting in no phase change (0 radians), unlike transverse waves which invert. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 0 radians, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Transverse and Longitudinal Waves

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