Waves

Sound Waves, Reflection and Characteristics

A steel rod of length 2 m has a fundamental frequency of longitudinal vibrations of 1.25 kHz. What is the speed of sound

**Longitudinal vibrations in rods** clamped at middle have fundamental with node at clamp and antinodes at ends, f₁ = v/(2L), v speed of sound in material (m/s). This relation allows v extraction from measured f₁ and length L, e.g., v = 2L·f₁. For rod clamped at middle, fundamental: v₁ = (v/2L) . 1250 = (v/2 × 2) ⇒ v = 1250 × 4 = 5000 m/s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 5000 m/s, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Sound Waves, Reflection and Characteristics

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Doppler Effect

A string fixed at both ends has a length of 1.6 m and a fundamental frequency of 62.5 Hz. What is the speed of the wave?

**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. Fundamental: v₁ = (v/2L) . 62.5 = (v/2 × 1.6) ⇒ v = 62.5 × 3.2 = 200 m/s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 200 m/s, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

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Doppler Effect

Two waves \( y_1 = 3 \sin (8x - 16t) \) and \( y_2 = 3 \sin (8x - 16t + \frac{2\pi}{3}) \) interfere. What is the amplit

**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'. Amplitude: A = 2a cos (Φ/2) , a = 3 m , Φ = (2π/3) . A = 2 × 3 cos (π/3) = 6 × (1/2) = 3 m . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 3 m, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

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Doppler Effect

A string of length 2 m and mass 0.01 kg is under a tension of 100 N. What is the time taken by a transverse pulse to tra

**Frequency shift** proportional to source speed relative to wave speed v. Understanding sign convention for approaching versus receding is key, with approaching increasing frequency and receding decreasing, central to Doppler applications. Linear mass density: μ = (mass/length) = (0.01/2) = 0.005 kg/m . Speed: v = √((T/μ)) = √((100/0.005)) = √(20000) ≈ 141.4 m/s . Time: t = (length/v) = (2/141.4) ≈ 0.014 s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 0.014 s, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

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Doppler Effect

A string of length 2.2 m fixed at both ends has a wave speed of 66 m/s. What is the frequency of its fourth harmonic?

**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. v_n = (n v/2L) . Fourth harmonic ( n = 4 ): v₄ = (4 × 66/2 × 2.2) = (264/4.4) = 60 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 60 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

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Doppler Effect

Two strings produce beats of 4 Hz. One has a frequency of 320 Hz. When the tension in the second string is increased, th

**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'. Let v₂ be the original frequency. |320 - v₂| = 4 ⇒ v₂ = 316 Hz or 324 Hz . Increasing tension increases frequency. If v₂ = 316 , new v₂’ > 316 , beat = 320 - v₂’ < 4 , becomes 2 Hz ( v₂’ = 318 ), consistent. If v₂ = 324 , beat increases, contradicts. So, v₂

Ref: NCERT > Physics Book > Waves > Doppler Effect

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Doppler Effect

A string fixed at both ends has a length of 1.4 m and a wave speed of 70 m/s. What is the frequency of its second harmon

**Frequency shift** proportional to source speed relative to wave speed v. Understanding sign convention for approaching versus receding is key, with approaching increasing frequency and receding decreasing, central to Doppler applications. For fixed ends: v_n = (n v/2L) . Second harmonic ( n = 2 ): v₂ = (2 × 70/2 × 1.4) = (140/2.8) = 50 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 50 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

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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 wave property is most affected when a sound wave encounters a change in the medium’s temperature?

**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'. Speed of sound in a gas increases with temperature ( v ∝ √(T) ), significantly altering its propagation, while frequency remains source-dependent and amplitude may vary less directly. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Speed, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

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Doppler Effect

A string fixed at both ends has a length of 1.5 m and a fundamental frequency of 50 Hz. What is the speed of the wave on

**Frequency shift** proportional to source speed relative to wave speed v. Understanding sign convention for approaching versus receding is key, with approaching increasing frequency and receding decreasing, central to Doppler applications. Fundamental frequency: v₁ = (v/2L) . 50 = (v/2 × 1.5) ⇒ 50 = (v/3) ⇒ v = 150 m/s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 150 m/s, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

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Doppler Effect

A string of length 3 m and mass 0.06 kg is under a tension of 150 N. What is the speed of a transverse wave on the strin

**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. Linear mass density: μ = (0.06/3) = 0.02 kg/m . Speed: v = √((T/μ)) = √((150/0.02)) = √(7500) ≈ 86.6 m/s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 86.6 m/s, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

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Doppler Effect

A pipe open at both ends has a length of 0.68 m and resonates with a source of frequency 750 Hz. What is the harmonic nu

**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'. v_n = (n v/2L) . 750 = (n × 340/2 × 0.68) = (n × 340/1.36) . 750 = n × 250 ⇒ n = (750/250) = 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 > Doppler Effect

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