#acoustics

Sound Waves, Reflection and Characteristics

A pipe closed at one end has a length of 0.25 m. What is the frequency of its third harmonic if the speed of sound is 34

**Reflection at boundaries** follows phase change rules: rigid boundary (fixed end) introduces π phase shift, inverting displacement y → -y, while free boundary reflects without phase change. Reflected wave derived by reversing propagation direction kx → -kx and applying phase shift, preserving k = 2π/λ and ω = 2πf. For closed pipe: v_n = (n + (1/2)) (v/2L) , n = 0, 1, 2, ldots . Third harmonic: n = 2 . v₂ = (2 + (1/2)) (340/2 × 0.25) = 2.5 × (340/0.5) = 1700 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields

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

View Details

Doppler Effect

A pipe 0.5 m long, open at both ends, resonates with a 340 Hz source. What is the harmonic number if the speed of sound

**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 open pipe: v_n = (n v/2L) . Given: v = 340 m/s , L = 0.5 m , v_n = 340 Hz . 340 = (n × 340/2 × 0.5) ⇒ 340 = 340n ⇒ n = 1 . First harmonic (fundamental mode). Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 1, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

View Details

Sound Waves, Reflection and Characteristics

A pipe closed at one end has a length of 0.85 m and resonates at its second harmonic with a speed of sound of 340 m/s. W

**Reflection at boundaries** follows phase change rules: rigid boundary (fixed end) introduces π phase shift, inverting displacement y → -y, while free boundary reflects without phase change. Reflected wave derived by reversing propagation direction kx → -kx and applying phase shift, preserving k = 2π/λ and ω = 2πf. For closed pipe: v_n = (n + (1/2)) (v/2L) , n = 1 for second harmonic. v₁ = (1 + (1/2)) (340/2 × 0.85) = 1.5 × (340/1.7) = 300 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 300 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

View Details

Sound Waves, Reflection and Characteristics

Why does the speed of sound increase with temperature in a gas?

**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. Speed of sound in a gas is v = √((gamma P/rho)) , and since P/rho ∝ T (ideal gas law), higher temperature increases molecular velocity, thus increasing v . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Molecular velocity increases, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

View Details

Beats Phenomenon

Two strings produce beats of 7 Hz. One has a frequency of 392 Hz. When the tension in the second string is increased, th

**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. |392 - v₂| = 7 ⇒ v₂ = 385 Hz or 399 Hz . Increasing tension increases frequency. If v₂ = 385 , new v₂’ > 385 , beat = 392 - v₂’ < 7 , becomes 5 Hz ( v₂’ = 387 ), consistent. If v₂ = 399 , beat increases, contradicts. So, v₂ = 385 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L)

Ref: NCERT > Physics Book > Waves > Beats Phenomenon

View Details

Vibrations of Air Columns - Open and Closed Pipes

A pipe closed at one end has a length of 0.5 m and a speed of sound of 340 m/s. What is the frequency of its fundamental

**Resonance in pipes** occurs when length accommodates standing wave pattern. Closed pipe L = (2n-1)λ/4, so f₁ = v/(4L). Given f₁ and v, length follows L = v/(4f₁), enabling length calculation from measured resonance frequency and sound speed 330-340 m/s. Fundamental: v₁ = (v/4L) . v₁ = (340/4 × 0.5) = (340/2) = 170 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 170 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

View Details

Wave Equation and Displacement Relation

Which wave type is most likely to propagate fastest in a solid 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. In solids, longitudinal waves (P-waves) travel faster than transverse waves (S-waves) because the bulk modulus plus shear modulus exceeds the shear modulus alone, increasing speed. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Longitudinal, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

View Details

Vibrations of Air Columns - Open and Closed Pipes

A pipe closed at one end has a length of 0.25 m and resonates at its fundamental frequency with a speed of sound of 340

**Resonance in pipes** occurs when length accommodates standing wave pattern. Closed pipe L = (2n-1)λ/4, so f₁ = v/(4L). Given f₁ and v, length follows L = v/(4f₁), enabling length calculation from measured resonance frequency and sound speed 330-340 m/s. Fundamental: v₁ = (v/4L) . v₁ = (340/4 × 0.25) = (340/1) = 340 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 340 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

View Details

Vibrations of Air Columns - Open and Closed Pipes

A pipe open at both ends has a length of 0.51 m and a speed of sound of 340 m/s. What is the frequency of its third harm

**Resonance in pipes** occurs when length accommodates standing wave pattern. Closed pipe L = (2n-1)λ/4, so f₁ = v/(4L). Given f₁ and v, length follows L = v/(4f₁), enabling length calculation from measured resonance frequency and sound speed 330-340 m/s. For open pipe: v_n = (n v/2L) . Third harmonic ( n = 3 ): v₃ = (3 × 340/2 × 0.51) = (1020/1.02) = 1000 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 1000 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

View Details

Superposition and Interference of Waves

What is the physical basis for the formation of beats in sound waves?

**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. Beats arise from the superposition of two waves with slightly different frequencies, causing periodic constructive and destructive interference, perceived as amplitude variation. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Superposition, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

View Details

Beats Phenomenon

What is the significance of the beat frequency when two sound waves interfere?

**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. The beat frequency, |f₁ - f₂| , is the rate at which amplitude modulates, perceived as periodic loudness variations, equal to the difference in wave frequencies. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Difference in frequencies, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Beats Phenomenon

View Details

Vibrations of Air Columns - Open and Closed Pipes

In a pipe closed at one end, which harmonics are absent compared to a pipe open at both ends?

**Air column vibrations** depend on end conditions. Pipe closed at one end has displacement node at closed end and antinode at open, allowing only odd harmonics, fundamental f₁ = v/(4L). Open pipe has antinodes at both ends, fₙ = n·v/(2L), all harmonics present, v sound speed. A pipe closed at one end has frequencies v_n = (2n - 1) (v/4L) , producing only odd harmonics (1, 3, 5, ..). A pipe open at both ends has all harmonics (1, 2, 3, ..), so even harmonics are absent in the closed pipe. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as

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

View Details