Sound Waves, Reflection and Characteristics

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.

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Sound Waves, Reflection and Characteristics

A steel rod of length 1.5 m has a fundamental frequency of longitudinal vibrations of 2 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) . 2000 = (v/2 × 1.5) ⇒ v = 2000 × 3 = 6000 m/s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 6000 m/s, illustrating frequency-length-speed interdependence and quantization by boundaries.

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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

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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.

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Sound Waves, Reflection and Characteristics

A pipe closed at one end has a length of 0.6 m and resonates at its second harmonic with a speed of sound of 360 m/s. Wh

**Sound wave reflection** at rigid wall behaves like string fixed end, displacement inverted. Equation of reflected wave includes sign change and direction reversal, amplitude unchanged but sign may flip, explaining standing wave formation with incident wave. For closed pipe: v_n = (n + (1/2)) (v/2L) , n = 1 for second harmonic. v₁ = (1 + (1/2)) (360/2 × 0.6) = 1.5 × (360/1.2) = 1.5 × 300 = 450 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 450 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

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Sound Waves, Reflection and Characteristics

A pipe closed at one end has a length of 0.4 m and a speed of sound of 320 m/s. What is the frequency of its fundamental

**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. Fundamental: v₁ = (v/4L) . v₁ = (320/4 × 0.4) = (320/1.6) = 200 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 200 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

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Sound Waves, Reflection and Characteristics

The speed of sound in air at STP is calculated using Newton’s formula as 280 m/s. If the actual speed is 331 m/s, what i

**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. Newton’s formula: v = √((P/rho)) = 280 m/s . Laplace correction: v = √((gamma P/rho)) = 331 m/s . Divide: (331/280) = √(gamma) . √(gamma) = 1.182 , so gamma = (1.182)² ≈ 1.4 . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 1.4, illustrating frequency-length-speed interdependence and quantization by boundaries.

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Sound Waves, Reflection and Characteristics

A pipe open at both ends has a length of 0.4 m and a speed of sound of 320 m/s. What is the frequency of its fourth harm

**Sound wave reflection** at rigid wall behaves like string fixed end, displacement inverted. Equation of reflected wave includes sign change and direction reversal, amplitude unchanged but sign may flip, explaining standing wave formation with incident wave. For open pipe: v_n = (n v/2L) . Fourth harmonic ( n = 4 ): v₄ = (4 × 320/2 × 0.4) = (1280/0.8) = 1600 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 1600 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

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Sound Waves, Reflection and Characteristics

A pipe open at both ends has a length of 0.25 m and a speed of sound of 340 m/s. What is the frequency of its second har

**Sound wave reflection** at rigid wall behaves like string fixed end, displacement inverted. Equation of reflected wave includes sign change and direction reversal, amplitude unchanged but sign may flip, explaining standing wave formation with incident wave. For open pipe: v_n = (n v/2L) . Second harmonic ( n = 2 ): v₂ = (2 × 340/2 × 0.25) = (680/0.5) = 1360 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 1360 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

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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.

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Sound Waves, Reflection and Characteristics

A wave \( y = 0.04 \sin (30x - 60t) \) reflects at an open boundary. What is the equation of the reflected wave?

**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. At open boundary, no phase change. Incident: y_i = 0.04 sin (30x - 60t) . Reflected: y_r = 0.04 sin (30x + 60t) . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields y = 0.04 sin (30x + 60t), illustrating frequency-length-speed interdependence and quantization by boundaries.

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Sound Waves, Reflection and Characteristics

A wave \( y = 0.07 \sin (20x - 60t) \) reflects at a rigid boundary. What is the equation of the reflected wave?

**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. At rigid boundary, phase changes by π . Incident: y_i = 0.07 sin (20x - 60t) . Reflected: y_r = -0.07 sin (20x + 60t) . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields y = -0.07 sin (20x + 60t), illustrating frequency-length-speed interdependence and quantization by boundaries.

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