#open boundary

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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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.

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

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Sound Waves, Reflection and Characteristics

A wave \( y = 0.03 \sin (20x - 40t) \) reflects at an open 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 open boundary, no phase change. Incident: y_i = 0.03 sin (20x - 40t) . Reflected: y_r = 0.03 sin (20x + 40t) . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields y = 0.03 sin (20x + 40t), illustrating frequency-length-speed interdependence and quantization by boundaries.

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

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Sound Waves, Reflection and Characteristics

A wave \( y = 0.02 \sin (40x - 120t) \) reflects at an open 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 open boundary, no phase change. Incident: y_i = 0.02 sin (40x - 120t) . Reflected: y_r = 0.02 sin (40x + 120t) . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields y = 0.02 sin (40x + 120t), illustrating frequency-length-speed interdependence and quantization by boundaries.

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

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Sound Waves, Reflection and Characteristics

A wave \( y = 0.03 \sin (15x - 45t) \) 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 = 0.03 sin (15x - 45t) . Reflected: y_r = 0.03 sin (15x + 45t) . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields y = 0.03 sin (15x + 45t), illustrating frequency-length-speed interdependence and quantization by boundaries.

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

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Sound Waves, Reflection and Characteristics

A wave \( y = 0.06 \sin (18x - 54t) \) 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 = 0.06 sin (18x - 54t) . Reflected: y_r = 0.06 sin (18x + 54t) . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields y = 0.06 sin (18x + 54t), illustrating frequency-length-speed interdependence and quantization by boundaries.

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

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