Waves – Displacement Relation in Progressive Wave: NEET MCQs with Answers

9 free questions on Waves – Displacement Relation in Progressive Wave for NEET (UG), starting with previous-year NEET and AIPMT questions. Try each one, then tap Show answer for the answer and a short explanation.

Syllabus covered: Displacement Relation in Progressive Wave: Displacement relation for a progressive wave.

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Question 1

Previous-year question
A uniform rope of length L and mass m₁ hangs vertically from a rigid support. A block of mass m₂ is attached to the free end of the rope. A transverse pulse of wavelength λ₁ is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is λ₂. The ratio λ₂/λ₁ is: [AIPMT, 2016]
  1. √(m₁/m₂)
  2. √((m₁ + m₂)/m₂)
  3. √(m₂/m₁)
  4. √((m₁ + m₂)/m₁)
Show answer
Answer: (B) √((m₁ + m₂)/m₂)
Explanation: The frequency of the pulse remains constant, so λ ∝ v ∝ √T. At the bottom, T₁ = m₂g; at the top, T₂ = (m₁ + m₂)g. Hence λ₂/λ₁ = √(T₂/T₁) = √((m₁ + m₂)/m₂).

Question 2

Previous-year question
A wave travelling in the +ve x direction having displacement along y direction as 1 m, wavelength 2π m and frequency of 1/π Hz is represented by: [NEET, 2013]
  1. y = sin(2πx + 2πt)
  2. y = sin(x − 2t)
  3. y = sin(2πx − 2πt)
  4. y = sin(10πx − 20πt)
Show answer
Answer: (B) y = sin(x − 2t)
Explanation: A = 1 m; k = 2π/λ = 2π/2π = 1 m⁻¹; ω = 2πf = 2π × (1/π) = 2 rad/s. For propagation along +x: y = A sin(kx − ωt) = sin(x − 2t).

Question 3

Previous-year question
The equation of a simple harmonic wave is given by: y = 3 sin(π/2)(50t − x), where x and y are in metres and t is in seconds. The ratio of maximum particle velocity to the wave velocity is: [AIPMT Mains, 2012]
  1. (3/2)π
  2. 3π
  3. (2/3)π
  4. 2π
Show answer
Answer: (A) (3/2)π
Explanation: Comparing with y = A sin(ωt − kx): A = 3 m, ω = 25π rad/s, k = π/2 m⁻¹. Maximum particle velocity = Aω; wave velocity = ω/k. Ratio = Aω/(ω/k) = Ak = 3 × π/2 = (3/2)π.

Question 4

Previous-year question
Two waves are represented by the equations y₁ = a sin(ωt + kx + 0.57) m and y₂ = a cos(ωt + kx) m, where x is in metre and t in seconds. The phase difference between them is: [AIPMT Pre., 2011]
  1. 0.57 radian
  2. 1.0 radian
  3. 1.25 radian
  4. 1.57 radian
Show answer
Answer: (B) 1.0 radian
Explanation: y₂ = a cos(ωt + kx) = a sin(ωt + kx + π/2), i.e., its phase constant is π/2 = 1.57 rad. Phase difference = 1.57 − 0.57 = 1.0 radian.

Question 5

Previous-year question
A transverse wave is represented by y = A sin(ωt − kx). For what value of the wavelength is the wave velocity equal to the maximum particle velocity? [AIPMT Pre., 2010]
  1. πA/2
  2. πA
  3. 2πA
  4. A
Show answer
Answer: (C) 2πA
Explanation: Wave velocity = ω/k; maximum particle velocity = Aω. Equating: ω/k = Aω ⇒ k = 1/A ⇒ 2π/λ = 1/A ⇒ λ = 2πA.

Question 6

Previous-year question
A wave in a string has an amplitude of 2 cm. The wave travels in the +ve direction of X-axis with a speed of 128 m/s and it is noted that 5 complete waves fit in 4 m length of the string. The equation describing the wave is: [AIPMT, 2009]
  1. y = (0.02) m sin(7.85x − 1005t)
  2. y = (0.02) m sin(7.85x + 1005t)
  3. y = (0.02) m sin(15.7x − 2010t)
  4. y = (0.02) m sin(15.7x + 2010t)
Show answer
Answer: (A) y = (0.02) m sin(7.85x − 1005t)
Explanation: λ = 4/5 = 0.8 m ⇒ k = 2π/λ = 7.85 m⁻¹. ω = vk = 128 × 7.85 = 1005 rad/s. For a wave moving in the +x direction, the phase is (kx − ωt). Hence y = 0.02 sin(7.85x − 1005t).

Question 7

A wave is described by y = (0.002 mm) sin[(60 s⁻¹)t + (3.0 m⁻¹)x]. Which statement about it is correct?
  1. Its amplitude is 0.004 mm.
  2. Its frequency is 30/π Hz.
  3. Its speed is 180 m/s.
  4. Its wavelength is 3.0 m.
Show answer
Answer: (B) Its frequency is 30/π Hz.
Explanation: Comparing with y = A sin(ωt + kx) gives A = 0.002 mm, ω = 60 s⁻¹ and k = 3.0 m⁻¹. The frequency is ν = ω/2π = 60/2π = 30/π Hz. The speed is ω/k = 20 m/s and the wavelength is 2π/k = 2π/3 m, about 2.1 m.
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Question 8

Two particles A and B lie in the path of a sinusoidal travelling wave, and at every instant their oscillations differ in phase by π. Which statement must be true?
  1. At every instant the displacements of A and B are equal in magnitude and opposite in direction.
  2. A and B reach their extreme positions a quarter period apart.
  3. The separation of A and B must be exactly one wavelength.
  4. A oscillates at half the frequency of B.
Show answer
Answer: (A) At every instant the displacements of A and B are equal in magnitude and opposite in direction.
Explanation: Every particle in the path of the wave oscillates at the frequency of the wave, so the two frequencies are equal. A phase difference of π gives y_B = −y_A at all times, so the two reach opposite extremes at the same moment. The separation must be an odd multiple of λ/2, which rules out one full wavelength.

Question 9

A sine wave travels along a long string. Two particles P and Q of the string are 5λ/2 apart, where λ is the wavelength. At a certain instant P has a displacement of +2 mm and is moving upward. At that instant Q has
  1. displacement +2 mm and is moving downward
  2. displacement +2 mm and is moving upward
  3. displacement −2 mm and is moving downward
  4. displacement −2 mm and is moving upward
Show answer
Answer: (C) displacement −2 mm and is moving downward
Explanation: The phase difference is (2π/λ)(5λ/2) = 5π, which is the same as π, so the particles are exactly out of step. At every instant their displacements and their velocities are equal in magnitude and opposite in direction. Q is therefore at −2 mm and moving downward.

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