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Longitudinal/transverse waves and progressive-wave relation: NEET MCQs with answers

66 questions on this topic of Oscillations and Waves (NEET 2027 syllabus). Here are 12; tap one to see its answer and explanation.

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easy

1. A sound wave travels through a gas with a speed of 340 m/s. If the frequency of the wave is 680 Hz, what is the phase difference between two points along the path of propagation separated by a distance of 12.5 cm?

  1. π/4 rad
  2. π rad
  3. π/2 rad
  4. 3π/4 rad
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Answer: (C) π/2 rad

The wavelength is λ = v / f = 340 / 680 = 0.5 m = 50 cm. The path difference is Δx = 12.5 cm. The phase difference is Δϕ = (2π / λ) × Δx = (2π / 50) × 12.5 = π/2 rad.

medium

2. In a longitudinal progressive wave traveling through a homogeneous fluid, consider a small volume element of the medium. The pressure variation ΔP at any point is directly related to the spatial derivative of particle displacement s(x, t) as ΔP = -B(∂s/∂x), where B is the bulk modulus. When the particle displacement s is maximum positive, the excess pressure ΔP at that point is:

  1. Maximum negative
  2. Maximum positive
  3. Zero
  4. Dependent on the wave frequency
Show answer and explanation

Answer: (C) Zero

For a wave s(x, t) = s₀ sin(kx - ωt), the spatial derivative is ∂s/∂x = k s₀ cos(kx - ωt). When s is at its maximum (s = +s₀), sin(kx - ωt) = 1, which means cos(kx - ωt) = 0. Therefore, the excess pressure ΔP = -B(∂s/∂x) = 0.

hard

3. A transverse wave propagating along the x-axis has a profile y(x) at t = 0 given by y = A sin(kx). If the wave travels in the negative x-direction with speed v, what is the expression for the particle acceleration a(x, t) at any position and time?

  1. a(x, t) = ω² A cos(kx + ωt)
  2. a(x, t) = -ω² A sin(kx + ωt)
  3. a(x, t) = -ω² A sin(kx - ωt)
  4. a(x, t) = -k² v A cos(kx + ωt)
Show answer and explanation

Answer: (B) a(x, t) = -ω² A sin(kx + ωt)

A wave moving in the negative x-direction has the general form y(x, t) = A sin(kx + ωt). Differentiating twice with respect to time gives ∂y/∂t = ω A cos(kx + ωt), and acceleration a(x, t) = ∂²y/∂t² = -ω² A sin(kx + ωt).

medium

4. A transverse sinusoidal wave is given by y(x, t) = A sin(ωt - kx). At a certain instant t, a particle at position x has a positive transverse velocity (∂y/∂t > 0). What can be deduced about the slope of the wave profile (∂y/∂x) at that point?

  1. The slope is zero
  2. The slope is positive
  3. The slope is negative
  4. The slope can be positive or negative depending on A
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Answer: (C) The slope is negative

The relation between particle velocity and wave velocity is (∂y/∂t) = -v × (∂y/∂x). For a wave propagating in the positive x-direction, the wave speed v is positive. Therefore, if particle velocity (∂y/∂t) is positive, the spatial slope (∂y/∂x) must be negative.

easy

5. A progressive harmonic wave is represented by the equation y(x, t) = 0.05 sin(40πt - 2πx), where x and y are in metres and t is in seconds. The wavelength and time period of the wave are respectively:

  1. 1 m and 0.05 s
  2. 2 m and 0.025 s
  3. 0.5 m and 0.05 s
  4. 1 m and 0.025 s
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Answer: (A) 1 m and 0.05 s

Comparing with y = A sin(ωt - kx), we have ω = 40π rad/s and k = 2π m⁻¹. The wavelength is λ = 2π/k = 2π/(2π) = 1 m. The time period is T = 2π/ω = 2π/(40π) = 0.05 s.

hard

6. A progressive wave of frequency 500 Hz has a velocity of 360 m/s. Two points on the wave have a phase difference of 60°. What is the minimum time interval between the moments when these two points achieve the same state of oscillation?

  1. 1.67 × 10⁻⁴ s
  2. 3.33 × 10⁻⁴ s
  3. 5.00 × 10⁻⁴ s
  4. 1.00 × 10⁻³ s
Show answer and explanation

Answer: (B) 3.33 × 10⁻⁴ s

The time period of the wave is T = 1 / f = 1 / 500 = 2 × 10⁻³ s. A phase difference of Δϕ = 60° = π/3 rad corresponds to a time lag of Δt = (Δϕ / 2π) × T = ( (π/3) / 2π ) × 2 × 10⁻³ = (1/6) × 2 × 10⁻³ = 3.33 × 10⁻⁴ s.

easy

7. Speed of sound in air at 0°C:

  1. 232 m/s
  2. 332 m/s
  3. 432 m/s
  4. 532 m/s
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Answer: (B) 332 m/s

v = 332 m/s at 0°C. Increases 0.61 m/s per °C.

Quick tip: 332 at 0°C. At 30°C: 332+18.3 ≈ 350 m/s.

NCERT Class 11, Chapter 15, page 368

easy

8. In a longitudinal progressive wave propagating through a gas, the regions of compression correspond to locations where:

  1. Pressure is maximum and density is minimum
  2. Pressure is maximum and particle velocity is zero
  3. Pressure is minimum and particle displacement is maximum
  4. Pressure and density are both maximum
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Answer: (D) Pressure and density are both maximum

In a longitudinal wave traveling through a gaseous medium, compression regions are formed where particles crowd together. Consequently, both the volume strain is maximum compressive, leading to maximum excess pressure and maximum density at these points.

medium

9. For a progressive wave y = a sin(kx - ωt), the wave travels:

  1. Along +x with speed ω/k
  2. Along -x with speed ω/k
  3. Along +x with speed k/ω
  4. It is a stationary wave
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Answer: (A) Along +x with speed ω/k

A wave of form f(kx - ωt) moves in the +x direction. Its speed is v = ω/k, with k = 2π/λ and ω = 2πν.

NCERT Class 11

medium

10. A sound wave propagates through an ideal gas. At a particular point, the medium undergoes rarefaction. During this phase:

  1. The density is at its equilibrium value and the pressure is minimum
  2. The density is maximum and the local pressure is minimum
  3. The density is minimum and the local pressure is maximum
  4. The density is minimum and the local pressure is minimum
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Answer: (D) The density is minimum and the local pressure is minimum

In a longitudinal sound wave, rarefaction corresponds to a region where particles have moved apart from each other. Consequently, the local density of the gas decreases to a minimum, and correspondingly the local pressure also reaches its minimum value.

easy

11. The equation of a progressive wave on a string is y(x, t) = 2.0 sin(10πt - 0.5πx), where x and y are in cm and t is in seconds. What is the maximum acceleration experienced by any segment of the string?

  1. 20π² cm/s²
  2. 200π² cm/s²
  3. 100π² cm/s²
  4. 50π² cm/s²
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Answer: (B) 200π² cm/s²

The maximum acceleration of a medium particle undergoing SHM in a progressive wave is a_max = ω²A. Here, ω = 10π rad/s and A = 2.0 cm. Therefore, a_max = (10π)² × 2.0 = 100π² × 2.0 = 200π² cm/s².

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12. Statement I: Transverse mechanical waves cannot propagate through the bulk of an ideal fluid because fluids have zero modulus of rigidity. Statement II: Longitudinal waves can travel through solids, liquids, and gases because all states of matter possess volume elasticity (bulk modulus).

  1. Both Statement I and Statement II are correct
  2. Statement I is correct but Statement II is incorrect
  3. Statement I is incorrect but Statement II is correct
  4. Both Statement I and Statement II are incorrect
Show answer and explanation

Answer: (A) Both Statement I and Statement II are correct

Transverse mechanical waves involve shearing strains, and since fluids cannot sustain static shear stress (shear modulus is zero), transverse waves cannot propagate through their bulk. In contrast, longitudinal waves involve compressive and tensile strains, which depend on the bulk modulus that exists in solids, liquids, and gases alike.

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