The Wave Equation MCQ Quiz - Objective Question with Answer for The Wave Equation - Download Free PDF

Calculation:

At point P, the wave moves from a lighter string (μ) to a denser string (4μ), hence:

y = α₁ y₀ cos(ωt + kx + π)

y = α₂ y₀ cos(ωt - kx)

Reflected wave: undergoes a phase change of π, so the equation becomes:

Transmitted wave: enters the denser medium, no phase change, so:

At point Q, the wave again moves from 4μ to 16μ (denser), so:

y = α₃ y₀ cos(ωt - kx + π)

y = α₄ y₀ cos(ωt - 4kx)

Reflected wave: undergoes a phase change of π, becomes:

Transmitted wave: in a medium with 4× linear mass density → the wave number becomes 4k:

Thus, options A and D are correct.

Calculation:
The displacement of a particle in Simple Harmonic Motion (SHM) is given by the equation:

x = 10sin(2t - π/6) metres

To find the velocity, we differentiate the displacement equation with respect to time.

The velocity (v) is given by:

v = dx/dt = d/dt [10sin(2t - π/6)]

Using the chain rule:

v = 10 × 2cos(2t - π/6)

v = 20cos(2t - π/6)

When the displacement is 6 m, we substitute this into the displacement equation:

6 = 10sin(2t - π/6)

sin(2t - π/6) = 6/10 = 0.6

2t - π/6 = sin⁻¹(0.6)

2t - π/6 = 0.6435 rad

2t = 0.6435 + π/6

2t = 0.6435 + 0.5236 = 1.1671 rad

t = 1.1671 / 2 = 0.5836 s

Now, substitute t = 0.5836 s into the velocity equation:

v = 20cos(2 × 0.5836 - π/6)

v = 20cos(1.1671 - 0.5236)

v = 20cos(0.6435) = 20 × 0.8 = 16 m/s

The velocity of the particle when its displacement is 6 m is 16 m/s.

Concept:

If particle displacement at time t is y = f(x), particle displacement at time t + Δt is f(x -vΔt).

Calculation:

So, at t = 0,

y = f(x) = \( {-1\over (x-6)^2}\)

And at t = 4,

y = f(x - v×4) = \( {-1\over (x-4v -6)^2}\)

So,

x-4v-6 = x-14

⇒ 4v = 8

∴ v = 2 m/s

We rewrite the given equations \( y = \dfrac{1}{1 + (x + v t)^2} \)

For \( t = 0 \), this becomes \( v = \dfrac{1}{1 + x^2} \) as given

For \( t = 2 \), this becomes \( y = \dfrac{1}{[1 + (x - 1)^2]} = \dfrac{1}{[1 + (x - 1)^2]} \)

\( \Rightarrow 2v = 1 \) or \( v = 0.5 \, m/s \)

Concept:

General equation of wave :

y = f ( x - v t )

Where y is the displacement of a particle

x is the position, 

v is the velocity of the wave

Calculation:

Given:

Displacement at t =0 is given by

 \(\rm {y}=3 \sin 2 π\left(-\frac{x}{3}\right)\)

v = 4 m/s

Using wave equation at t = 0,

y = f( x - v × 0) = 3 sin 2π ( -x / 3 )

f(x) = 3 sin 2π ( -x / 3 )

Now, at t = 4, 

y = f ( x - v × 4)

⇒ y = f ( x - 4× 4 )

⇒ y = f ( x - 16 )

⇒ y = f ( x - 16 )

⇒ y = 3 sin 2π ( - ( x -16 )  / 3 )

∴  option 1) is correct

CONCEPT:

• Wave-number (k): It is defined as the number of complete cycles over its wavelength.

\(⇒ k = \frac{{2\pi }}{\lambda }\)

• Its measure is rad/m
• The wave equation of a wave travelling in the x-direction is

\(⇒ y = A\sin \omega \left( {t - \frac{x}{v}} \right) = A\sin \left( {\omega t - kx} \right) = A\sin 2\pi \left( {\frac{t}{T} - \frac{x}{\lambda }} \right) = A\sin k\left( {vt - x} \right)\;\)

• Wavelength: 

​\(⇒ \lambda = vT = v.\frac{{2\pi }}{\omega } = \frac{v}{\nu } ⇒ v = \nu \lambda \)

• Wavenumber: 

​\(⇒ k = \frac{{2\pi }}{\lambda } = \frac{{2\pi \nu }}{v} = \frac{\omega }{v}\)

CALCULATION:

Given, the wave function: 

⇒ y = (5 mm) sin [(1 cm-1)x - (60 s-1)t]= (5mm)sin[(1cm−1)x−(60s−1)t].y=(5mm)sin[(1cm−1)x−(60s−1)t].\(⇒ y = A\sin \left( {kx - \omega t} \right) ⇒ y = \left( {5mm} \right)\sin \left[ {\left( {1\;c{m^{ - 1}}} \right)x - \left( {60{s^{ - 1}}} \right)t} \right]\;\)

⇒ k = 1 cm^-1

• Option 4 is correct

CONCEPT:

• Transverse Wave: A wave in which the medium particles move in a perpendicular direction to the direction that the wave moves.

• Equation of transverse wave is given in the form

⇒ y(x, t) = Asin(kx − ωt + ϕ)

Where the amplitude is A, ω is the angular frequency (ω = 2π/T), k is the wave-number (k = 2π/λ), ϕ is the phase, and y is changing with respect to position x and time t.

• Wavelength (λ): ​The minimum distance of separation between two particles which are in the same phase is called a wavelength.
• The velocity of wave: The velocity of a wave is given by

\(\Rightarrow v=\frac{λ}{T}=\frac{λ 2π}{T2π}=\frac{ω }{k}\)

where  ω is the angular frequency (ω = 2π/T) and k is the wave-number (k = 2π/λ)

EXPLANATION:

• Wavenumber: It is a constant term denoted by k.

\(\Rightarrow k=\frac{2\pi}{\lambda}\)

• So option 3 is correct

CONCEPT:

• Mechanical waves: The waves which need a medium to travel are called mechanical waves.
  • These waves are governed by Newton's laws.

For example: sound, waves in a string, water wave, etc.

EXPLANATION:

• Mechanical waves are governed by Newton's Laws. So option 1 is correct.

CONCEPT:​

• Equation of wave is given in the form

y(x, t) = Asin(kx − ωt + ϕ)

where the amplitude is A, ω is the angular frequency (ω = 2πf; f is frequency), k is the wave-number (k = 2π/λ), ϕ is the phase, and y is changing with respect to position x and time t.​

Maximum Acceleration a = A ω2 

Acceleration a = A ω2 

CALCULATION:

Given equation is \(y= 3 \sin 2 π \left(\dfrac{t}{0.04} - \dfrac{x}{0.01}\right)\)

compare it with standard equation y = A sin (ωt – kx) 

CONCEPT:

The equation of wave is given in the form

y = A sin(ωt + ϕ)

where the amplitude is A, ω is the angular frequency (ω = 2π/T), ϕ is the initial phase, and y is changing position with respect to time t.​

CALCULATION:

Given that:

x =  a sin(ωt + π/6)

Initial phase (ϕ₁) = π/6

x = a cos ωt = a sin (ωt + π/2)

Initial phase (ϕ₂) = π/2

So phase difference = ϕ₂ - ϕ₁ = π/2 - π/6 = π/3

Hence option 1 is correct.

CONCEPT:

The general equation of wave motion is given by:

y =  Asin (kx - ωt)

where A = amplitude, ω = angular frequency, k = angular wavenumber, t = time, and x is position

• Wavelength (λ): Distance between two nearest particles vibrating in the same phase.
• Frequency (f): Number of vibrations complete by a particle in one second.
• Time Period (T): Time taken by the wave to travel a distance equal to one wavelength.
• Amplitude (A): Maximum displacement of vibrating particles from the mean position.

The relationship between frequency (f), wavelength (λ), and velocity (v) of a wave is given by:

f × λ = v 

CALCULATION:

Given that:

y = 2 sin (3x - t/4)

By comparing above equation with the given standard equation we get,

A = 2; ω = 1/4 rad/s; k = 3 rad/m

We know that ω = 2πf and f = 1/T

ω = 2π/T

T = 2π/ω

T = 2π × (4/1) = 25.13 s

• Hence option 2 is correct

The correct answer is option 2) i.e. 3 m/s

CONCEPT:

• Progressive waves: A wave that is capable of traveling in a medium from one point to another is called a progressive wave. They are also known as traveling waves.
  • The two types of progressive waves are longitudinal and transverse waves.

The displacement of a progressive harmonic wave is given by the equation:

y = Asin(kx - ωt + ϕ)

Where y is the displacement of the wave at time t, A is the amplitude or maximum displacement of the wave, k is the wavenumber (k = \(\frac{2π}{λ}\)), ω is the angular frequency (ω = 2πf).

CALCULATION:

Given that: y = 15 x 10-2 sin (300t – 100x)

Comparing with y = Asin(kx - ωt + ϕ),

A = 15 x 10-2 m

k = 100 rad/m

ω = 300 rad/s

We know ω = 2πf and k = \(\frac{2π}{λ}\) ⇒ kλ = \(\frac{\omega}{f}\) 

⇒ \(\frac{\omega}{k}\) = fλ = v (∵ speed of a wave, v = fλ)

Therefore velocity, v = \(\frac{\omega}{k}\) = \(\frac{300}{100}\) = 3 m/s

CONCEPT:

• Progressive waves: A wave that is capable of travelling in a medium from one point to another is called a progressive wave. They are also known as travelling waves.
  • The two types of progressive waves are longitudinal and transverse waves.
• The displacement of a progressive harmonic wave is given by the equation:

⇒ y = Asin(kx - ωt + ϕ)

Where y is the displacement of the wave at time t, A is the amplitude or maximum displacement of the wave, k is the wavenumber (\(k =\frac{2π}{λ}\)), ω is the angular frequency (ω = 2πf).

• The velocity (v) of a wave related to its wavelength (λ) and frequency (f) as follows:

​\(⇒ f =\frac{v}{\lambda}\)

CALCULATION:

Wave equation: y = 3 sin (4t - π/6)

Here, ω = 4, and amplitude (A) = 3

• As we know, the maximum velocity of the particle can be calculated as

⇒ V_max = ωA 

⇒ V_max = 3 × 4 = 12 m/s

Concept:

• Simple harmonic motion occurs when the restoring force is directly proportional to the displacement from equilibrium.

​F α -x

Where F = force and x = the displacement from equilibrium.

Force (F) = - k x

Where k = restoring force, x = distance from the equilibrium position, F = force it experiences towards mean position

• The equation of SHM is given by:

x = A sinωt

where x is the distance from the mean position at any time t, A is amplitude, t is time, and ω is the angular frequency.

​Calculation:

• Equation of a progressive sound wave is 

\(\Rightarrow y=0.5\,sin\frac{2π }{3.2}(64t-x)\).......(1)

Where A = amplitude, ω = Angular frequency, x = Displacement 

• A plane progressive wave is given by

​\(\Rightarrow y=a\,sin(\omega t-\frac{2π x}{\lambda})\)........(2)

On comparing equation 1 and 2, we get

⇒ \(\omega = \frac{128 \pi}{3.2}\)

 As we know, frequency is equal to 

\(\Rightarrow f = \frac{ω}{2\pi}=\frac{128\pi}{(3.2)2\pi}=20.Hz\)

The approximate value is 20 Hz.

CONCEPT:

• Transverse Wave: A wave in which the medium particles move in a perpendicular direction to the direction that the wave moves.

Equation of transverse wave is given in the form

y(x, t) = Asin(kx − ωt + ϕ)

where the amplitude is A, ω is the angular frequency (ω = 2π/T), k is the wave-number (k = 2π/λ), ϕ is the phase, and y is changing with respect to position x and time t.

• Wavelength (λ): ​The minimum distance of separation between two particles which are in the same phase is called a wavelength.

The velocity of wave: The velocity of a wave is given by

\(v=\frac{\lambda}{T}=\frac{\lambda 2\pi}{T2\pi}=\frac{ω }{k}\)

where  ω is the angular frequency (ω = 2π/T), k is the wave-number (k = 2π/λ),

CALCULATION:

Given equation is y = 0.05 sin π(2t – 0.02x)

compare it with standard equation y = A sin (ωt – kx)