Test: Waves - Transverse ,longitudinal & Displacement relation in a progressive(15 Oct) - JEE MCQ

Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave.
Matter is made of atoms, and atoms are made protons, neutrons and electrons. These are not macroscopic particles.

Let the waves be
y1 = Asin(wt)
y2 = Asin(wt + φ)
If φ = π the interference is destructive.
Destructive interference occurs when the maxima of two waves are 180 degrees out of phase: a positive displacement of one wave is cancelled exactly by a negative displacement of the other wave. The amplitude of the resulting wave is zero. The dark regions occur whenever the waves destructively interfere.
Hence B is the correct answer.

A longer wavelength on a fluid interface will result in gravity–capillary waves which are influenced by both the effects of surface tension and gravity, as well as by fluid inertia. On the open ocean, much larger ocean surface waves (seas and swells) may result from coalescence of smaller wind-caused ripple-waves.

Speed of sound in air is 343 m/s or we can say approx 300m/s
And speed of light is approx 300,000,000 m/s
Clearly the ratio is 10⁶

Angular frequency is associated with the number of revolutions an object performs in a certain unit of time. In that sense is related to frequency but in terms of how many times it turns a full period of motion in radians units.
The formula of angular frequency is given by:
Angular frequency = 2 π / (period of oscillation)
ω = 2π / T = 2πf
Where we have:
ω: angular frequency
T: period
f: frequency
Hence C is correct.

The linear velocity of a wave traveling in a stretched string is given by the equation,

Where T is the tension in the stretched string, m is the mass of the string and L is the length of the string.

The general equation of a wave is y = Asin(ωt + kx + ϕ)

Where y is the displacement of the wave at time t, A is the amplitude, ϕ is the phase and k is the angular wavenumber.

CALCULATION:

Given that:

Linear density = mass per unit length = m/L = 1.3 x 10⁻⁴ kg/m

Wave equation, y = 0.021 sin (x + 30t) ⇒ k =1, ω = 30 rad/s

Equation of simple harmonic wave:

Equation: Y = Y₀ sin (ωt - kx), where ω = angular frequency, k = propagation constant 
The relation between the angular frequency and the time period is,  
The relation between the propagation constant and the wavelength is, 

Calculation:

The equation of a simple harmonic wave is given by y = 5 sin (50πt – πx/2)

Here, ω = 50 π 

The period of the wave in the second will be 0.04.

The equation of wave is given in the form

y = A sin(ωt + ϕ)

(ωt + ϕ) is the phase at time t

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.​

• The amplitude of SHM (A): maximum displacement from the mean position.
• frequency (f): no. of oscillations in one second.
• Angular frequency (ω) of SHM is given by:

where T is the time period.

Explanation:

Given Equation is:

y = a sin (3πnt + α)

Comparing with the standard equation of wave

(3πnt + α) is the phase at time t.

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:
  

Where y is the displacement of the wave at time t, A is the amplitude or maximum displacement of the wave, k is the is the angular frequency (ω = 2πf).

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

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:

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  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