Two waves represented by the following equations are travelling in the same medium: \(y_1 = 5 \sin 2\pi(75t – 0.25x)\) and \(y_2 = 10 \sin 2\pi(150t – 0.50x)\). The intensity ratio \(I_1/I_2\) of the two waves is:
1. 1 : 2
2. 1 : 4
3. 1 : 8
4. 1 : 16
View Answer
The intensity of a wave is proportional to the square of its amplitude and frequency, \(I \propto A^2 f^2\). Substituting \(A_1=5, f_1=75\) and \(A_2=10, f_2=150\) gives \(\frac{I_1}{I_2} = \left(\frac{5}{10}\right)^2 \left(\frac{75}{150}\right)^2 = \frac{1}{16}\).
A wave travelling in the positive x-direction having displacement amplitude along y-direction as 1 m, wavelength \(2\pi\text{ m}\) and frequency of \(\frac{1}{\pi}\text{ Hz}\) is represented by
1. \(y = \sin(10\pi x - 20\pi t)\)
2. \(y = \sin(2\pi x + 2\pi t)\)
3. \(y = \sin(x - 2t)\)
4. \(y = \sin(2\pi x - 2\pi t)\)
View Answer
Wave equation is \(y = A\sin(kx - \omega t)\). Here, \(A = 1\text{ m}\), \(k = \frac{2\pi}{\lambda} = 1\text{ m}^{-1}\), and \(\omega = 2\pi f = 2\text{ rad/s}\). Thus, \(y = \sin(x - 2t)\).
Match List-I with List-II where list-I denotes nature and medium of wave and list-II denotes the expression for speed of wave. (All symbols have their usual meaning)
List-I
a. Transverse wave on a stretched string
b. Longitudinal wave in a metallic bar
c. Longitudinal wave in a fluid
List-II
(i) \( \sqrt{\frac{Y}{\rho}} \)
(ii) \( \sqrt{\frac{B}{\rho}} \)
(iii) \( \sqrt{\frac{T}{\mu}} \)
Choose the correct option:
1. a(ii), b(iii), c(i)
2. a(i), b(ii), c(iii)
3. a(iii), b(i), c(ii)
4. a(iii), b(ii), c(i)
View Answer
Speed of a wave on a string is \( \sqrt{T/\mu} \) (a matches iii). Speed of a longitudinal wave in a metallic bar is \( \sqrt{Y/\rho} \) (b matches i). Speed of a longitudinal wave in a fluid is \( \sqrt{B/\rho} \) (c matches ii).
Assertion (A): When a wave enters from one medium to another, its frequency is not changed.
Reason (R): Speed of a wave in a medium is property of the source.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
The frequency of a wave is determined by the source that generates it and remains constant as the wave propagates from one medium to another. The speed of a wave, however, is a characteristic property of the medium it is traveling through, not the source. Thus, Assertion A is true, but Reason R is false.
Assertion (A): Two waves moving in a uniform string having uniform tension cannot have different velocities.
Reason (R): Elastic and inertial properties of string are same for all waves in same string. Moreover, velocity of wave in a string depends on its elastic and inertial properties only.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
The speed of a transverse wave on a string is given by \( v = \sqrt{T/\mu} \), where \( T \) is the tension (elastic property) and \( \mu \) is the linear mass density (inertial property). If the string is uniform (constant \( \mu \)) and has uniform tension (constant \( T \)), then \( v \) must be constant for all waves propagating on it. Both A and R are true, and R correctly explains A.
Assertion (A): Every small part of string does SHM in sinusoidal travelling wave.
Reason (R): In this small segment of string total energy is conserved.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
In a sinusoidal traveling wave, each particle of the string oscillates in simple harmonic motion (SHM) perpendicular to the direction of wave propagation. So, Assertion A is true. However, for a *traveling* wave, energy is continuously transmitted along the string. Therefore, the total energy within a small segment of the string is *not* conserved, as energy flows into and out of the segment. Reason R is false.
Assertion (A): In a sinusoidal travelling wave on a string potential energy of deformation of string element at extreme position is maximum.
Reason (R): The particles in sinusoidal travelling wave perform SHM.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
In a wave, potential energy is stored due to deformation (strain). At extreme positions (maximum displacement), the deformation is maximum, leading to maximum potential energy. So (A) is true.
Particles in a transverse wave undergo simple harmonic motion. So (R) is true. However, (R) does not explain why potential energy is maximum at extreme positions; it's a general characteristic of the particle motion. Therefore, (R) is not the correct explanation of (A).
Assertion (A): \(Y = 2A sin kx cos \omega t\) refers to a travelling wave along -ve x-direction.
Reason (R): When a continuous travelling wave interacts with its reflection from a rigid support, forms a standing wave.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
The equation \(Y = 2A sin kx cos \omega t\) represents a standing wave, not a travelling wave. Thus (A) is false. When a travelling wave reflects from a boundary and superposes with the incident wave, a standing wave is formed. Thus (R) is true. Since (A) is false and (R) is true, none of the standard options (A true, R true; A true, R false; A false, R true; A false, R false) perfectly matches. However, given the provided options, and (A) being false, option (4) is selected as it states (A) is false, despite (R) being true.