Advertisements
Advertisements
Question
For the harmonic travelling wave y = 2 cos 2π (10t – 0.0080x + 3.5) where x and y are in cm and t is second. What is the phase difference between the oscillatory motion at two points separated by a distance of What is the phase difference between the oscillation of a particle located at x = 100 cm, at t = T s and t = 5 s?
Solution
Given, wave functions are y = 2 cos 2π (10t – 0.0080x + 3.5)
= 2 cos(20πt – 0.016πx + 7π)
Now, the standard equation of a travelling wave can be written as y = a cos(ωt – kx + `phi`)
On comparing with the above equation, we get
a = 2 cm
ω = 20π rad/s
k = 0.016π
Path difference = 4 cm
T = `(2π)/ω = (2π)/(20π) = 1/10`s
∴ At x = 100 cm, t = T
`phi`1 = 20πT – 0.016π(100) + 7π
= `20π(1/10) - 16π + 7π`
= 2π – 1.6π + 7π .......(i)
Again, at x = 100 cm, t = 5s
`phi`2 = 20π(5) – 0.016π(100) + 7π
= `100π - (0.016 xx 100)π + 7π`
= 100π – 1.6π + 7π .......(ii)
∴ From equations (i) and (ii), we get
Δ`phi` = Phase difference = `phi_1 - phi_2`
= (100π – 1.6π + 7π) – (2π – 1.6π + 7π)
= 100π – 2π
= 98π rad
APPEARS IN
RELATED QUESTIONS
Consider two waves passing through the same string. Principle of superposition for displacement says that the net displacement of a particle on the string is sum of the displacements produced by the two waves individually. Suppose we state similar principles for the net velocity of the particle and the net kinetic energy of the particle. Such a principle will be valid for
Two waves, each having a frequency of 100 Hz and a wavelength of 2⋅0 cm, are travelling in the same direction on a string. What is the phase difference between the waves (a) if the second wave was produced 0⋅015 s later than the first one at the same place, (b) if the two waves were produced at the same instant but first one was produced a distance 4⋅0 cm behind the second one? (c) If each of the waves has an amplitude of 2⋅0 mm, what would be the amplitudes of the resultant waves in part (a) and (b) ?
A sonometer wire having a length of 1⋅50 m between the bridges vibrates in its second harmonic in resonance with a tuning fork of frequency 256 Hz. What is the speed of the transverse wave on the wire?
Answer briefly.
State and explain the principle of superposition of waves.
The wavelength of light used in young.'s double slit experiment is λ. The intensity at a point on the screen is I where the path difference is λ/6. If I0 denotes the maximum intensity, then the ratio of I and I0 is ______.
The displacement of an elastic wave is given by the function y = 3 sin ωt + 4 cos ωt. where y is in cm and t is in second. Calculate the resultant amplitude.
For the harmonic travelling wave y = 2 cos 2π (10t – 0.0080x + 3.5) where x and y are in cm and t is second. What is the phase difference between the oscillatory motion at two points separated by a distance of 0.5 m
For the harmonic travelling wave y = 2 cos 2π (10t – 0.0080x + 3.5) where x and y are in cm and t is second. What is the phase difference between the oscillatory motion at two points separated by a distance of `λ/2`
In the interference of two sources of intensities I0 and 9I0 the intensity at a point where the phase difference is `pi/2` is ______.
In Young's double-slit experiment, the intensity at a point where the path difference is `lambda/4` is l. If the maximum intensity is l0, and then the ratio of `l_0/l` is ______.
`(cos45^circ = 1/sqrt2 = sin45^circ)`
