Advertisements
Advertisements
Question
For the travelling harmonic wave
y (x, t) = 2.0 cos 2π (10t – 0.0080x + 0.35)
Where x and y are in cm and t in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of `(3λ)/4`.
Advertisements
Solution 1
Equation for a travelling harmonic wave is given as:
y (x, t) = 2.0 cos 2π (10t – 0.0080x + 0.35)
= 2.0 cos (20πt – 0.016πx + 0.70 π)
Where,
Propagation constant, k = 0.0160 π
Amplitude, a = 2 cm
Angular frequency, ω= 20 π rad/s
Phase difference is given by the relation:
`phi = kx = (2pi)/lambda`
For `x = (3lambda)/4`
`phi = (2pi)/lambda xx (3lambda)/4`
`= 1.5 pi` rad
Solution 2
The given equation can be drawn be rewritten as under
y(x, t) `= 2.0 cos [2pi (10"t" - 0.0080x) + 2pi xx 0.35]`
or y(x, t) `= 2.0 cos [2pi xx 0.0080((10"t")/0.0080 - x) + 0.7pi]`
Comparing this equation with the standard equation of a travelling harmonic wave.
`(2pi)/lambda = 2pi xx 0.0080` or `lambda = 1/0.0080 "cm" = 125` cm
The phase difference between oscillatory motion of two points seperated by a distance `trianglex` is given by
`trianglephi = (2pi)/lambda trianglex`
When `trianglex = (3lambda)/4 = (3xx125)/4` cm, then
`triangle phi = (2phi)/125 xx (3xx125)/4`
`= (3pi)/2 "rad"`
RELATED QUESTIONS
You have learnt that a travelling wave in one dimension is represented by a function y= f (x, t)where x and t must appear in the combination x – v t or x + v t, i.e. y = f (x ± v t). Is the converse true? Examine if the following functions for y can possibly represent a travelling wave:
(a) `(x – vt )^2`
(b) `log [(x + vt)/x_0]`
(c) `1/(x + vt)`
(i) For the wave on a string described in Exercise 15.11, do all the points on the string oscillate with the same (a) frequency, (b) phase, (c) amplitude? Explain your answers. (ii) What is the amplitude of a point 0.375 m away from one end?
A steel rod 100 cm long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod is given to be 2.53 kHz. What is the speed of sound in steel?
Show that the particle speed can never be equal to the wave speed in a sine wave if the amplitude is less than wavelength divided by 2π.
Velocity of sound in air is 332 m s−1. Its velocity in vacuum will be
Two wave pulses travel in opposite directions on a string and approach each other. The shape of one pulse is inverted with respect to the other.
A sonometer wire supports a 4 kg load and vibrates in fundamental mode with a tuning fork of frequency 416. Hz. The length of the wire between the bridges is now doubled. In order to maintain fundamental mode, the load should be changed to
The displacement of the particle at x = 0 of a stretched string carrying a wave in the positive x-direction is given f(t) = A sin (t/T). The wave speed is v. Write the wave equation.
A wave travels along the positive x-direction with a speed of 20 m s−1. The amplitude of the wave is 0⋅20 cm and the wavelength 2⋅0 cm. (a) Write the suitable wave equation which describes this wave. (b) What is the displacement and velocity of the particle at x= 2⋅0 cm at time t = 0 according to the wave equation written? Can you get different values of this quantity if the wave equation is written in a different fashion?
A wave travelling on a string at a speed of 10 m s−1 causes each particle of the string to oscillate with a time period of 20 ms. (a) What is the wavelength of the wave? (b) If the displacement of a particle of 1⋅5 mm at a certain instant, what will be the displacement of a particle 10 cm away from it at the same instant?
Following figure shows two wave pulses at t = 0 travelling on a string in opposite directions with the same wave speed 50 cm s−1. Sketch the shape of the string at t = 4 ms, 6 ms, 8 ms, and 12 ms.
The string of a guitar is 80 cm long and has a fundamental frequency of 112 Hz. If a guitarist wishes to produce a frequency of 160 Hz, where should the person press the string?
What is the interference of sound waves?
A string 1 m long is fixed at one end. The other end is moved up and down with a frequency of 20 Hz. Due to this, a stationary wave with four complete loops gets produced on the string. Find the speed of the progressive wave which produces the stationary wave.
Use the formula `v = sqrt((gamma P)/rho)` to explain why the speed of sound in air is independent of pressure.
At what temperatures (in °C) will the speed of sound in air be 3 times its value at O°C?
The amplitude of wave disturbance propagating in the positive x-direction given is by `1/(1 + x)^2` at time t = 0 and `1/(1 + (x - 2)^2)` at t = 1 s, where x and y are in 2 metres. The shape of wave does not change during the propagation. The velocity of the wave will be ______ m/s.
Two perfectly identical wires kept under tension are in unison. When the tension in the wire is increased by 1% then on sounding them together 3 beats are heard in 2 seconds. What is the frequency of each wire?
