Advertisements
Advertisements
प्रश्न
The equation of a wave travelling on a string stretched along the X-axis is given by
\[y = A e {}^- \left( \frac{x}{a} + \frac{t}{T} \right)^2 .\]
(a) Write the dimensions of A, a and T. (b) Find the wave speed. (c) In which direction is the wave travelling? (d) Where is the maximum of the pulse located at t = T? At t = 2 T?
Advertisements
उत्तर
Given,
Equation of the wave travelling on a string stretched along the X-axis:
\[y = Ae {}^\left( \frac{x}{a} + \frac{t}{T} \right) {{}^-}^2 \]
(a) The dimensions of A (amplitude), T (time period) and \[a = \frac{\lambda}{2\pi}\] , which will have the dimensions of the wavelength, are as follows:
\[\left[ A \right] = \left[ M^0 L^1 T^0 \right]\]
\[\left[ T \right] = \left[ M^0 L^0 T^{- 1} \right]\]
\[\left[ a \right] = \left[ M^0 L^1 T^0 \right]\]
(b) Wave speed, \[\nu = \frac{\lambda}{T} = \frac{a}{T} \left[ \lambda = a \right]\]
(c) If \[y = f \left( t + \frac{x}{\nu} \right)\] , then the wave travels in the negative direction; and if \[y = f \left( t - \frac{x}{\nu} \right)\] ,then the wave travels in the positive direction.
Thus, we have:
`"y"="Ae"^[[(x/a)+(t/T)]^(-2)]`
\[ = A e^{- \frac{1}{T} \left[ t + \left( \frac{xT}{a} \right) \right]^2} \]
\[ = A e^{- \frac{1}{T}\left[ t + \frac{x}{V} \right]} \]
\[ = A e^{- f\left[ t + \frac{x}{V} \right]}\]
Hence, the wave is travelling is the negative direction.
(d) Wave speed, \[v = \frac{a}{t}\] Maximum pulse at t = T = \[\left( \frac{a}{T} \right) \times T = a\] (Along the negative x - axis)
Maximum pulse at t = 2T = \[\left( \frac{a}{T} \times 2T \right) = 2a \] (Along the negative x - axis)
Therefore, the wave is travelling in the negative x-direction.
APPEARS IN
संबंधित प्रश्न
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?
A train, standing at the outer signal of a railway station blows a whistle of frequency 400 Hz in still air. (i) What is the frequency of the whistle for a platform observer when the train (a) approaches the platform with a speed of 10 m s–1, (b) recedes from the platform with a speed of 10 m s–1? (ii) What is the speed of sound in each case? The speed of sound in still air can be taken as 340 m s–1.
A train, standing in a station-yard, blows a whistle of frequency 400 Hz in still air. The wind starts blowing in the direction from the yard to the station with at a speed of 10 m s–1. What are the frequency, wavelength, and speed of sound for an observer standing on the station’s platform? Is the situation exactly identical to the case when the air is still and the observer runs towards the yard at a speed of 10 m s–1? The speed of sound in still air can be taken as 340 m s–1.
Show that for a wave travelling on a string
\[\frac{y_{max}}{\nu_{max}} = \frac{\nu_{max}}{\alpha_{max}},\]
where the symbols have usual meanings. Can we use componendo and dividendo taught in algebra to write
\[\frac{y_{max} + \nu_{max}}{\nu_{max} - \nu_{max}} = \frac{\nu_{max} + \alpha_{max}}{\nu_{max} - \alpha_{max}}?\]
Two strings A and B, made of same material, are stretched by same tension. The radius of string A is double of the radius of B. A transverse wave travels on A with speed `v_A` and on B with speed `v_B`. The ratio `v_A/v_B` is ______.
A wave pulse, travelling on a two-piece string, gets partially reflected and partially transmitted at the junction. The reflected wave is inverted in shape as compared to the incident one. If the incident wave has wavelength λ and the transmitted wave λ'
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 wave propagates on a string in the positive x-direction at a velocity \[\nu\] \[t = t_0\] is given by \[g\left( x, t_0 \right) = A \sin \left( x/a \right)\]. Write the wave equation for a general time t.
A wire of length 2⋅00 m is stretched to a tension of 160 N. If the fundamental frequency of vibration is 100 Hz, find its linear mass density.
Figure shows an aluminium wire of length 60 cm joined to a steel wire of length 80 cm and stretched between two fixed supports. The tension produced is 40 N. The cross-sectional area of the steel wire is 1⋅0 mm2 and that of the aluminium wire is 3⋅0 mm2. What could be the minimum frequency of a tuning fork which can produce standing waves in the system with the joint as a node? The density of aluminium is 2⋅6 g cm−3 and that of steel is 7⋅8 g cm−3.
A 2⋅00 m-long rope, having a mass of 80 g, is fixed at one end and is tied to a light string at the other end. The tension in the string is 256 N. (a) Find the frequencies of the fundamental and the first two overtones. (b) Find the wavelength in the fundamental and the first two overtones.
A man standing unsymmetrical position between two mountains and fires a gun. He hears the first echo after 1.5 s and the second echo after 2.5 s. If the speed of sound in air is 340 m/s, then the distance between the mountains will be ______
An organ pipe of length 0.4 m is open at both ends. The speed of sound in the air is 340 m/s. The fundamental frequency is ______
What is the interference of sound waves?
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 4 m.
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 0.5 m.
A steel wire has a length of 12 m and a mass of 2.10 kg. What will be the speed of a transverse wave on this wire when a tension of 2.06 × 104N is applied?
A wave of frequency υ = 1000 Hz, propagates at a velocity v = 700 m/sec along x-axis. Phase difference at a given point x during a time interval M = 0.5 × 10-3 sec is ______.
