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 \([\frac{x + vt}{x_0} ] \)
(c) \(\frac{1}{(x + vt)}\)
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 \([\frac{x + vt}{x_0} ] \)
(c) \(\frac{1}{(x + vt)}\)
Solution and Explanation
No;
Does not represent a wave Represents a wave Does not represent a wave The converse of the given statement is not true. The essential requirement for a function to represent a travelling wave is that it should remain finite for all values of x and t.
Explanation:
For x = 0 and t = 0, the function (x – vt) 2 becomes 0.
Hence, for x = 0 and t = 0, the function represents a point and not a wave.
For x = 0 and t = 0, the function
log \((\frac{x+vt}{x_0})=logo=∞\)
Since the function does not converge to a finite value for x = 0 and t = 0, it represents a travelling wave.
For x = 0 and t = 0, the function
log \(\frac{1}{x+vt}=\frac{1}{0}=∞\)
Since the function does not converge to a finite value for x = 0 and t = 0, it does not represent a travelling wave.
Top CBSE Class XI Physics Questions
- An oxygen cylinder of volume 30 litre has an initial gauge pressure of 15 atm and a temperature of 27 °C. After some oxygen is withdrawn from the cylinder, the gauge pressure drops to 11 atm and its temperature drops to 17 °C. Estimate the mass of oxygen taken out of the cylinder (R = 8.31 J mol–1 K–1, molecular mass of O2 = 32 u).
- Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity 25.0 m3 at a temperature of 27 °C and 1 atm pressure.
- An air bubble of volume 1.0 cm3 rises from the bottom of a lake 40 m deep at a temperature of 12 °C. To what volume does it grow when it reaches the surface, which is at a temperature of 35 °C ?
- The ceiling of a long hall is 25 \(\text m\) high. What is the maximum horizontal distance that a ball thrown with a speed of 40\(\text m \;\text s^{-1}\) can go without hitting the ceiling of the hall ?
- Estimate the average thermal energy of a helium atom at (i) room temperature (27 °C), (ii) the temperature on the surface of the Sun (6000 K), (iii) the temperature of 10 million kelvin (the typical core temperature in the case of a star).
Top CBSE Class XI Waves Questions
- For the wave described in Exercise 14.8, plot the displacement (y) versus (t) graphs for x = 0, 2 and 4 cm. What are the shapes of these graphs? In which aspects does the oscillatory motion in travelling wave differ from one point to another: amplitude, frequency or phase ?
The transverse displacement of a string (clamped at its both ends) is given by
y(x, t) = 0.06 sin \((\frac{2π}{3 }x)\) cos (120 πt)
where x and y are in m and t in s. The length of the string is 1.5 m and its mass is 3.0 ×10–2 kg.
Answer the following :
(a) Does the function represent a travelling wave or a stationary wave?
(b) Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave?
(c) Determine the tension in the string
- (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?
Given below are some functions of x and t to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all:
(a) y = 2 cos (3x) sin (10t)
(b) \(y=2\sqrt{x-vt}\)
(c) y = 3 sin (5x – 0.5t) + 4 cos (5x – 0.5t)
(d) y = cos x sin t + cos 2x sin 2t
Top CBSE Class XI Questions
- The total number of protons in $ 10\,g$ of calcium carbonate is $ (N_{0}=6.023\times 10^{23})$ :
- Justify giving reactions that among halogens, fluorine is the best oxidant and among hydrohalic compounds, hydroiodic acid is the best reductant.
- An oxygen cylinder of volume 30 litre has an initial gauge pressure of 15 atm and a temperature of 27 °C. After some oxygen is withdrawn from the cylinder, the gauge pressure drops to 11 atm and its temperature drops to 17 °C. Estimate the mass of oxygen taken out of the cylinder (R = 8.31 J mol–1 K–1, molecular mass of O2 = 32 u).
- Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity 25.0 m3 at a temperature of 27 °C and 1 atm pressure.
- An air bubble of volume 1.0 cm3 rises from the bottom of a lake 40 m deep at a temperature of 12 °C. To what volume does it grow when it reaches the surface, which is at a temperature of 35 °C ?
Concepts Used:
Waves
Waves are a disturbance through which the energy travels from one point to another. Most acquainted are surface waves that tour on the water, but sound, mild, and the movement of subatomic particles all exhibit wavelike properties. inside the most effective waves, the disturbance oscillates periodically (see periodic movement) with a set frequency and wavelength.
Types of Waves:
Transverse Waves -
Waves in which the medium moves at right angles to the direction of the wave.
Examples of transverse waves:
- Water waves (ripples of gravity waves, not sound through water)
- Light waves
- S-wave earthquake waves
- Stringed instruments
- Torsion wave
The high point of a transverse wave is a crest. The low part is a trough.
Longitudinal Wave -
A longitudinal wave has the movement of the particles in the medium in the same dimension as the direction of movement of the wave.
Examples of longitudinal waves:
- Sound waves
- P-type earthquake waves
- Compression wave