Find the best tutors and institutes for Class 11 Tuition
Q14.19(e):
Explain why (or how): the shape of a pulse gets distorted during propagation in a dispersive medium.
Solution :
Theoretical Foundation: Wave Superposition and Dispersion
A wave pulse is not a pure monochromatic wave of a single frequency $\omega$. According to the Fourier Theorem, any complex pulse $f(x, t)$ can be mathematically represented as a linear superposition of an infinite number of harmonic (sinusoidal) waves, each characterized by a distinct angular frequency $\omega$ and a corresponding wave number $k$.
The total pulse at time $t$ and position $x$ is given by the integral:
$f(x, t) = \int_{-\infty}^{\infty} A(k) e^{i(kx - \omega(k)t)} dk$
[Per the principle of superposition and Fourier analysis].
Step 1: The Role of the Dispersion Relation
In a non-dispersive medium, the phase velocity $v_p = \frac{\omega}{k}$ is constant for all frequencies. Consequently, every constituent component of the pulse travels at the same velocity $v$. The pulse shape remains invariant as it moves through space. [Ref: Wave Equation derivation for non-dispersive media].
In a dispersive medium, the dispersion relation is non-linear, meaning $\omega$ is a non-linear function of $k$ (i.e., $\frac{d^2\omega}{dk^2} \neq 0$). Thus, the phase velocity depends on the frequency: $v_p(k) = \frac{\omega(k)}{k}$.
Step 2: Differential Propagation Velocities
Since the pulse is a packet of waves with a range of wave numbers $[k, k + dk]$, each constituent frequency component travels at its own phase velocity $v_p(\omega)$. Because these components move at different speeds, they do not maintain their relative spatial positions within the envelope of the pulse.
Step 3: Analytical Mechanism of Distortion
As the wave packet propagates, the individual spectral components undergo a relative phase shift $\Delta \phi = (k_1 - k_2)x - (\omega_1 - \omega_2)t$.
- Phase Mismatch: Because $\frac{\omega_1}{k_1} \neq \frac{\omega_2}{k_2}$, the constructive interference that originally formed the "peak" of the pulse at $t=0$ is gradually lost.
- Group Velocity Dispersion (GVD): The pulse envelope travels at the group velocity $v_g = \frac{d\omega}{dk}$. In dispersive media, $v_g$ is also a function of $k$. Different parts of the pulse envelope move at different speeds, leading to pulse broadening or "chirping."
Summary of Physical Process
The distortion is the direct result of Differential Phase Velocity. When different Fourier components of a pulse propagate with unequal phase velocities, the temporal alignment of these components is disrupted. This violates the condition for a rigid translation of the wave packet, manifesting macroscopically as a spreading and reshaping of the pulse profile. [Per the theory of Wave Packet Dynamics].
Final Solution: The shape of a pulse is distorted because a dispersive medium causes different frequency components of the pulse to propagate at different phase velocities, resulting in the loss of coherence in the superposition that maintains the original pulse envelope.
More Questions from Class 11 Physics Waves EXERCISES
- Q14.1: A string of mass 2.50 kg is under a tension of 200 N. The length of the stretched string is 20.0 m. If the transverse jerk is struck at one end of the string, how long does the disturbance take to reach the other end?
- Q14.10(a): For the travelling harmonic wave (y(x, t)) = (2.0 \cos 2\pi (10t – 0.0080 x + 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,
- Q14.10(b): For the travelling harmonic wave (y(x, t)) = (2.0 \cos 2\pi (10t – 0.0080 x + 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,
- Q14.10(c): For the travelling harmonic wave (y(x, t)) = (2.0 \cos 2\pi (10t – 0.0080 x + 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 (\frac{\lambda}{2}),
- Q14.10(d): For the travelling harmonic wave (y(x, t)) = (2.0 \cos 2\pi (10t – 0.0080 x + 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 (\frac{3\lambda}{4})
- Q14.11(a): The transverse displacement of a string (clamped at its both ends) is given by (y(x, t)) = (0.06 \sin(\frac{2\pi x}{3}) \cos (120 \pi t)) 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 (\times)10^{-2} kg. Does the function represent a travelling wave or a stationary wave?
- Q14.11(b): The transverse displacement of a string (clamped at its both ends) is given by (y(x, t)) = (0.06 \sin(\frac{2\pi x}{3}) \cos (120 \pi t)) 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 (\times)10^{-2} kg. Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave ?
- Q14.11(c): The transverse displacement of a string (clamped at its both ends) is given by (y(x, t)) = (0.06 \sin(\frac{2\pi x}{3}) \cos (120 \pi t)) 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 (\times)10^{-2} kg. Determine the tension in the string.
- Q14.12(i)(a): For the wave on a string described, do all the points on the string oscillate with the same frequency? Explain your answers.
- Q14.12(i)(b): For the wave on a string described, do all the points on the string oscillate with the same phase? Explain your answers.
- Q14.12(i)(c): For the wave on a string described, do all the points on the string oscillate with the same amplitude? Explain your answers.
- Q14.12(ii): For the wave on a string described, what is the amplitude of a point 0.375 m away from one end?
- Q14.13(a): 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: (y) = (2 \cos (3x) \sin (10t))
- Q14.13(b): 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: (y) = (2\sqrt{x - vt})
- Q14.13(c): 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: (y) = (3 \sin (5x – 0.5t) + 4 \cos (5x – 0.5t))
- Q14.13(d): 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: (y) = (\cos x \sin t + \cos 2x \sin 2t)
- Q14.14(a): A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is 3.5 (\times) 10^{-2} kg and its linear mass density is 4.0 (\times) 10^{-2} kg (m^{-1}). What is the speed of a transverse wave on the string?
- Q14.14(b): A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is 3.5 (\times) 10^{-2} kg and its linear mass density is 4.0 (\times) 10^{-2} kg (m^{-1}). What is the tension in the string?
- Q14.15: A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340 Hz) when the tube length is 25.5 cm or 79.3 cm. Estimate the speed of sound in air at the temperature of the experiment. The edge effects may be neglected.
- Q14.16: A steel rod 100 cm long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod are given to be 2.53 kHz. What is the speed of sound in steel?
- Q14.17: A pipe 20 cm long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a 430 Hz source ? Will the same source be in resonance with the pipe if both ends are open? (speed of sound in air is 340 m (s^{-1})).
- Q14.18: Two sitar strings (A) and (B) playing the note ‘Ga’ are slightly out of tune and produce beats of frequency 6 Hz. The tension in the string (A) is slightly reduced and the beat frequency is found to reduce to 3 Hz. If the original frequency of (A) is 324 Hz, what is the frequency of (B)?
- Q14.19(a): Explain why (or how): in a sound wave, a displacement node is a pressure antinode and vice versa,
- Q14.19(b): Explain why (or how): bats can ascertain distances, directions, nature, and sizes of the obstacles without any “eyes”,
- Q14.19(c): Explain why (or how): a violin note and sitar note may have the same frequency, yet we can distinguish between the two notes,
- Q14.19(d): Explain why (or how): solids can support both longitudinal and transverse waves, but only longitudinal waves can propagate in gases, and
- Q14.2: A stone dropped from the top of a tower of height 300 m splashes into the water of a pond near the base of the tower. When is the splash heard at the top given that the speed of sound in air is 340 m (s^{-1}) ? ((g) = 9.8 m (s^{-2}))
- Q14.3: A steel wire has a length of 12.0 m and a mass of 2.10 kg. What should be the tension in the wire so that speed of a transverse wave on the wire equals the speed of sound in dry air at 20 (^{\circ})C = 343 m (s^{-1}).
- Q14.4(a): Use the formula (v) = (\sqrt{\frac{\gamma P}{\rho}}) to explain why the speed of sound in air is independent of pressure,
- Q14.4(b): Use the formula (v) = (\sqrt{\frac{\gamma P}{\rho}}) to explain why the speed of sound in air increases with temperature,
- Q14.4(c): Use the formula (v) = (\sqrt{\frac{\gamma P}{\rho}}) to explain why the speed of sound in air increases with humidity.
- Q14.5(a): 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 \pm v t)). Is the converse true? Examine if the following functions for (y) can possibly represent a travelling wave : ((x – vt)^{2})
- Q14.5(b): 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 \pm v t)). Is the converse true? Examine if the following functions for (y) can possibly represent a travelling wave : (\log [\frac{(x + vt)}{x_{0}}])
- Q14.5(c): 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 \pm v t)). Is the converse true? Examine if the following functions for (y) can possibly represent a travelling wave : (\frac{1}{(x + vt)})
- Q14.6(a): A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a water surface, what is the wavelength of the reflected sound? Speed of sound in air is 340 m (s^{-1}) and in water 1486 m (s^{-1}).
- Q14.6(b): A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a water surface, what is the wavelength of the transmitted sound? Speed of sound in air is 340 m (s^{-1}) and in water 1486 m (s^{-1}).
- Q14.7: A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the wavelength of sound in the tissue in which the speed of sound is 1.7 km (s^{-1}) ? The operating frequency of the scanner is 4.2 MHz.
- Q14.8(a): A transverse harmonic wave on a string is described by (y(x, t)) = (3.0 \sin (36 t + 0.018 x + \frac{\pi}{4})) where (x) and (y) are in cm and (t) in s. The positive direction of (x) is from left to right. Is this a travelling wave or a stationary wave ? If it is travelling, what are the speed and direction of its propagation ?
- Q14.8(b): A transverse harmonic wave on a string is described by (y(x, t)) = (3.0 \sin (36 t + 0.018 x + \frac{\pi}{4})) where (x) and (y) are in cm and (t) in s. The positive direction of (x) is from left to right. What are its amplitude and frequency ?
- Q14.8(c): A transverse harmonic wave on a string is described by (y(x, t)) = (3.0 \sin (36 t + 0.018 x + \frac{\pi}{4})) where (x) and (y) are in cm and (t) in s. The positive direction of (x) is from left to right. What is the initial phase at the origin ?
- Q14.8(d): A transverse harmonic wave on a string is described by (y(x, t)) = (3.0 \sin (36 t + 0.018 x + \frac{\pi}{4})) where (x) and (y) are in cm and (t) in s. The positive direction of (x) is from left to right. What is the least distance between two successive crests in the wave ?
- Q14.9: For the wave described, 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 ?
- Q15.1: A string of mass 2.50 kg is under a tension of 200 N. The length of the stretched string is 20.0 m. If the transverse jerk is struck at one end of the string, how long does the disturbance take to reach the other end?
- Q15.10(a): For the travelling harmonic wave $y(x, t) = 2.0 \cos 2\pi (10t – 0.0080 x + 0.35)$ 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 (a) 4 m,
- Q15.10(b): For the travelling harmonic wave $y(x, t) = 2.0 \cos 2\pi (10t – 0.0080 x + 0.35)$ 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 (b) 0.5 m,
- Q15.10(c): For the travelling harmonic wave $y(x, t) = 2.0 \cos 2\pi (10t – 0.0080 x + 0.35)$ 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 (c) $\lambda/2$,
- Q15.10(d): For the travelling harmonic wave $y(x, t) = 2.0 \cos 2\pi (10t – 0.0080 x + 0.35)$ 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 (d) $3\lambda/4$
- Q15.11(a): The transverse displacement of a string (clamped at its both ends) is given by $y(x, t) = 0.06 \sin(\frac{2\pi x}{3}) \cos(120 \pi t)$ where $x$ and $y$ are in m and $t$ in s. The length of the string is 1.5 m and its mass is $3.0 \times 10^{-2}$ kg. Answer the following : (a) Does the function represent a travelling wave or a stationary wave?
- Q15.11(b): The transverse displacement of a string (clamped at its both ends) is given by $y(x, t) = 0.06 \sin(\frac{2\pi x}{3}) \cos(120 \pi t)$ where $x$ and $y$ are in m and $t$ in s. The length of the string is 1.5 m and its mass is $3.0 \times 10^{-2}$ kg. Answer the following : (b) Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave ?
- Q15.11(c): The transverse displacement of a string (clamped at its both ends) is given by $y(x, t) = 0.06 \sin(\frac{2\pi x}{3}) \cos(120 \pi t)$ where $x$ and $y$ are in m and $t$ in s. The length of the string is 1.5 m and its mass is $3.0 \times 10^{-2}$ kg. Answer the following : (c) Determine the tension in the string.
- Q15.12(i)(a): For the wave on a string described in Exercise 15.11, do all the points on the string oscillate with the same (a) frequency? Explain your answers.
- Q15.12(i)(b): For the wave on a string described in Exercise 15.11, do all the points on the string oscillate with the same (b) phase? Explain your answers.
- Q15.12(i)(c): For the wave on a string described in Exercise 15.11, do all the points on the string oscillate with the same (c) amplitude? Explain your answers.
- Q15.12(ii): What is the amplitude of a point 0.375 m away from one end?
- Q15.13(a): 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) $y = 2 \cos(3x) \sin(10t)$
- Q15.13(b): 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: (b) $y = 2 \sqrt{x-vt}$
- Q15.13(c): 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: (c) $y = 3 \sin(5x - 0.5t) + 4 \cos(5x - 0.5t)$
- Q15.13(d): 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: (d) $y = \cos x \sin t + \cos 2x \sin 2t$
- Q15.14(a): A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is $3.5 \times 10^{-2}$ kg and its linear mass density is $4.0 \times 10^{-2}$ kg $m^{-1}$. What is (a) the speed of a transverse wave on the string, and
- Q15.14(b): A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is $3.5 \times 10^{-2}$ kg and its linear mass density is $4.0 \times 10^{-2}$ kg $m^{-1}$. What is (b) the tension in the string?
- Q15.15: A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340 Hz) when the tube length is 25.5 cm or 79.3 cm. Estimate the speed of sound in air at the temperature of the experiment. The edge effects may be neglected.
- Q15.16: A steel rod 100 cm long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod are given to be 2.53 kHz. What is the speed of sound in steel?
- Q15.17: A pipe 20 cm long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a 430 Hz source ? Will the same source be in resonance with the pipe if both ends are open? (speed of sound in air is 340 m $s^{-1}$).
- Q15.18: Two sitar strings A and B playing the note ‘Ga’ are slightly out of tune and produce beats of frequency 6 Hz. The tension in the string A is slightly reduced and the beat frequency is found to reduce to 3 Hz. If the original frequency of A is 324 Hz, what is the frequency of B?
- Q15.19(a): Explain why (or how): (a) in a sound wave, a displacement node is a pressure antinode and vice versa,
- Q15.19(b): Explain why (or how): (b) bats can ascertain distances, directions, nature, and sizes of the obstacles without any “eyes”,
- Q15.19(c): Explain why (or how): (c) a violin note and sitar note may have the same frequency, yet we can distinguish between the two notes,
- Q15.19(d): Explain why (or how): (d) solids can support both longitudinal and transverse waves, but only longitudinal waves can propagate in gases, and
- Q15.19(e): Explain why (or how): (e) the shape of a pulse gets distorted during propagation in a dispersive medium.
- Q15.2: A stone dropped from the top of a tower of height 300 m splashes into the water of a pond near the base of the tower. When is the splash heard at the top given that the speed of sound in air is 340 m $s^{-1}$ ? ($g$ = 9.8 m $s^{-2}$)
- Q15.20(i)(a): 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 $s^{-1}$,
- Q15.20(i)(b): 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 (b) recedes from the platform with a speed of 10 m $s^{-1}$?
- Q15.20(ii): A train, standing at the outer signal of a railway station blows a whistle of frequency 400 Hz in still air. (ii) What is the speed of sound in each case ? The speed of sound in still air can be taken as 340 m $s^{-1}$.
- Q15.21: 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 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 $s^{-1}$
- Q15.3: A steel wire has a length of 12.0 m and a mass of 2.10 kg. What should be the tension in the wire so that speed of a transverse wave on the wire equals the speed of sound in dry air at 20 °C = 343 m $s^{-1}$.
- Q15.4(a): Use the formula $v = \sqrt{\frac{\gamma P}{\rho}}$ to explain why the speed of sound in air (a) is independent of pressure,
- Q15.4(b): Use the formula $v = \sqrt{\frac{\gamma P}{\rho}}$ to explain why the speed of sound in air (b) increases with temperature,
- Q15.4(c): Use the formula $v = \sqrt{\frac{\gamma P}{\rho}}$ to explain why the speed of sound in air (c) increases with humidity.
- Q15.5(a): 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. $y = f (x \pm v t)$. Is the converse true? Examine if the following functions for $y$ can possibly represent a travelling wave : (a) $(x - vt)^2$
- Q15.5(b): 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. $y = f (x \pm v t)$. Is the converse true? Examine if the following functions for $y$ can possibly represent a travelling wave : (b) $\log[\frac{(x + vt)}{x_0}]$
- Q15.5(c): 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. $y = f (x \pm v t)$. Is the converse true? Examine if the following functions for $y$ can possibly represent a travelling wave : (c) $\frac{1}{(x + vt)}$
- Q15.6(a): A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a water surface, what is the wavelength of (a) the reflected sound, (b) the transmitted sound? Speed of sound in air is 340 m $s^{-1}$ and in water 1486 m $s^{-1}$.
- Q15.6(b): A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a water surface, what is the wavelength of (b) the transmitted sound? Speed of sound in air is 340 m $s^{-1}$ and in water 1486 m $s^{-1}$.
- Q15.7: A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the wavelength of sound in the tissue in which the speed of sound is 1.7 km $s^{-1}$ ? The operating frequency of the scanner is 4.2 MHz.
- Q15.8(a): A transverse harmonic wave on a string is described by $y(x, t) = 3.0 \sin (36 t + 0.018 x + \pi/4)$ where $x$ and $y$ are in cm and $t$ in s. The positive direction of $x$ is from left to right. (a) Is this a travelling wave or a stationary wave ? If it is travelling, what are the speed and direction of its propagation ?
- Q15.8(b): A transverse harmonic wave on a string is described by $y(x, t) = 3.0 \sin (36 t + 0.018 x + \pi/4)$ where $x$ and $y$ are in cm and $t$ in s. The positive direction of $x$ is from left to right. (b) What are its amplitude and frequency ?
- Q15.8(c): A transverse harmonic wave on a string is described by $y(x, t) = 3.0 \sin (36 t + 0.018 x + \pi/4)$ where $x$ and $y$ are in cm and $t$ in s. The positive direction of $x$ is from left to right. (c) What is the initial phase at the origin ?
- Q15.8(d): A transverse harmonic wave on a string is described by $y(x, t) = 3.0 \sin (36 t + 0.018 x + \pi/4)$ where $x$ and $y$ are in cm and $t$ in s. The positive direction of $x$ is from left to right. (d) What is the least distance between two successive crests in the wave ?
- Q15.9: For the wave described in Exercise 15.8, 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 ?
CBSE Solutions for Class 11 Physics Waves
Chapters in CBSE - Class 11 Physics
Top Tutors who teach Waves
Top Tutor
Vayusenabad, Delhi
5 yrs of
Exp
5 students
118 hrs of classes completed
400
per hour
Online Classes
One of the key challenges in teaching commerce is the diversity of students’ learning styles. Some students may excel in theory, while others may struggle with calculations or interpreting financial statements. As a commerce teacher, you need to be adaptable—using visual aids, group work, and real-world scenarios to make abstract concepts more understandable. For example, showing a video on how a business grows can bring the business theory to life in a way that reading a textbook may not. You might begin a lesson by discussing real-world examples, such as current economic trends, business case studies, or financial news. This helps students connect theoretical concepts to practical situations. For example, when teaching accounting principles, you might ask students to work through a mock company’s balance sheet or profit and loss statement, making the material more tangible. Regular quizzes, assignments, and group projects can be used to assess both theoretical knowledge and practical application of commerce concepts. For instance, group projects in business studies might involve students creating a business plan or analyzing a company’s performance, fostering teamwork and real-world problem-solving.
He classes are really interesting, ma'am makes the classes really fun so that understanding becomes easier.
Top Tutor
Talwandi Sector-B, Kota
10 yrs of
Exp
13 students
891 hrs of classes completed
1850
per hour
Online Classes
Student's Home
Tutor's home
My teaching exp.11 year in most reputed coaching in india. Mentor of jee top rank air-12 yatis agrawal, 19, 28 many more in top 100 in jee adavnced. Many of students got 100%tile in jee mains along with 100% marks in board exams. I also provide complete relevent study material like worksheet, practice sheet , dpps , assignment etc. Daily quizz based on assignment. Periodically test papers topic wise, minor and major as per shedule. I have helped thousands of students overcome their fear of maths.
He is a very nice sir and knows his knowledge well, he is also well educated and good with his content.
Top Tutor
Karampura, Delhi
10 yrs of
Exp
5 hrs of classes completed
2500
per hour
Online Classes
Student's Home
I teach and help students to achieve immediate and long term goals. My experience as a personal tutor and the founder of an online educational startup 'MinDely' has helped him gain knowledge of child psychology and understanding students needs and takes measures accordingly. I plan the lessons as per the students ability to cope up and moreover upon their needs. My teaching technique is student friendly and easy to understand which helps students understand it better. Weekly test conducted by him helps the parents to know the students performance.
Sir explains concepts clearly and makes an effort to make sure that i understand the logic behind formulas rather than just memorising them.The examples discussed in class are helpful,which makes the subject easier to understand. He is patient with doubts which creates a positive learning environment. Since this subject is relatively new and has no official NCERT textbook his self prepared material helps alot.
Super Tutor
Sector 15, New Chandigarh
13 yrs of
Exp
48 students
3180 hrs of classes completed
400
per hour
Online Classes
Go through my profile to know about me Chemistry 11 is a pivotal, often challenging, transition from general science to in-depth, analytical chemistry, focusing on atomic structure, stoichiometry, and chemical reactions. The experience involves intensive lab work, rigorous mathematical problem-solving (mole concept), and understanding molecular behavior, moving from conceptual understanding to practical, hands-on experimentation.
She was an excellent tutor; thanks to her guidance, I achieved outstanding marks in my 12th-grade board exams. Thank you is such a small word for all your help—thank you so much, Ma'am! ❤️✨
Top Tutor
Behala, Kolkata
6 yrs of
Exp
4 students
1162 hrs of classes completed
1000
per hour
Online Classes
Student's Home
Tutor's home
As a passionate mathematics teacher, I have always enjoyed imparting knowledge to others. I am a B.Tech graduate in Electronics and Communication. I have been teaching mathematics for over four years, and my love for the subject has grown stronger with time. I started teaching as a part-time tutor while pursuing my engineering degree and fell in love with the profession. After completing my degree, I worked with Byjus, one of the leading Edtech companies in India, as a mathematics teacher for two years. While I enjoyed my time at Byju's, I could do more as a teacher and wanted to explore new avenues. I joined Vedantu, an online learning platform, where I have been teaching mathematics for the last two years. I started my journey as a math teacher out of pure passion. Mathematics has always been my favourite subject, and I enjoyed solving problems and equations from a young age. As I grew older, I realised I had a natural talent for teaching, and I decided to pursue it as a career. Currently, I provide mathematics classes for students from sixth to twelfth for all boards, including ICSE, CBSE, and West Bengal. I teach offline and online classes, each lasting for one and a half hours. I use interactive tools such as Zoom, Pentab, and One Note to make my lessons engaging and interactive for my students. To help my students excel in mathematics, I provide them with reference books, notes, and in-built assignments to test their knowledge. I also conduct monthly tests to assess their progress and identify improvement areas. Teaching on UrbanPro has allowed me to interact with students from different parts of the country and help them realise their full potential in mathematics. I want to continue making a positive impact in the lives of my students and help them develop a love for mathematics, just like I did
Sirs class is very interactive, explains each topic in detail making it crystal clear for the student to understand and learn. Always approachable and is ready to clear doubts anytime.
