CLASS 11th
Waves
Waves
01. Wave Motion Ÿ Ÿ
When a particle moves through space, it carries energy with itself. (Wave motion) to transport energy from one part to space to other without any bulk motion of material together with it.
Examples of waves Ripples on a pond (water waves), visible light, radio and TV signals
02. Classification of waves Based on medium necessity The waves which require medium for their propagation are called mechanical waves. In the propagation of mechanical waves elasticity and density of the medium is important therefore mechanical waves are known as elastic waves.
Based on energy propagation Based on direction of propagation Based on the motion of particles of medium Mechanical transverse waves produce in such type of medium which have shearing property
03. Speed of transverse wave on string As a wave travels along the x-axis, the points on the string oscillate back and forth in the y-direction. sin The maximum velocity of a small segment of the string is max NOTE ☞
Creating a wave of larger amplitude increases the speed of particles in the medium, but it does not change the speed of the wave through the medium.
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Waves
04. Characteristics of wave motion The disturbance travels through the medium due to repeated periodic oscillations. The energy is transferred from place to another without any actual transfer of the particles of the medium. There is a regular phase difference between one particle and the next. The velocity with which a wave travels is called as wave velocity. The wave velocity remains constant in a given medium
05. Some Important Terms Connected with Wave Motion Wavelength The distance between any two nearest particles, medium, vibrating in the same phase. Frequency Number of vibrations (Number of complete wavelengths) complete by a particle in one second. Time period Time taken by wave to travel a distance equal to one wavelength. Amplitude Maximum displacement of vibrating particle from its equilibrium position. Angular wave number It is defined as Wave number It is defined as
06. The General Equation of Wave Motion ...(i) ± The general solution of this equation is of the from ...(ii) Thus, any function of x and t and which satisfies equation (i) or which can be written as equation (ii) represent a wave. The only condition is that it should be finite everywhere and at all times. Speed of wave (v) is given by
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Waves
07. Equation of a Plane Progressive Wave
sin sin ...(i) sin Also ...(ii) sin ...(iii) This is the equation of a simple harmonic wave travelling along +x direction. If the wave is travelling along the –x direction then inside the brackets in the above equations, instead of minus sign there will be plus sign.
08. Intensity of Wave The amount of energy flowing per unit area and per unit time is called the intensity of wave. It is represented by I. Its units are or watt/metre2. i.e. ∝ and ∝
If P is the power of an isotropic point source, then intensity at a distance r is given by, or ∝ (for a line source) If P is the power of a line source, then intensity at a distance r is given by, or ∝ (for a line source) As, ∝ ∝ (for a point source) and ∝ Therefore, (for a line source)
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Waves
09. Superposition Principle Two or more waves can propagate in the same medium without affecting the motion of one another. If several waves propagate in a medium simultaneously, then the resultant displacement of any particle of the medium at any instant is equal to the vector sum of the displacements produced by individual wave. are the displacement of particle at a particular time due to individual waves, If then the resultant displacement is given by
Principle of superposition holds for all types of waves.
10. Interference of Waves sin and sin sin
Two waves By principle of superposition Where
sin cos and tan cos
As intensity
∝
Constructive interference (maximum intensity) max Destructive interference (minimum intensity) min
11. Velocity of Transverse Wave × Mass of per unit length where d = Density of matter
Velocity of transverse wave in any wire Ÿ Ÿ
it is called tension law. If m is constant then, ∝ If m is constant then, ∝ ← is called law of mass
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∵ or
Waves
Ÿ
Ÿ
If T is constant & taken wire of different radius for same material then ∝ ← it is called law of radiation If T is constant & taken wire of same radius for different material. Then ∝ ← law of density.
12. Stationary Waves Wave propagating in such a medium will be reflected at the boundary and produce a wave of the same kind travelling in the opposite direction. The superposition of two waves will give rise to stationary wave. Formation of stationary wave is possible only and only in bounded medium.
13. Analytical Method for Stationary Waves From rigid end Equation for progressive wave in positive x-direction sin After reflection from rigid end sin sin By principle of superposition. sin sin sin cos This is equation of stationary wave reflected from rigid end Amplitude = 2a sin kx Velocity of particle sin sin Strain cos cos Elasticity Change in pressure∆ Node Antinode
From free end Equation for progressive wave in positive x-direction sin After reflection from free end sin By Principle of superposition sin sin sin cos Velocity of particle cos cos Strain sin sin Change in pressure ∆ Antinode Node Amplitude cos
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Waves
14. Stationary Wave are of Two Types (a) Transverse stationary wave (stretched string) (b) Longitudinal stationary wave (organ pipes)
Transverse Stationary Wave ⇒ ⇒
Fundamental Harmonic
⇒ ⇒
Second Harmonic
Law of length For a given string, under a given tension, the fundamental frequency of vibration is inversely proportional to the length of the string, i.e., ∝ (T and m are constant)
Law of tension The fundamental frequency of vibration of stretched string is directly proportional to the square root of the tension in the string, provided that length and mass per unit length of the string are kept constant. ∝ and m are constant
Law of mass The fundamental frequency of vibration of a stretched string is inversely proportional to the square root of its mass per unit length provided that length of string and tension in the string and T are constant are kept constant, i.e., ∝
15. Sonometer Sonometer consists of a hollow rectangular box of light wood. One end of the experimental wire is faster to one end of the box. The wire passes over a frictionless pulley P at the other end of the box. The wire stretched by a tension T. The box serves the purpose of increasing the loudness of the sound produced by the vibrating wire. If the length of the wire between the two bridges is then the frequency of vibration is
16. Speed of Longitudinal (Sound) Waves For solid medium
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Where E = Y = Young‘s modulas
Waves For liquid Medium
Where E = B, where B = volume elasticity coefficient of liquid
For gas medium Adiabatic Elasticity so that
or
or
And from kinetic-theory of gases
(i)
s
So
Effect of temperature (a) For any gas medium (b) For air sec sec
(ii)
Effect of Relative Humidity . We With increase in humidity, density decreases so in the light of conclude that with rise in humidity velocity of sound increases. This is why sound travels faster in humid air.
(iii) Effect of Pressure
So pressure has no effect on velocity of sound in a gas as long as temperature remain constant.
(iv) Effect of Motion of Air If air is blowing then the speed of sound changes.
(v)
Effect of Frequency There is no effect of frequency on the speed of sound.
17. Vibration of Air Column in Organ Pipes When two longitudinal waves of same frequency and amplitude travel in a medium in opposite directions then by superposition, standing waves are produced. These waves are produced in air columns in cylindrical tube of uniform diameter. These sound producing tubes are called organ pipes.
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Waves (i)
Vibration of air column in closed organ pipe At the closed end there is node since particles does not have freedom to vibrate whereas at open end there is an antinode because particles have greatest freedom to vibrate.
(a)
(c)
(b)
Hence frequency of overtones is given by
(ii) Vibration of air columns in open organ pipe
(b)
(a)
The tube which is open at both ends is called an open organ pipe. Now the pipe is open at both ends by which an antinode is formed at open end. When open organ pipe vibrate in mth overtone then
18. Doppler Effect for Sound Waves The apparent change in the frequency of sound when the source of sound, the observer and the medium are in relative motion. Assumptions
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Waves (a) (b)
Velocity of the source, observer and medium are along the line joining the positions of the source and the observer. Velocity of the source and the observer is less than velocity of sound.
In sound it depends on whether the source or observer of both are in motion ± ′ ±
Notations → actual frequency → actual wave length → velocity of sound → velocity of observer
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′ → observed frequency (apparent frequency) ′ → observed (apparent) wave length → velocity of source → wind velocity
Waves
AIIMS Exercise (1) 1. A transverse wave is represented by the equation y = y0 sin (vt‒x) for what value of λ is the maximum particle velocity equal to two times the wave velocity? (a) λ = (b) λ =
(c) λ = (d) λ =
2. A wave y = a sin(ωt‒kx) on a string meets with another wave producing a node x = 0. Then the equation of the unknown wave is : (a) y = a sin(ωt+kx) (b) y = ‒a sin(ωt+kx) (c) y = a sin(ωt‒kx) (d) y = ‒a sin(ωt‒kx) 3. When a stretched wire and a tuning fork are sounded together, 5 beats per second are produced, when length of wire is 95 cm or 100 cm. Then the frequency of fork is : (a) 90 (b) 100
(c) 105 (d) 195
4. A glass tube of 1.0 metre length is filled with water. The water can be drained out slowly at the bottom of the tube. If a vibrating tuning fork of frequency 500 Hz is brought at the upper end of the tube and the velocity of sound is 330 m/s, then the total number of resonances obtained will be : (a) 4 (b) 3
(c) 2 (d) 1
5. Two wires of the same material and radii r and 2r respectively are welded together end to end. The combination is used as a sonometer wire and kept under tension T. The welded point is mid-way between the two bridges. When stationary waves are set up in the composite wire, the joint is a node. Then the ratio of the number of loops formed in the thinner to thicker wire is : (a) 2 : 3 (b) 1 : 2
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(c) 2 : 1 (d) 5 : 4
Waves 6. The linear density of a vibrating string is 10‒4 kg m‒1. A transverse wave is propagating on the string, which is described by the equation y = 0.02 sin 9(x + 30t) where x and y are in metres and time t is in seconds. The tension in the string is : (a) 0.09 N (b) 0.36 N
(c) 0.9 N (d) 3.6 N
7. A police car moving at 22 m/s chases a motor cyclist. The police car sounds its horn at 176 Hz, while both of them move towards a stationary siren of frequency 165 Hz. Calculate the speed of motor cycle if it is given that he does not observe any beats. (velocity of sound = 330 m/s) (a) 33 m/s (b) 22 m/s
(c) zero (d) 11 m/s
8. A stretched wire of some length under a tension is vibrating with its fundamental frequency. Its length is decreased by 45% and tension is increased by 21%. Now its fundamental frequency : (a) increases by 50% (b) increases by 100% (c) decreases by 50% (d) decreases by 25% 9. A radar sends a ratio signal of frequency 9 × 109 Hz towards an aircraft approaching the radar. If the reflected wave shows a frequency shift of 3 × 103 Hz, the speed with which the aircraft is approaching the radar in ms‒1 [velocity of the ratio signal = 3 × 108 ms‒1] (a) 150 (b) 100
(c) 50 (d) 25
10. A sound absorber attenuates the sound level by 20 dB. The intensity decreases by a factor of : (a) 10 (b) 100 11. The (a) (b) (c) (d)
motion is given This represents This represents This represents This represents
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(c) 1000 (d) 10000 by y = 2e2x.e3t, where y and x are in metres and t is in second. a progressive wave travelling in ‒ve x-direction. a progressive wave travelling in +ve x-direction. a stationary wave. a wave pulse.
Waves In each of the following questions, a statement of Assertion (A) is followed by a corresponding statement of Reason (R). Of the following statements, choose the correct one. (a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not correct explanation of A. (c) A is true but R is false. (d) A is false but R is true. (e) Both A and R are false. 12. (A) : Radiowaves can be polarised. (R) : Sound waves in air are longitudinal in nature. 13. (A) : Velocity of sound decreases with decrease in humidity in air. (R) : With decrease in humidity, the elasticity of air decreases.
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