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Chapter Contents Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456 Introduction Transverse and Longitudinal Waves Displacement Relation for a Progressive Wave The Principle of Superposition of Waves Reflection of Waves Beats Doppler Effect Chapter 15 Waves Introduction Most of us experience the phenomenon of wave propagation when we drop a stone in a pond of still water. These waves move outwards in expanding circles until they reach the shore. It seems as if the water is moving outward from the point of disturbance. If we examine carefully the motion of a leaf floating on the disturbed water, we see that the leaf moves up and down about its original position but there is no displacement of leaf towards the shore. Thus we can say that energy is transferred but there is no transfer of medium. Such a pattern in which there is no actual transfer or flow of matter as a whole but energy is transmitted from one part of a medium to another part is called wave. Waves travelling in a medium are closely connected to harmonic oscillations. In this chapter, we will discuss the type of propagation of waves through different media, and the factors affecting the speed of waves in them. We’ll also discuss the superposition of waves and Doppler’s effect in sound. There are mainly three types of waves : 1. Mechanical waves 2. Electromagnetic waves 3. Matter waves 1. Mechanical waves require a material medium for their propagation. They cannot travel through vacuum. Sound waves, water waves, and waves on a spring etc. are some examples of mechanical waves. 2. Electromagnetic waves do not require a medium for their propagation. They can travel through vacuum. X-rays, radio waves, light etc., are electromagnetic waves. 3. Matter waves are associated with constituents of matter such as electrons, protons, neutrons, atoms and molecules. In this chapter, we will discuss mechanical waves and their characteristic properties in details.
44 Waves NEET Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456 TRANSVERSE AND LONGITUDINAL WAVES Mechanical waves are further divided into two parts : 1. Transverse waves 2. Longitudinal waves 1. Transverse waves : Transverse waves are the waves in which the constituents of the medium oscillate perpendicular to the direction of wave propagation. If we give an upward jerk to one end of a long rope that has its opposite end fixed, a single wave pulse is formed and travels along the rope with a fixed speed. Pulse But if we give continuous periodic up and down jerks to one end Harmonic wave Crest Trough of the rope, a sinusoidal wave is produced on the rope. It travels in the form of crest and trough. One crest and one trough makes a wave. length of one wave called wave length. 2. Longitudinal waves : Longitudinal waves are the waves in which the constituents of the medium oscillates along the direction of wave propagation. It travels in the form of compression and rarefaction. S. No. Transverse Longitudinal 1 Particles of the medium vibrate at right angles to the direction of wave motion 2 Particle velocity is always perpendicular to wave velocity Particles of the medium vibrate in the direction of wave motion Particle velocity is parallel or antiparallel to wave velocity 3 These can be polarised Cannot be polarised 4 Do not exist in gases as they do not possess shear modulus or modulus of rigidity Can exist in a solid, liquid or gas DISPLACEMENT RELATION FOR A PROGRESSIVE WAVE If, during propagation of a wave in a medium, the particles of the medium perform simple harmonic motion then the wave is called a ‘simple harmonic progressive wave’ and a plane progressive wave travelling in +x direction is given by the equation : y = a sin (t – kx ± ) while the equation of a plane progressive wave travelling in –x direction is given by y = a sin (t + kx ± ) Note : Here 2 , T    [T is time period] is called angular frequency 2 k , v      [ is wavelength and v is wave velocity]. This 'k' is called “angular wave number” or propagation constant.
NEET Waves 45 Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456 Wave Function It is a mathematical description of the disturbances created by a wave. For a string, the wave function is a (vector) displacement, whereas for sound waves, it is a (scalar) pressure or density fluctuations. In xy frame y = f (x) y = y, x = x – vt so, y = f (x – vt) In general y = f (x ± vt) O x = 0 x vt = y x vt O x y A pulse is observed from a stationary ( ) xy frame and a moving ( ) frame x y  (i) If y = f (x + vt), then wave is moving in negative x-direction with velocity v. (ii) If y = f(x – vt), then wave is moving in positive x-direction with velocity v. (iii) If y = f (x ± vt) 2, y f x vt     or f (x ± vt) 3 are valid wave equation. (iv) y f x vt     , y = f (x2 ± v2t) or f (x3 ± v3t) are not wave equation. Differential Equation of Wave It has been shown analytically that any function of space and time which satisfies the equation 2 2 2 22   1    y y x v t represents a wave. Here y is the wave function and doesn’t necessarily denote y co-ordinate. y = A sint or y = A sinkx, don’t satisfy the above equation so do not represent waves, while functions y = A sin(t – kx), y = Alog(at + bx) y = A sinkx sint or y = Asin(t – kx) + Bcos(t + kx) satisfy the above equation, so represent waves. Equations of the form y = f(at ± bx) represent travelling or progressive waves. Following are the common travelling wave equations : Travelling y = Alog(at + bx), y =   ax bt  , y = (ax – bt) 2,  2 – – B x vt y Ae  , y = Asin(ax – bt) 2, y = acos2(t – kx) or y = acost sin(t – kx) But y = A sin(2x2 – 3t 2), y = A sin(ax2 + bt), y = coskx. sint, don’t represent traveling or progressive waves.
46 Waves NEET Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456 If a traveling wave is a sine or cosine function it is known as harmonic wave. In our exam we are asked just to recognize the equations of spherical and cylindrical progressive waves. sin –   A y t kr r   represents spherical progressive waves, while y = sin –   A t kr r  represents cylindrical progressive waves. Maximum exam questions are related to one dimensional plane progressive waves which is given by y A t kx    sin  ∓ Just remember the meaning of each term and the various forms in which the equation can be written as follows : (i) y in general is disturbance produced by the wave but right now remember it is just the displacement of medium particles from its mean position. It’ll be given in centimeter or meter. (ii) A represents the amplitude or maximum displacement of the medium particle from the mean position, will be given in centimeter or metre. (iii)  is known as angular or circular frequency of medium particle in oscillation under SHM. Its unit will be radian per sec.  = 2 = 2 T  , where  is natural frequency given in hertz or per sec or cycle per sec. T is time period given in second. (iv) t is the variable time from when the wave begins. Given in second. (v) –ve sign between t and x indicates that the wave is travelling in +ve, x-direction. (vi) k is called propagation constant or angular wave number and given by k = 2  , where  is the wavelength. Note that 1  is also written as  or wave number which indicates the number of waves per metre. (vii) x is the position of the medium particle from where the wave has started. (viii) The constant  is called phase constant or initial phase. (xi) Medium particle velocity vp = cos  dy A t kx dt   ∓ So, maximum particle velocity vpmax = A (x) vp = – vw × slope of wave. p w dy v v dx   Example 1 : The frequency of a tuning fork is 150 Hz and distance travelled by the sound, produced in air is 20 cm in one vibration. Calculate the speed of sound in air. Solution : v =  Distance travelled by the sound in one vibration equals wavelength of sound wave  = 20 cm = 1 m 5  Speed of sound in air, 1 150 30 m/s 5 v  