In this chapter we will start the discussion on wave phenomena. We will study the following topics:

Transcription

1 Chapter 16 Waves I In this chapter we will start the discussion on wave phenoena. We will study the following topics: Types of waves Aplitude, phase, frequency, period, propagation speed of a wave Mechanical waves propagating along a stretched string Wave equation Principle of superposition of waves Wave interference Standing waves, resonance (16 1)

2 A wave is defined as a disturbance that is self-sustained and propagates in space with a constant speed Waves can be classified in the following three categories: 1. Mechanical waves. These involve otions that are governed by Newton s laws and can exist only within a aterial ediu such as air, water, rock, etc. Coon exaples are: sound waves, seisic waves, etc.. Electroagnetic waves. These waves involve propagating disturbances in the electric and agnetic field governed by Maxwell s equations. They do not require a aterial ediu in which to propagate but they travel through vacuu. Coon exaples are: radio waves of all types, visible, infra-red, and ultraviolet light, x-rays, gaa rays. All electroagnetic waves propagate in vacuu with the sae speed c = 300,000 k/s 3. Matter waves. All icroscopic particles such as electrons, protons, neutrons, atos etc have a wave associated with the governed by Schroedinger s equation. (16 )

3 Transverse and Longitudinal waves (16 3) Waves can be divided into the following two categories depending on the orientation of the disturbance with r respect to the wave propagation velocity v. If the disturbance associated with a particular wave is perpendicular to the wave propagation velocity, this wave is called " transverse". An exaple is given in the upper figure which depicts a echanical wave that propagates along a string. The oveent of each particle on the string is along the y-axis; the wave itself propagates along the x-axis. A wave in which the associated disturbance is parallel to the wave propagation velocity is known as a " lonitudinal wave". An exaple of such a wave is given in the lower figure. It is produced by a piston oscillating in a tube filled with air. The resulting wave involves oveent of the air olecules along the axis of the tube which is also the direction of the wave propagation velocity v r.

4 (16 4) Consider the transverse wave propagating along the string as shown in the figure. The position of any point on the string can be described by a function y = h( x, t). Further along the chapter we shall see that function h has to have a specific for to describe a wave. Once such suitable ( ω ) function is: y( x, t) = y sin kx - t Such a wave which is described by a sine (or a cosine) function is known as " haronic wave". The various ters that appear in the expression for a haronic waveare identified in the lower figure Function y( x, t) depends on x and t. There are two ways to visualize it. The first is to "freeze" tie (i.e. set o ). This is like taking a snapshot of the t = to y = y ( x to ) = x. In this case y = y ( x, t. ) wave at., The second is to set x o t = t o

5 (16 5) The aplitude ( ω ) y( x, t) = y sin kx t y is the absolute value of the axiu displaceent fro the equilibriu position. The phase is defined as the arguent of the sine function. The wavelength λ is the shortest distance between two repetitions of the wave at a fixed tie. ( kx ωt ) We fix t at t = 0. We have the condition: y( x,0) = y( x + λ,0) ( ) = ( + λ ) = ( + λ ) 1 1 y sin kx1 y sin k x1 y sin kx1 k π Since the sine function is periodic with period π kλ = π k = λ A period T is the tie it takes (with fixed x ) tthe sine function to coplete one oscillation. We take x = 0 y(0, t) = y(0, t + T ) π y sin ( ωt ) = y sin ω ( t + T ) = y sin ( ωt + ωt ) ωt = π ω = T

6 (16 6) v = ω k The speed of a traveling wave In the figure we show two snapshots of a haronic wave taken at ties t and t + t. During the tie interval t the wave has traveled a distance x. The wave speed x v =. One ethod of finding v is to iagine that t we ove with the sae speed along the x-axis. In this case the wave will see to us that it does not change. ( ω ) Since y( x, t) = y sin kx t this eans that the arguent of the sine function is constant. kx ωt = constant. We take the derivative with respect to t. dx dx ω dx ω k ω = 0 = The speed v = = dt dt k dt k A haronic wave that propagates along the negative x-axis is described by the equation: ( ω ) ( ω ) y( x, t) = y sin kx + t. The function y( x, t) = h kx t describes a general wave that propagates along the positive x-axis. A general wave that propagates along the ( ) negative x-axis is described by the equatio n: y( x, t) = h kx +ωt

7 v = τ µ Wave speed on a stretched string Below we will deterine the speed of a wave that propagates along a string whose linear ass density is µ. The tension on the string is equal to τ. Consider a sall section of the string of length l. The shape of the eleent can be approxiated to be an arc of a circle of radius R ( τ θ ) whose center is at O. The net force in the direction of O is F = sin. l l Here we assue that θ = 1 sin θ ; θ = F = τ ( eqs.1) R R v v The force is also given by Newton's second law: F = = ( µ l ) (eqs.) R R If we copare equations 1 and we get: Note : v l µ = τ v = R R ( l ) The speed v depends on the tension τ and the ass density µ but not on the wave frequency f. τ µ (16 7)

8 Rate of energy transission Consider a transverse wave propagating along a string which is described by the equation: ( ω ) y( x, t) = y sin kx - t. The transverse velocity y u = = -ω y cos ( kx - ωt) At point a both t y and u are equal to zero. At point b both y and u have axia. 1 In general the kinetic energy of an eleent of ass d is given by: dk = dv 1 1 dk = dx - cos ( - ) The rate at which konetic energy propagates ω y kx ωt dk 1 along the string is equal to = µ vω y cos ( kx - ωt ) The average rate dt dk 1 1 = µ vω y cos ( kx - ω t ) = µ vω y As in the case of the oscillating dt avg 4 avg spring-ass syste du dk du dk = P = + dt dt dt dt avg avg avg avg avg = (16 8) 1 µ vω y

9 y 1 y = t v t θ The wave equation (16 9) Consider a string of ass density µ and tension τ A transverse wave propagates along the string. The transverse otion is described by y( x, t) Consider an eleent of length dx and ass d = µ dx The forces 1 the net force along the y-axis is given by the equation: ( ) F = F sinθ F sinθ = τ sinθ sin θ Here we ynet y assue that θ1 = 1 and θ = 1 sinθ1 ; tanθ1 = x and sinθ F ; = F = τ tanθ y y y = Fynet τ = x x x 1 1 Fro Newton's second law we have: = = = y y y Fynet day µ dx τ t x x 1 y y y y y x x 1 y µ y 1 y τ µ dx x x = = = = 1 t dx t τ t v t

10 y ( x, t) = y ( x, t) + y ( x, t) 1 The principle of superposition for waves y µ y 1 y The wave equation = = even though t τ t v t it was derived for a transverse wave propagating along a string under tension, is true for all types of waves. This equation is "linear" which eans that if y and are solutions of the wave equation, the function c y + c y is also a solution. Here c and c are constants. The principle of superposition is a direct consequence of the linearity of the wave equation. This principle can be expressed as follows: y Consider two waves of the sae type that overlap at soe point P in space. Assue that the functions y ( x, t) and y ( x, t) describe the displaceents 1 if the wave arrived at P alone. The displaceent at P when both waves are present is given by: y ( x, t) = y ( x, t) + y ( x, t) 1 Note : Overlapping waves do not in any way alter the travel of each other (16 10)

11 (16 11) Interference of waves Consider two haronic waves of the sae aplitude and frequency which propagate along the x-axis. The two waves have a phase difference φ. We will cobine these waves using the principle of superposition. The phenoenon of cobing waves is knwon as interference and the two waves are said to interfere. The displaceent of the two waves sin ( kx ωt ) y x t = y ( kx ωt + φ ) y = y1 + y (, ) sin ( ω ) + sin ( kx ω + φ ) are given by the functions: y1 ( x, t) = y and (, ) sin. y x t = y kx t y t φ φ y ( x, t ) = y cos sin kx ωt + The resulting wave has the sae frequency as the original waves, and its aplitude y φ φ = y cos Its phase is equal to

12 (16 1) Constructive interference The aplitude of two interefering waves is given by: φ y = y cos It has its axiu value if φ = 0 In this case y = y The displaceent of the resulting wave is: y φ ( x, t ) = [ y ] sin kx ωt + This phenoenon is known as fully constructive interference

13 (16 13) Destructive interference The aplitude of two interefering waves is given by: φ y = y cos It has its iniu value if φ = π In this case y = 0 The displaceent of the resulting wave is: ( x t ) y, = 0 This phenoenon is known as fully destructive interference

14 (16 14) Interediate interference The aplitude of two interefering waves is given by: y = φ y cos When interference is neither fully constructive nor fully destructive it is called interediate interference π An exaple is given in the figure for φ = 3 In this case y = y The displaceent of the resulting wave is: π y ( x, t ) = [ y ] sin kx ωt + 3 Note : Soeties the phase difference is expressed as a difference in wavelength λ In this case reebre that: π radians 1λ

15 (16 15) Phasors A phasor is a ethod for representing a wave whose ( ) = ( ω ) diasplaceent is: y x, t y sin kx t 1 1 The phasor is defined as a vector with the following properties: 1. Its agnitude is equal to the wave's aplitude y 1. The phasor has its tail at the origin O and rotates in the clockwise direction about an axis through O with angular speed ω. Thus defined, the projection of the phasor on the y-axis (i.e. its y-coponent) is equal to ( ) ( ω φ ) 1 ( ω ) y sin kx t A phasor diagra can be used to represent ore than one waves. (see fig.b). The displaceent of the second wave is: y x, t = y sin kx t + The phasor of the second wave fors an angle φ with the phasor of the first wave indicating that it lags behind wave 1 by a phase angle φ.

16 (16 16) Wave addition using phasors Consider two waves that have the sae frequency but different aplitudes. They also have a phase difference φ. The displaceents of the two waves y1 ( x t ) = y1 ( kx ωt ) ( ) = ( ω + φ ) are:, sin and y x, t y sin kx t. The superposition of the two waves yields a wave that has the saeangular frequency ω ( ω β ) and is described by: y = y sin kx t + Here y is the wave aplitude and β is the phase angle. To deterine y and β we add the two phasors representing the waves as vectors (see fig.c). Note: The phasor athod can be used to add vectors that have different aplitudes.

17 ( ) [ ] y x, t = y sin kx cosωt Standing Waves : Consider the superposition of two waves that have the sae frequency and aplitude but travel in opposite directions. The displaceents y1 ( x t ) = y ( kx ωt ) y ( x t) = y sin ( kx + ωt ) y ( x t ) = y1 ( x t ) + y ( x t ) (, ) sin ( ω ) sin ( ω ) [ sin ] cosω of two waves are:, sin,, The displaceent of the resulting wave,,, y x t = y kx t + y kx + t = y kx t This is not a traveling wave but an oscillation that has a position dependent aplitude. It is known as a standing wave. (16 17)

18 The displaceent of a standing wave is given by the equation: ( ) [ ] y x, t = y sin kx cosωt The position dependant aplitude is equal to y sin kx a a Nodes : These are defined as positions where the standing wave aplitude vanishes. They occur when kx = nπ n = 0,1,, π λ x = nπ xn = n n = 0,1,,... λ n n n n a (16 18) Antinodes : These are defined as positions where the standing wave aplitude is axiu. 1 They occur when kx = n + π n = 0,1,,... π 1 1 λ x = n + π x n = n + n = 0,1,,... λ Note 1: The distance between ajacent nodes and antinodes is λ/ Note : The distance between a node and an ajacent antinode is λ/4

19 A A A (16 19) B B B Standing waves and resonance Consider a string under tension on which is claped at points A and B separated by a sistance L. We send a haronic wave traveling the thr right. the wave is reflected at point B and the reflected wave travels to the left. The left going wave reflects back at point A and creates a thrird wave traveling to the right. Thus we have a large nuber of overlapping waves half of which travel to the right and the rest to the left. For certain frequencies the interference produces a standing wave. Such a standing wave is said to be wave occurs are known as the at resonance. The frequencis at which the standing resonant frequencie s of the syste.

20 A (16 0) B Resonances occur when the resulting standing wave satisfies the boundary condition of the proble. These are that the Aplitude ust be zero at point A and point B and arise fro the fact that the string is claped at both points and therefore cannot ove. A A B B The first resonance is shown in fig.a. The standing wave has two nodes at points A and B. Thus L = λ = L. The second standing wave is shown 1 in fig.b. It has three nodes (two of the at A and B) λ1 λ In this case L = = λ λ = L The third standing wave is shown in fig.c. It has four nodes (two of the at A and B) λ In this case L = 3 = λ λ3 = L The general expression for the resonant 3 L v v wavelengths is: λn = n = 1,,3,... the resonant frequencies fn = = n n λ L n