9.1 Waves in Two Dimensions

Transcription

1 INVESTIGATION 9.. Transmsson, Reflecton, and Refracton of Water Waes n a Rpple Tank How can we ew waes to study them? How are waes transmtted, reflected, and refracted n a rpple tank? 9. Waes n Two Dmensons It s dffcult to study the propertes of waes for sound, lght, and rado because we cannot ew the waes drectly. Howeer, f we use a rpple tank, not only can we ew the waes drectly, but we can create most condtons needed to demonstrate the propertes of transerse waes n ths two-dmenson space. Inestgaton 9.., n the Lab Acttes secton at the end of ths chapter, prodes you wth an opportunty to study the propertes of waes n a rpple tank n order to better understand and predct smlar behaours and relatonshps for other waes. Transmsson A wae orgnatng from a pont source s crcular, whereas a wae orgnatng from a lnear source s straght. We confne ourseles for the moment to waes from sources wth a constant frequency. As a wae moes away from ts constant-frequency source, the spacng between successe crests or successe troughs the waelength remans the same proded the speed of the wae does not change. A contnuous crest or trough s referred to as a. To show the drecton of trael, or transmsson, of a, an arrow s drawn at rght angles to the (Fgure ). Ths lne s called a wae. Sometmes we refer to wae s nstead of s when descrbng the behaour of a wae. drecton of trael Fgure In both cases the wae s at 90 to the. transerse wae perodc dsturbance where partcles n the medum oscllate at rght angles to the drecton n whch the wae traels the leadng edge of a contnuous crest or trough wae a straght lne, drawn perpendcular to a, ndcatng the drecton of transmsson straght wae When the speed decreases, as t does n shallow water, the waelength decreases (Fgure ), ce waelength s drectly proportonal to speed (λ ). When the frequency of a source s ncreased, the dstance between successe crests becomes smaller, ce waelength s nersely proportonal to frequency λ f. Both proportonaltes are consequences of the unersal wae equaton, f λ. Ths equaton holds for all types of waes one-dmensonal, two-dmensonal, and three-dmensonal. The wae traellng n deep water has a speed f λ. Smlarly, f λ for the wae traellng n shallow water. In a rpple tank, the frequency of a water wae s determned by the wae generator and does not change when the speed changes. Thus f f. If we dde the frst equaton by the second equaton, we get f λ f λ Howeer, f f. Therefore, crcular wae λ λ 444 Chapter 9 NEL

2 Secton 9. deep shallow λ λ Fgure Perodc straght waes traellng from deep water to shallow water (left to rght) SAMPLE problem A water wae has a waelength of.0 cm n the deep secton of a tank and.5 cm n the shallow secton. If the speed of the wae n the shallow water s cm/s, what s ts speed n the deep water? Soluton λ.0 cm λ.5 cm cm/s? λ λ λ λ. 0 cm.5 cm cm/s 6 cm/s The speed of the wae n deep water s 6 cm/s. Practce Understandng Concepts. The speed and the waelength of a water wae n deep water are 8.0 cm/s and.0 cm, respectely. The speed n shallow water s 0.0 cm/s. Fnd the correspondng waelength.. A wae traels 0.75 tmes as fast n shallow water as t does n deep water. Fnd the waelength of the wae n deep water f ts waelength s.7 cm n shallow water. 3. In queston, what are the respecte frequences n deep and shallow water? Answers.. cm. 3.6 cm Hz; 9.0 Hz NEL Waes and Lght 445

3 angle of ncdence ( ) the angle between the ncdent and the barrer, or the angle between the ncdent and the angle of reflecton ( r ) the angle between the reflected and the barrer, or the angle between the reflected and the Reflecton from a Straght Barrer A straght traels n the wae drecton perpendcular to the, but how wll t behae when encounterng obstacles? When a straght runs nto a straght reflecte barrer, head on, t s reflected back along ts orgnal path (Fgure 3). If a wae encounters a straght barrer oblquely (.e., at an angle other than 90 ), the s lkewse reflected oblquely. The angle formed by the ncdent and the s equal to the angle formed by the reflected and the. These angles are called the angle of ncdence ( ) and the angle of reflecton ( r ), respectely (Fgure 4). Reflecton leaes waelength, speed, and frequency unchanged. refracton the bendng effect on a wae s drecton that occurs when the wae enters a dfferent medum at an angle barrer barrer ncdent waes reflected waes hgher speed (deep water) ncdent wae R refracted wae R Fgure 3 A straght meetng a straght barrer head on s reflected back along ts orgnal path. ncdent wae reflected wae lower speed (shallow water) r lower speed (shallow water) ncdent wae R refracted wae R ncdent r reflected hgher speed (deep water) Fgure 5 When water waes trael oblquely nto a slower medum, the wae bends toward the. If the new medum s a faster one, the wae bends away from the. Fgure 4 When a wae encounters a straght barrer oblquely, rather than head on, the angle of ncdence equals the angle of reflecton. Refracton When a wae traels from deep water to shallow water n such a way that t meets the between the two depths straght on, no change n drecton occurs. On the other hand, f a wae meets the at an angle, the drecton of trael does change. Ths phenomenon s called refracton (Fgure 5). 446 Chapter 9 NEL

4 Secton 9. We usually use wae s to descrbe refracton. The s a lne drawn at rght angles to a at the pont where an ncdent wae strkes the. The angle formed by an ncdent wae and the s called the angle of ncdence,. The angle formed by the and the refracted wae s called the angle of refracton, R. When a wae traels at an angle nto a medum n whch ts speed decreases, the refracted wae s bent (refracted) toward the, as n Fgure 5. If the wae traels at an angle nto a medum n whch ts speed ncreases, the refracted wae s bent away from the, as n Fgure 5. Fgure 6 shows geometrcally that s equal to the angle between the ncdent wae front and the and that R s equal to the angle between the refracted and the. In the rpple tank, t s easer to measure the angles between the wae s and the, that s, and R. To analyze s refracted at a, the angles of ncdence and refracton can be determned ug the equatons λ xy and R λ xy, respectely (Fgure 7). ncdent a 90 a 90 Fgure 6 a R b refracted R R b 90 R b 90 R R ncdent x R y refracted a straght lne drawn perpendcular to a barrer struck by a wae angle of refracton ( R ) the angle between the and the refracted, or between the refracted and the Fgure 7 The rato of the es ges R = λ xy λ xy whch reduces to R λ = λ λ For a specfc change n medum, the rato has a constant alue. Recall Snell s law from λ optcs, R. Ths equaton can be conerted to n R. The constant of proportonalty (n) and the ndex of refracton (n) are one and the same thng. NEL Waes and Lght 447

5 Consequently, we can wrte R = n absolute ndex of refracton the ndex of refracton for lght pasg from ar or a acuum nto a substance Ths relatonshp holds for waes of all types, ncludng lght, whch we wll see shortly. When lght passes from a acuum nto a substance, n s called the absolute ndex of refracton. (See Table for a lst of absolute ndexes of refracton.) The alue for the absolute ndex of refracton s so close to the alue from ar to a substance that we rarely dstngush between them. In ths text, when we refer to the ndex of refracton, we wll be referrng to the absolute ndex of refracton. Table Approxmate Absolute Indexes of Refracton for Varous Substances* Substance Absolute Refracte Index acuum ar ce.3 water.333 ethyl alcohol.36 turpentne.47 glass.50 Plexglas.5 crown glass.5 polystyrene.59 carbon dsulphde.68 flnt glass.66 zrcon.93 damond.47 gallum phosphde 3.50 *Measured wth a waelength of 589 nm. Values may ary wth physcal condtons. You wll also recall that we dered a general equaton for Snell s law that apples to any two substances: n n where n s the ndex of refracton n the frst medum, n s the ndex of refracton n the second medum, and and are angles n each respecte medum. For waes we found that λ, whch we can generalze to s n λ.but s n R λ λ from the unersal wae equaton, f λ, we can show that λ ce f s constant. Therefore, we can wrte λ s n λ λ n n The followng sample problems wll llustrate the applcaton of these relatonshps n both n the rpple tank and for lght. 448 Chapter 9 NEL

6 Secton 9. SAMPLE problem A 5.0 Hz water wae, traellng at 3 cm/s n deep water, enters shallow water. The angle between the ncdent n the deep water and the between the deep and shallow regons s 50. The speed of the wae n the shallow water s 7 cm/s. Fnd the angle of refracton n the shallow water the waelength n shallow water Soluton f 5.0 Hz cm/s? 7 cm/s s n 7 cm/s 3 cm/s 4.9, or The angle of refracton s 4. λ but f f f 5.0 Hz 7 cm/ s 5.0 Hz λ 5.4 cm The waelength n shallow water s 5.4 cm. SAMPLE problem 3 For a lght traellng from glass nto water, fnd the angle of refracton n water, f the angle of ncdence n glass s 30.0 the speed of lght n water Soluton From Table, n g n.50 g 30.0 n w n.333 w? s n n n g nw s n w ng s n ẇ w.333 w 34.3 The angle of refracton n water s NEL Waes and Lght 449

7 Answers 4... (c) cm/s cm,.8 cm (c) cm/s; 8. cm/s regon A 0 Fgure 8 For queston 7 30 regon B n a n.00 n w n.333 c m/s? n n n n (.00)( m/s) m/s The speed of lght n water s m/s. Practce Understandng Concepts 4. A wae n a rpple tank passes from a deep to a shallow regon wth 60 and 45. Calculate the ratos n the two meda of the waelengths, the speeds, and (c) the frequences. 5. Water waes traellng at a speed of 8 cm/s enter deeper water at 40. Determne the speed n the deeper water f = A 0.0-Hz water wae traels from deep water, where ts speed s 38.0 cm/s, to shallow water, where ts speed s 8.0 cm/s and = 30. Fnd the ndex of refracton, the waelengths n the two meda, and (c) the angle of refracton n the shallow water. 7. A plane wae generator wth a frequency of 6.0 Hz creates a water wae of waelength.0 cm n regon A of a rpple tank (Fgure 8). The angle between the wae crests and the straght between regons A and B s 30. In regon B the angle s 0. Use Snell s law to determne the refracte ndex of the two regons. Fnd the speed n each regon. 8. Lght traels from crown glass nto ar. (Refer to Table for the ndexes of refracton.) The angle of refracton n ar s Calculate the angle of ncdence n the crown glass. 9. If the ndex of refracton for damond s.4, what wll the angle of refracton be n damond for an angle of ncdence of 60.0 n water? Fgure 9 At hgher angles of ncdence, there s reflecton as well as refracton. You can see such partal reflecton partal refracton on the rght. total nternal reflecton the reflecton of lght n an optcally denser medum; t occurs when the angle of ncdence n the denser medum s greater than a certan crtcal angle Partal Reflecton Partal Refracton When refracton occurs, some of the energy usually reflects as well as refracts. Ths phenomenon was referred to n optcs as partal reflecton partal refracton, a descrpton that can also be used when referrng to ths behaour n waes. We can demonstrate the same behaour n a rpple tank, wth waes traellng from deep to shallow water, proded we make the angle of ncdence large, as n Fgure 9. The amount of reflecton s more notceable when a wae traels from shallow to deep water, where the speed ncreases and agan becomes more pronounced as the angle of ncdence ncreases. Fgure 0 shows that an ncdent angle s reached where the wae s refracted at an angle approachng 90. For stll larger ncdent angles there s no refracton at all, wth all the wae energy beng reflected; ths behaour of lght s referred to as total nternal reflecton. Ths phenomenon s analogous to the total nternal reflecton of lght. 450 Chapter 9 NEL

8 Secton 9. (c) θ θ c θ θ < θ c θ = θ c θ > θ c We hae remarked that the frequency of a wae does not n general change when ts speed changes. Snce λ, you mght expect that the ndex of refracton and the λ amount of bendng would not change for waes of dfferent frequences, proded the medum remans the same (e.g., water of the same depth n both cases). Fgure, howeer, shows that ndexes of refracton do, n general, depend on waelength. In Fgure, the low-frequency (long-waelength) waes are refracted, as ndcated by a rod placed on the screen below the transparent rpple tank. The rod s exactly parallel to the refracted s. In Fgure, the frequency has been ncreased (the waelength decreased), wth the rod left n the same poston. The rod s no longer parallel to the refracted s. It appears that the amount of bendng, and hence the ndex of refracton, s affected slghtly by the frequency of a wae. We can conclude that, ce the ndex of refracton represents a rato of speeds n two meda, the speed of the waes n at least one of those meda must depend on ther frequency. Such a medum, n whch the speed of the waes depends on the frequency, s called a dsperse medum. Fgure 0 Partal refracton partal reflecton At the crtcal angle (c) Total nternal reflecton Fgure The refracton of straght waes, wth a rod marker placed parallel to the refracted s. The refracted s of the hgher frequency waes are no longer parallel to the marker. NEL Waes and Lght 45

9 We stated preously that the speed of waes depends only on the medum. Ths statement now proes to be an dealzaton. Neertheless, the dealzaton s a good approxmaton of the actual behaour of waes, ce the dsperson of a wae s the result of mnute changes n ts speed. For many applcatons, t s acceptable to make the assumpton that frequency does not affect the speed of waes. SUMMARY Waes n Two Dmensons The waelength of a perodc wae s drectly proportonal to ts speed. The frequency of a perodc wae s determned by the source and does not change as the wae moes through dfferent meda or encounters reflecte barrers. All perodc waes obey the unersal wae equaton, f λ. The ndex of refracton for a par of meda s the rato of the speeds or the rato of the waelengths n the two meda λ. λ Snell s law n s n R holds for waes and for lght. When a wae passes from one medum to another, the waelength changes and partal reflecton partal refracton can occur. Secton 9. Questons Understandng Concepts. Straght s n the deep regon of a rpple tank hae a speed of 4 cm/s and a frequency of 4.0 Hz. The angle between the s and the straght of the deep regon s 40. The wae speed n the shallow regon beyond the s 5 cm/s. Calculate the angle the refracted makes wth the the waelength n the shallow water. The followng obseratons are made when a straght perodc wae crosses a between deep and shallow water: 0 s cross the eery 5.0 s, and the dstance across 3 s s 4.0 cm n deep water and 8.0 cm n shallow water. Calculate the speed of the wae n deep water and n shallow water. Calculate the refracte ndex. 3. Straght s wth a frequency of 5.0 Hz, traellng at 30 cm/s n deep water, moe nto shallow water. The angle between the ncdent n the deep water and the straght between deep and shallow water s 50. The speed of the wae n the shallow water s 7 cm/s. Calculate the angle of refracton n the shallow water. Calculate the ndex of refracton. (c) Calculate the waelength n the shallow water. 4. Straght s n the deep end of a rpple tank hae a waelength of.0 cm and a frequency of Hz. The wae fronts strke the of the shallow secton of the tank at an angle of 60 and are refracted at an angle of 30 to the. Calculate the speed of the wae n the deep water and n the shallow water. 5. The speed of a sound wae n cold ar (0 C) s 30 m/s; n warm ar (37 C), the speed s 354 m/s. If the wae front n cold ar s nearly lnear, fnd R n the warm ar f s A straght separates two bodes of rock. Longtudnal earthquake waes, traellng through the frst body at 7.75 km/s, meet the at an angle of ncdence of 0.0. The wae speed n the second body s 7.7 km/s. Calculate the angle of refracton. 7. Under what condtons do wae s n water and lght s exhbt total nternal reflecton? 8. Lght traels from ar nto a certan transparent materal of refracte ndex.30. The angle of refracton s 45. What s the angle of ncdence? 9. A of lght passes from water, wth ndex of refracton.33, nto carbon dsulphde, wth ndex of refracton.63. The angle of ncdence s Calculate the angle of refracton. 45 Chapter 9 NEL