Lu 9 No-Nomal Id of Wavs a Ifas I hs lu you wll la: Wha happs wh wavs s a fa bw wo dff mda omg a a agl Rflo ad asmsso of wavs a fas Applao of -fld ad H-fld bouday odos Toal al flo Bws s agl C 303 Fall 005 Faha Raa Coll Uvsy µ o Wavs a Ifas T ad TM Wavs µ o Tasvs l (T wav H Tasvs Mag (TM wav H Pla of Id: Th pla oag h d wavvo ad a vo ha s omal o h fa s alld h pla of d ( h fgu abov h -z pla s h pla of d T Wav: If h -fld of h wav s ppdula o h pla of d h h wav s alld a T-wav TM Wav: If h H-fld of h wav s ppdula o h pla of d h h wav s alld a TM-wav C 303 Fall 005 Faha Raa Coll Uvsy
Tasvs l (T wav T Wav - Wavvos H θ θ H µ o µ o H zz [ s( os( θ z ω µ o ω zz [ s( θ os( θ z ω µ o ω zz [ s( θ os( θ z ω µ o ω C 303 Fall 005 Faha Raa Coll Uvsy T Wav Fs Bouday Codo H Tasvs l (T wav θ θ H µ o µ o H ( y. y. s z< 0 ( y. z> 0 s θ Us bouday odos: [ ( os( θ z [ ( os( θ z [ s( θ os( θ z ( A z 0 h -fld paalll o h fa mus b ouous aoss h fa fo all Ths gvs: s( θ s( θ s( θ C 303 Fall 005 Faha Raa Coll Uvsy
T Wav Phas Mahg Codo H Tasvs l (T wav θ θ H µ o µ o H s ( θ s( θ s( θ ( θ s( θ s( θ s Th fs qualy gvs ( usg : s Th oly way h abov bouday odo a b sasfd fo all s f all h - dpd phas faos a h sam (hs s alld phas mahg ( s( θ θ θ C 303 Fall 005 Faha Raa Coll Uvsy agl of d quals h agl of flo T Wav Sll s Law H Tasvs l (T wav θ θ H µ o µ o H ( θ s( θ s( θ s Th sod qualy gvs: ( θ s( θ s ω s s s ( θ ω s( θ ( θ s( θ ( θ ( s θ s( θ Sll s Law ( C 303 Fall 005 Faha Raa Coll Uvsy 3
T Wav Sod Bouday Codo H Tasvs l (T wav θ θ H µ o µ o H ( A z 0 h H-fld ompo paalll o h fa mus b ouous fo all ( ( ( H y. y. [ s( os( θ z z< 0 η η H( ( y s( θ os( θ z. z> 0 η [ s( θ os( θ z os os η C 303 Fall 005 Faha Raa Coll Uvsy [ ( ( ( ( s θ s θ θ θ ( θ s( θ os os η η ( θ os( θ η η η ( T Wav Rflo ad Tasmsso Coffs H Tasvs l (T wav θ θ H µ o µ o H Th soluo s: Tasmsso off Rflo off T Γ ( θ ( θ ( θ ( θ ( θ ( θ ( θ ( θ η os η os η os η os η os η os η os η os ( θ ( θ ( θ ( θ ( θ ( θ ( θ ( θ os z os z os z z os os z os z os z z os C 303 Fall 005 Faha Raa Coll Uvsy 4
TM Wav - Wavvos H Tasvs Mag (TM wav θ θ H µ o H µ o zz [ s( os( θ z ω µ o ω zz [ s( θ os( θ z ω µ o ω zz [ s( θ os( θ z ω µ o ω C 303 Fall 005 Faha Raa Coll Uvsy TM Wav Fs Bouday Codo H Tasvs Mag (TM wav θ θ H µ o H µ o ( H y H y H.. [ s( os( θ z z< 0 H( y H. ( ( z> 0 s θ os θ z [ s( θ os( θ z Us bouday odos: ( A z 0 h H-fld paalll o h fa mus b ouous aoss h fa fo all [ Ths gvs: s( θ s( θ s( θ H H H C 303 Fall 005 Faha Raa Coll Uvsy 5
TM Wav Phas Mahg Codo H Tasvs Mag (TM wav θ θ H µ o H µ o s H Th oly way h abov bouday odo a b sasfd fo all s f all h - dpd phas faos a h sam (hs s alld phas mahg ( θ s( θ s( θ s Th fs qualy gvs ( usg : s ( s( θ θ θ s θ H s θ ( θ ( ( H C 303 Fall 005 Faha Raa Coll Uvsy agl of d quals h agl of flo TM Wav Sll s Law H Tasvs Mag (TM wav θ θ H µ o H µ o ( θ s( θ s( θ s Th sod qualy gvs: ( θ s( θ s ω s s H s H H H ( θ ω s( θ ( θ s( θ ( θ ( H s θ H s( θ Sll s Law ( C 303 Fall 005 Faha Raa Coll Uvsy 6
TM Wav Sod Bouday Codo H Tasvs Mag (TM wav θ θ H µ o H µ o ( A z 0 h -fld ompo paalll o h fa mus b ouous fo all ( ( ( y. H y. η η H z < 0 ( ( y H [ s( os( θ z. η z> 0 s( θ os( θ z [ s( θ os( θ z os [ ( ( ( ( ( ( H s θ H H θ θ η s θ s os θ η os θ η ( θ ( η H η H os( θ η H ( os C 303 Fall 005 Faha Raa Coll Uvsy TM Wav Rflo ad Tasmsso Coffs H Tasvs Mag (TM wav θ θ H µ o H µ o Th soluo s: TM H T H Tasmsso off Rflo off TM H Γ H ( θ ( θ ( θ ( θ ( θ ( θ ( θ ( θ η os η os η os η os η os η os η os η os ( θ ( θ ( θ ( θ ( θ ( θ ( θ ( θ os z os z os z z os os z os z os z z os C 303 Fall 005 Faha Raa Coll Uvsy 7
Sll s Law ( θ s( θ s θ If < h θ < ad h asmd wav bds owads h omal θ θ If > h θ > ad h asmd wav bds away fom h omal C 303 Fall 005 Faha Raa Coll Uvsy Toal Ial Rflo Cal Agl If > h θ > If s asd, h θ wll vually bom 90 o Th valu of fo whh θ s 90 o s alld h al agl θ s ( θ s( θ s ( θ s s( θ π Wha f s asd byod θ? θ θ Wh s asd byod θ h wav s o asmd bu s omplly (00% fld a h fa ba o h mdum of d Ths phomo s alld oal al flo happs fo boh T ad TM wavs C 303 Fall 005 Faha Raa Coll Uvsy 8
Toal Ial Rflo Phas Mahg > ad > θ W d o osd mo dal wha happs wh h agl of d s ga ha h al agl θ Th phas mahg odo gvs us: s ω s W also ow h dspso lao fo mdum : ω ω ω z C 303 Fall 005 Faha Raa Coll Uvsy s( θ ( θ ( θ Toal Ial Rflo vas Wav > ad > θ If s lag ha θ h wav mdum s vas h z-do [ Th pvous wo quaos mply: s ( θ z ω ω z Fo > θ o a w: ω z s ( θ z Th -fld (assumg T wav mdum s h: z ( y z z> 0 ω s ( θ -v wh > θ Th z-ompo of h wavvo has bom omplly magay Th fld s vas h z-do mdum C 303 Fall 005 Faha Raa Coll Uvsy 9
Toal Ial Rflo vas Wav > ad > θ Th -fld (assumg T wav mdum s: z z ( z φ y y z z> 0 z (, y z os( ω φ z> 0 Th wav s popagag alog h fa ( h -do bu dayg (whou spaal osllaos h z-do C 303 Fall 005 Faha Raa Coll Uvsy Toal Ial Rflo Rflo Coff > ad > θ If s lag ha θ h wav s omplly fld ba o h mdum of d Fo > θ o a w: z ω θ ( s z Th z-ompo of h wavvo has bom omplly magay Th flo off fo h -fld (assumg T wav s: z ϕ Γ z z z z Th phas ϕ of h flo z off Γ oal al z z z z z flo s alld h Goosz Hash phas-shf Γ C 303 Fall 005 Faha Raa Coll Uvsy 0
Γ T Wavs T ad TM Wavs: Rflo Coffs z z z z TM H Γ H TM Wavs z z z z Quso: Ca o v g h flo off o go o zo (vy dsabl o g d of uwad flos ops? ω z ω z ( f ( Phas mahg Bu z z Γ s v zo fo T wavs Bu o a hav Γ0 fo TM wavs f: z z os θ os θ ( ( C 303 Fall 005 Faha Raa Coll Uvsy TM Wavs: Bws s Agl O a hav Γ0 fo TM wavs f: Fo θ B ( θ os( θ os θ Sll s law gvs: ( θ s( θ s Th abov wo quaos wll hav a soluo f ad oly f: ( θ os( θ ad os( θ s( θ s Ths happs wh: θ θ π Th agl of d fo whh hs happs s alld h Bws s agl θ B : a ( θ θ a B C 303 Fall 005 Faha Raa Coll Uvsy