Chaper 4 Linear Surface Graviy Waves 3., Dispersion, Group Velociy, and Energy Propagaion 4. Descripion In many aspecs of wave evoluion, he concep of group velociy plays a cenral role. Mos people now i as he speed of energy propagaion, bu i is acually much more. We herefore begin by discovering group velociy in some simple examples. As previously saed, dispersive waves have phase speeds (c =!/ in one dimension) ha depend on heir wavelenghs. This propery gives rise o wo ineresing behaviors: ) Waves creaed by a localized disurbance will no hemselves ac by wavenumber as hey move away from he source. ) The energy carried by dispersive waves moves a a speed di eren from he phase speed. Propery ) is no surprising in view of he definiion of dispersion, and propery ) also seems reasonable if one acceps ha he energy is usually associaed wih he squared ampliude of he moion, and ha he sum of squares is no he same as he square of sums. I is somewha surprising, however, ha, if he dispersive waves are creaed suddenly near x =0aime =0,heplaceolooforwavesofwavenumber a ime is no a x = c bu raher a x = c g where c g = d! (4.) d is called he group velociy. This same velociy becomes associaed wih he speed of energy propagaion when one loos more closely a propery ). These facs hin ha he group 8
velociy, and no he phase velociy, is he fundamenal speed ha characerizes dispersive waves. 4. A wo-wave example Propery ) is exhibied in is simples form by a flow made up of precisely wo equalampliude plane waves. We choose deep waer (i.e., shor) graviy waves in one space dimension for our example. The dispersion relaion is! = g A wo-wave soluion is =Re Ae i( x! ) + Ae i( x! ) where! = p g and! = p g. By he formula cos cos = [cos( + )+cos( )] his is =[Acos( x!)] cos( x!) where = + =! =! +!! =!! The ineresing case is when. Then he facor A(x, ) =A cos( x!) can be inerpreed as he slowly varying ampliude of a carrier wave wih he wavenumber and frequency! The equaion for A(x, ) iscalledheenvelopeof he carrier wave. envelope carrier wave 9
Cress and roughs of he carrier wave move a he phase speed!/, bu he envelope moves a speed!/ which limis on as c g = d! d! 0. In he deep-waer graviy wave example, c = r g while c g = r g so ha he envelope moves a half he speed of he cress. Thus he propagaion speed c g of he wave envelope (or wave group) is he same as he speed associaed wih he wave energy flux F = c g E in he equaion (4.8), Two di eren approaches give he same resul for c g. 4.3 Many waves: Saionary Phase @E @ + @ @x (c ge) =0, (4.) We coninue o consider one space dimension, bu generalize o he case of an arbirary iniial condiion. Le (x, ) be he free surface elevaion, and le he iniial condiion on be given as (x, 0) = f(x). We will assume ha f(x) isnonzeroonlynearx =0. We inroduce he Fourier ransform of and f by (x, ) = f(x) = Z Z ˆ (, )e ix d F ()e ix d The inverse Fourier ransforms are ˆ (, ) = Z (x, )e ix and similarly. We noe ha ˆ (, ) =ˆ (, ) because (x, ) isreal. Now we could space-ransform he whole graviy wave problem and solve for ˆ (, ). Bu we have already done somehing lie his for each wavenumber componen, in our analysis 30
of plane waves in he wave equaion (Chaper ), Thus, we now ha he general soluion for waves in deep waer is (x, ) = where!() p g and Z d A()e i(x!()) + B()e i(x+!()) The coe ˆ (, ) = A()e i!() + B()e i!() ciens A() and B() arechosenomeeheiniialcondiions. Fromheiniial condiion (x, 0) = f(x) weobain A()+B() =F () To deermine boh A() and B() we obviously need an addiional iniial condiion. This is expeced, because he complee iniial condiions for any mechanical sysem are he locaion and velociy of paricles. surface be iniially a res,. We ae, for our addiional iniial condiion, ha he @ @ =0 As wih he wave equaion (), his implies so ha i!a()+i!b() =0 A() =B() = F () Thus he exac soluion o our problem is (x, ) = Z d F () e i(x!()) + e i(x+!()). F The form (F) is no convenien. We herefore see an approximaion o (F) ha is valid for large x and bused upon he assumpion ha f(x) 6= 0onlywihinafiniedisanceofx =0. Now (F) haswoverysimilarermswhichwewillreainpreciselyhesameway. To avoid unnecessary wriing, we will emporarily preend ha only one erm is presen, viz. (x, y) (x, ) = Z F ()e i(x!()) () 3
A he end we will discuss boh erms ogeher. However, i should already be obvious ha he erm we are eeping is he only imporan erm for large posiive x, because he negleced erm propagaes in he = x direcion. where Wrie () in he form (x, ) = Z F ()exp[i ()] d (4.3) () = x!() (4.4) Now, le!wih x/ fixed (bu arbirary). This means ha x ges really big as well. Wihin he inegral (4.3) for large enough, even small changes in () wih will cause rapid oscillaions in exp[i ]. If F () issmooh,heseoscillaionswillproducecanceling conribuions o he inegral. This is rue excep where d /d = 0 () =0,becauseif 0 =0 hen changes in produce no change in (). We herefore assume ha as!,he dominan conribuions o (4.3) comefromallhewavenumbers 0 a which and approximae d ( 0 ) d = x d!( 0 ) d =0 (4.5) (x, ) X Z 0 + F ( 0 )e i () d (4.6) 0 where he summaion is over all 0 ha saisfy (4.6). However, near 0 0 () = ( 0 )+ 0 ( 0 )( 0 )+ 00 ( 0 )( 0 ) +... apple 0 x!( 0 ) +0!00 ( 0 )( 0 ) Here we assume ha! 00 ( 0 ) 6= 0andsopaferhaerm. (The case where! 00 ( 0 )=0 will be considered laer. Then (4.6) becomes (x, ) = X 0 F ( 0 )e i( 0x!( 0 )) Z 0 + 0 apple exp To evaluae he inegral in (4.7), change he inegraion variable o i!00 ( 0 )( 0 ) d (4.7) Then he inegral becomes r =( 0 )!00 ( 0 ) q!00 ( 0 ) Z + exp i sgn (! 00 ( 0 )) d (7) 3
where = r!00 ( 0 ) (8) Bu as!,!for any, and he inegraion limis in (7) may be replaced by (, ). This is very saisfying, because was an arbirary quaniy. I only remains o evaluae Z e i sgn(! 00 ( 0 )) d (9) Some care is required, because his inegral exiss only if carefully inerpreed. Suppose! 00 ( 0 ) > 0andreplace(9)by Z e (i+ ) d (0) where > 0 is real. Now here is no problem wih convergence since e (i+ ) = e! 0 as! ± Then Z e (i+ ) d = p p i +! p e i 4 as! 0 where we selec he branch of p i + wih posiive real par. Treaing he case! 00 ( 0 ) < 0 similarly, we find ha Z e i sgn(! 00 ( 0 )) = p exp i 4 sgn!00 ( 0 ). Combining all resuls, where A(x, ; 0 )= (x, ) X 0 A(x, ; 0 )e i( 0x!( 0 ) p F (0 ) q w00 ( 0 ) 4 sgn!00 ( 0 )) near x = c g ( 0 ), 9 >= >; () The approximaion mehod used o ge () is called he mehod of saionary phase. The version presened here is somewha less rigorous han ha given by Lighhill (p. 48-5), who poins ou ha he procedure can also be done as a special case of he mehod of seepes descens. Lighhill s approach avoids he ric for evaluaing (9). Where has he assumpion ha f(x) =0awayfromx =0acuallybeenused? Inhe assumpion ha F () issmooh! ForifF () has infinielysharp corners, hen even he rapid oscillaions of e i may no cause cancellaion. 33
The fac ha f(x) =0for x su cienly large forces F () obesmoohinhesense ha rapid changes in any funcion are associaed wih high wavenumber in is Fourier ransform. We nex examine he physical conen of (). Le x>0belargeandfixed. Aanyime, he local soluion near x is given by () where 0 saisfy x = c g ( 0 ). Now since!() = p g, c g () = p g. For each x and, here is only one soluion for 0 : he wavenumber 0 observed a ime and locaion x is jus ha wavenumber whose group velociy is x. Alernaively, imagine an observer wih a special preference for he wavenumber 0, who is willing o su er any inconvenience o always be where he local wavenumber is 0. This observer mus move a he velociy G g ( 0 ), and for him 0 in () is ruly a consan. However, since c g ( 0 )= c( 0) for surface waves, cress and roughs will move righward relaive o he observer, no maer wha 0. Going bac o our full soluion (F), we easily see ha he second erm, whose analysis we deferred, maes no conribuion on x 0. For x 0, he firs erm is negligible, and he second erm maes a conribuion analogous o he one we have wored ou. 4.4 Slowly varying waverain I is possible o reach many of he conclusions of he previous secion by beginning from he assumpion of a slowly varying waverain and posulaing ha he plane wave dispersion relaion is saisfied locally. By slowly varying i is mean ha he wavenumber and frequency change only gradually over many oscillaions. These slow variaions can come abou eiher because he fluid is in he final sages of dispersion (as in he previous secion), or because he medium iself exhibis a slow variaion (in fluid deph, for example). We suppose ha he wave can be wrien in he form A(x, )e i (x,) () 34
and define @ @x and! @ @ o be he local wavenumber and frequency. Noe ha, in he special case of a plane wave, = x! and,! are consans. More generally if @A A @x, @ @x, @!! @, ec. hen () is locally sinusoidal. Direcly from he definiions (-3), i follows ha @ @ + @! @x () (3) =0 (4) This is ofen inerpreed as a conservaion equaion for wave cress. (Why?) Now if A,,! change slowly enough, i is plausible ha!,saisfy he dispersion relaion for plane waves in each localiy. We herefore assume!(x, ) = ((x, )) (5) where! = () is he dispersion relaion for plane waves. This assumpion is he basis for all resuls o follow. (We use he noaion o emphasize explici funcional dependence.) For surface waves, () = p g anh(h). By (4) and (5) @ @ + @! @x = @ @ + d @ d @x =0 @ () @ + C @ g =0 (6) @x This saes ha an observer moving a he group velociy would always see he same local wavenumber, as shown previously in a special case. Similarly () @! @ = d @ d @ = @ @ + C @ g @x d @! d @x! =0 (7) Thus he local frequency also propagaes a he group velociy c g. 35
Homewor. (From Pedlosy) Consider a wave of he form = Ae i (x,) where (x, ) = g 4x (a) Wha is he x wavenumber? (b) Wha is he frequency (c) Under wha condiions can his be hough of as slowly varying? (d) Wha speed would an observer need o move a o see a consan frequency and wavenumber? (e) A wha speed would an observer need o ravel o say on a paricular cres?. (from Kundu e al., 7.) The e ec of viscosiy on he energy of linear deep-waer surface waves can be deermined from he wave moions velociy componens and he viscous dissipaion. (a) For incompressible flow, he viscous dissipaion of energy per uni mass of fluid is = Sij, where S ij is he srain-rae ensor and m is he fluids viscosiy. Calculae from he wave soluions in Lecure for deep waer waves. (b) Verically inegrae he wave o calculae D w = Z 0 (z)dz Now a wave energy conservaion equaion ha included viscous e ecs would loo lie @E @ + @ @x (Ec g)= D w (4.8) (c) Assuming @ =0,haisabalancebeweenfluxdivergenceanddissipaion,wrie acolosedformsoluionforwaveenergyorwaveampliudeasafuncionofx (again in deep waer). (d) Using =0 6 m s, deermine he disance necessary for a 50% ampliude reducion for waves wih period T =sandt =0s. (e) Thining abou Waves across he Pacific : find he disance from New Zealand o Souhern California. How much loss of wave energy (or ampliude) would you expec for he long-period (T =0s)waves? 36