7 Wave Equation in Higher Dimensions

Transcription

1 7 Wave Equation in Highe Dimensions We now conside the initial-value poblem fo the wave equation in n dimensions, u tt c u 0 x R n u(x, 0 φ(x u t (x, 0 ψ(x whee u n i u x i x i. (7. 7. Method of Spheical Means Ref: Evans, Sec..4.; Stauss, Sec. 9. We begin by intoducing a method to solve (7. in odd dimensions. Fist, we intoduce some notation. Fo x R n, let B(x, Ball of adius about x B(x, Bounday of ball of adius about x α(n Volume of unit ball in R n nα(n Suface Aea of unit ball in R n. With this notation, the volume of the ball of adius about x R n, witten as Vol(B(x,, is given by α(n n and the suface aea of the ball of adius about x R n, witten as S.A.(B(x,, is given by nα(n n. Fo f : R n R, we define the aveage of f ove B(x, as B(x, f(y dy Vol(B(x, B(x, f(y dy We define the aveage of f ove B(x, as f(y ds(y f(y ds(y S.A.(B(x, whee ds(y denotes the suface measue of B(x, in R n. α(n n B(x, f(y dy. f(y ds(y, nα(n n Example. Fo n 3, Vol(B(x, 4 3 π3. Theefoe, fo f : R 3 R, the aveage of f ove B(0, is given by f(y dy 3 π π f(ρ, θ, φρ sin φ dρ dθ dφ. 4π 3 B(0, B(0, 0 0 Fo n 3, S.A.(B(x, 4π. Theefoe, fo f : R 3 R, the aveage of f ove B(0, is given by f(y ds(y π π f(, θ, φ sin φ dθ dφ π π f(, θ, φ sin φ dθ dφ. 4π 0 0 4π 0 0 0

2 Ou plan to solve (7. is the following. Fix a point x R n. Fo > 0, we define u(x;, t u(y, t ds(y, the aveage of u(, t ove B(x,. Fo 0, we define u(x; 0, t u(x, t. Fo < 0, we define u(x;, t u(x;, t. We claim that fo u smooth, u is a continuous function of, and, theefoe, lim u(x;, t u(x, t. 0+ In ode to solve (7., we will assume u is a solution of (7. and look fo an equation u solves. Note: We will assume c. Fo c, we can make a change of vaiables to deive the solution fom the solution in the case c. Lemma. If u solves u tt u 0, x R n, t 0 u(x, 0 φ(x u t (x, 0 ψ(x, then u(x;, t solves (n u tt u u 0, 0 < <, t 0 u(x;, 0 φ(x; φ(y ds(y u t (x;, 0 ψ(x; ψ(y ds(y fo evey x R n. Poof. u(x;, t B(0, u(y, t ds(y u(x + z, t ds(z.

3 Theefoe, u (x;, t B(0, nα(n n nα(n n nα(n n u(x + z, t z ds(z u(y, t y x u (y, t ds(y ν B(x, B(x, ds(y u (y, t ds(y ν u(y, t dy u tt (y, t dy by the Divegence Theoem, and using the fact that u solves the wave equation, u tt u 0. Theefoe, u (x;, t u nα(n n tt (y, t dy B(x, which implies Theefoe, Theefoe, which implies Theefoe, n u (x;, t u tt (y, t dy. nα(n B(x, ( n u (x;, t nα(n n nα(n n n n u tt (x;, t. u tt (y, t ds(y u tt (y, t ds u tt (y, t ds(y ( n u (x;, t n u tt (x;, t, (n n u + n u n u tt. u tt u (n u 0 3

4 and Similaly, as claimed. u(x;, 0 u(y, 0 ds φ(y ds φ(x;. u t (x;, 0 ψ(x; Solution fo n 3. We now conside the case of the wave equation in thee dimensions. solution of (7. fo n 3. As befoe define the function u(x;, t such that u(x;, t u(y, t ds(y. Assume u is a Next intoduce a function v(x;, t such that v(x;, t u(x;, t and new functions g(x; and h(x; such that g(x; φ(x; h(x; ψ(x; φ( ds(y ψ( ds(y. Lemma 3. Fo each x R n, the function v(x;, t solves the one-dimensional wave equation on the half-line with Diichlet bounday conditions, v tt v 0 0 < <, t 0 v(x;, 0 g(x; 0 < < v t (x;, 0 h(x; 0 < < v(x; 0, t 0 t 0. Poof. v tt u tt [ u + ] u u + u (u + u (u v. 4

5 Next, Similaly, Now, v(x;, 0 u(x;, 0 u(y, 0 ds(y φ(y ds(y φ(x, g(x; v t (x;, 0 h(x;. v(x; 0, t 0 u(x; 0, t 0. Theefoe, v(x;, t solves the one-dimensional wave equation on a half-line with Diichlet bounday conditions, as claimed. Now we use this fact to constuct the solution of (7.. By d Alembet s fomula, we know that fo 0 t, the solution v(x;, t is given by Now and Theefoe, Now implies Similaly, v(x;, t [g(x; + t g(x; t ] + v(x;, t u(x, t lim 0 + lim 0 + u(x, t lim u(x;, t 0 + v(x;, t u(x;, t. +t { [g(x; t + g(x; t ] + d g(x; t + h(x; t. dt g(x; φ(x; +t +t h(x; y dy. +t g(x; t tφ(x; t t φ(y ds(y. h(x; t tψ(x; t t ψ(y ds(y. } h(x; y dy 5

6 Theefoe, the solution of the wave equation in R 3 (with c is given by u(x, t [ ] t φ(y ds(y + t ψ(y ds(y. If φ is smooth, the solution can be simplified futhe. In paticula, fo φ smooth, we have d dt g(x; t d ( t φ(y ds(y dt d ( t φ(x + tz ds(z dt B(0, φ(x + tz ds(z + t φ(x + tz z ds(z B(0, B(0, ( y x φ(y ds(y + t φ(y ds(y t φ(y ds(y + φ(y (y x ds(y. And, h(x; t tψ(x; t t ψ(y ds(y. Theefoe, we have u(x, t [φ(y + φ(y (y x + tψ(y] ds(y. We note that in R 3,. nα(nt n 4πt Theefoe, the solution of the IVP fo the wave equation in R 3 (with c and φ smooth is given by u(x, t [φ(y + φ(y (y x + tψ(y] ds(y. (7. 4πt This is known as Kichoff s fomula fo the solution of the initial value poblem fo the wave equation in R 3. Remak. Above we found the solution fo the wave equation in R 3 in the case when c. If c, we can simply use the above fomula making a change of vaiables. In paticula, conside the initial-value poblem v tt c v 0 x R n v(x, 0 φ(x v t (x, 0 ψ(x. 6 (7.3

7 though a point, its infomation is immediately fogotten. This popety, the stong vesion of Huygens pinciple, is valid not only fo n 3, but in all odddimensional spaces. It does not apply, howeve, to the wave equation in spaces of even dimensionality. Thee, even though infomation still popagates at speed one, it does not do it though shap fonts, leaving instead a tace behind as it passes though a point. Hence, when a tsunami shakes the (D ocean, it leaves significant wave action behind its leading font. We shall see now that this is the case in two dimensions, though an application of Hadamad s method of descent. The same methodology applies in all even dimensions n d, once we have the geneal solution to the initial value poblem in the odd-dimensional space n d The method of descent The method of descent, also due to Hadamad, consists simply in thinking of any solution to the wave equation in even (n d dimensions as a solution in one moe dimension which does not depend on one of the space vaiables. In two dimensions, in paticula, we can wite u(x, y, t ũ(x, y, z, t, whee ũ is a solution to the thee dimensional wave equation with initial data that do not depend on z: ũ(x, y, z,0 g(x, y, z g(x, y, ũ t (x, y, z,0 h(x, y, z h(x, y. Fo ũ we have the exact fomula (53, so the same applies to u. Howeve, by definition, the coesponding G(x, and H(x, ae the spheical means ove thee dimensional balls of functions g(x and h(x that do not depend on z. Then we have G(x, g(sds g(sj da, B(x, S(x, whee B is the suface of a thee dimensional sphee, S is the suface of a two dimensional cicle, and J is the Jacobian J s x that pojects one aea element onto the othe. Fo ou puposes, it is enough to notice that now the fomula fo u involves integals ove the inteio of cicles of adius t, not just thei cicumfeence. Hence the stong vesion of Huygens pinciple does not apply in two dimensions: the solution to the wave equation at point x and time t depends on all the initial data within a cicle of adius t aound x, not just on thei values and deivatives on the cicumfeence y x t. 7

8 Suppose v is a solution of (7.3. Then define u(x, t v(x, t. Then c implies u is a solution of u tt u c v tt v 0 u tt u xx 0 x R n u(x, 0 φ(x u t (x, 0 c ψ(x. Theefoe, u is given by Kichoff s fomula above. Now by making the change of vaiables t t, we see that c v(x, t u(x, c t, and we aive at the solution fo (7.3, v(x, t [φ(y + φ(y (y x + tψ(y] ds(y. 4πc t 7. Method of Descent B(x,ct In this section, we use Kichoff s fomula fo the solution of the wave equation in thee dimensions to deive the solution of the wave equation in two dimensions. This technique is known as the method of descent. This technique can be used in geneal to find the solution of the wave equation in even dimensions, using the solution of the wave equation in odd dimensions. Solution fo n. Suppose u is a solution of the initial value poblem fo the wave equation in two dimensions, u tt u 0, x R, t 0 u(x, 0 φ(x u t (x, 0 ψ(x. We will find a solution in the -D case, by using the solution to the 3-D poblem. u(x, x, t be the solution to the -D poblem. Define Theefoe, ũ(x, x, x 3, t u(x, x, t. ũ(x, x, x 3, 0 u(x, x, 0 φ(x, x ũ t (x, x, x 3, 0 u(x, x, 0 ψ(x, x. Clealy, ũ(x, x, x 3, t is a solution of the 3D wave equation with initial data φ(x, x and ψ(x, x, ũ tt ũ x x ũ x x ũ x3 x 3 0 ũ(x, x, x 3, 0 φ(x, x, x 3 φ(x, x ũ t (x, x, x 3, 0 ψ(x, x, x 3 ψ(x, x. 7 Let

9 Now we can solve the 3D wave equation using Kichoff s fomula. In paticula, ou solution is given by ũ(x, x, 0, t [ φ(y + φ(y (y x + t ψ(y] ds(y whee B(x, t is the ball of adius t in R 3 about the point x (x, x, 0. Now we note that φ(y ds(y φ(y ds(y 4πt φ(y( + γ(y / dy πt B(x,t whee B(x, t is the ball in R of adius t about the point x (x, x and γ(y (t y x /. Theefoe, y x γ(y (t y x / which implies Theefoe, Similaly, and ( ( + γ(y / t /. t y x φ(y ds(y tφ(y dy. πt B(x,t (t y x / t ψ(y ds(y t ψ(y dy πt B(x,t (t y x / φ(y (y x ds(y t φ(y (y x dy. πt B(x,t (t y x / Theefoe, the solution of the initial-value poblem fo the wave equation in R (with c is given by u(x, t tφ(y + t ψ(y + t φ(y (y x dy. (7.4 πt B(x,t (t y x / Again, by making a change of vaiables, we see that the solution of the wave equation in two dimensions is given by u(x, t ctφ(y + ct ψ(y + ct φ(y (y x dy. πc t (c t y x / B(x,ct 8

10 7.3 Huygen s Pinciple Note that fo the initial-value poblem fo the wave equation in thee dimensions, the value of the solution at any point (x, t R 3 (0, depends only on the values of the initial data on the suface of the ball of adius ct about the point x R 3 ; that is, on B(x, ct. That is to say, distubances all tavel at exactly speed c. This is known as Huygens s pinciple. In contast, in two dimensions, the value of the solution u at the point (x, t depends on the initial data within the ball of adius ct about the point x R. Signals don t all tavel at speed c. In fact, as we will see, fo n 3 and odd, Huygens s pinciple holds. That is, all signals tavel at exactly speed c. In even dimensions, howeve, that is not the case. 7.4 Wave Equation in R n, n > 3 Ref: Evans, Sec..4. Note: In this section, we assume c. Fo c, we can make a change of vaiables to find the solution. Odd dimensions. Fo the case of odd dimensions, we use the method of spheical means as we did fo the case of n 3. Let n k +. Let x R n. Define ( k v(x;, t ( k u(x;, t ( k g(x; ( k φ(x; h(x; k ( k ψ(x;. ( Notice that fo k, these definitions educe to those functions intoduced in the case n 3. Fist, we will show that v(x;, t solves the wave equation on the half-line with Diichlet bounday conditions. Lemma 4. Fo each intege k, fo each x R n, the function v(x;, t defined above solves v tt v 0 > 0 v(x;, 0 g(x; v t (x;, 0 h(x; v(x; 0, t 0. The poof elies on the following lemma. Lemma 5. Let φ : R R be C k+. Then fo k,,.... ( d d ( k d ( k φ( d 9 ( d d k ( k dφ d (

11 of the \holy Tinity" of patial dieential equations is the second{ode wave equation, the One example of a hypebolic PDE. In n dimensions the equation takes the fom canonical is the n. A wave speed c can be included by a whee c on the ight-hand side. Since ( is of second ode in t, a well-posed initial-value poblem facto wave equation descibes linea, nondispesive wave popagation. The Fo example, Figue pesents a pai of images show the outwad spead of a cicula pulse that D. At t 0 we begin with a cone in adius 0: with u t (0 0. of t, the cone has spead At : Popagation Fig. a cicula pulse of wave equation aises in numeous applications. The classical D example is the vibation of an The sting (! ef, and in D this becomes the vibation of an ideal membane o dum (! ef. ideal 3D, the most famous example is the popagation of sound waves in a gas o liquid. Indeed, In ( is often called the acoustic wave equation to distinguish it fom the moe complicated equation wave equation (! ef, whee the pesence of stiness as well as compessibility leads to elastic appeaance of two distinct kinds of waves. the hypebolic, the wave equation has nite speed of popagation fo all infomation namely, Being the equation as witten in (. A cuious popety known as Huygens' pinciple is as follows. fo dimensions n 3; 5; 7; 9; : : : ; all infomation popagates unde ( at speed exactly, neve In Thus, the light fom a bulb ashed at t 0 passes the obseve at a late time as a pue slowe. function. In dimensions n ; ; 4; 6; 8; : : : ; on the othe hand, a nite faction of the enegy delta tavel moe slowly than at speed, so the obseve sees a delta function ash followed by a may tail. To illustate this phenomenon, Figue shows the esult at time t of the initial decaying u t (x; 0 maxf0; 0jxjg in dimensions ; ; 3; 4; 5; 6, whee jxj (x + + x n. condition an unbounded domain, the wave equation is eadily investigated by Fouie analysis. Sepaation In vaiables leads to the obsevation that fo any n-vecto k, known as the wave numbe, thee ae of k x k x + + k n x n, so long as! jkj. This condition elating the fequency to the whee numbe is the dispesion elation fo (. By a Fouie integal, geneal solutions to ( can wave obtained by the supeposition of plane waves (, and unde suitable technical assumptions, all be can be witten this way. solutions a bounded domain, sepaation of vaiables in ( leads to oscillatoy solutions of the fom In i!j t j (x, whee the functions j (x ae eigenfunctions of the Laplacian opeato fo (! ef. e The allowed fequencies! j as seies athe than integals. If is a ectangle, a disk, o a ball, the eigenfunctions supepositions tigonometic functions, Bessel functions, o spheical hamonics, espectively. ae technique in the study of the wave equation is Hadamad's method of descent. The idea Anothe is that any solution in dimension n can be thought of as a solution in dimension n + that hee to be invaiant with espect to one coodinate. In paticula, solutions in even dimensions happens be obtained fom solutions in the odd dimension one highe, which ae elatively elementay can applications of the wave equation, boundaies and vaiable coecients ae impotant, including In in the sound speed. Among the phenomena that aise ae eection, efaction, and discontinuities Just as the eld of uid mechanics can be descibed without too much exaggeation diaction. the study of the Navie{Stokes equations (! ef, so the eld of acoustics is moe o less the as of the wave equation. Thee ae enough subtleties hee to ll books, and caees even if we study ou attention to the fascinating subeld of the physics of musical instuments. conne popagation of light and sound 5. Wave equation! n n jxj 0 u tt u; ( n 4 n 3 n 5 n 6 Fig. : Huygens' pinciple: zeo tails in odd dimensions n 3 fo this equation would nomally involve two initial conditions such as u(x; 0 and u t (x; 0. plane wave solutions of ( of the fom u(x; t e i(!t+kx ; ( a concentic ing of to adius ex- oute actly :. t now belong to a discete set, and geneal solutions can be obtained via t 0 supepositions of expanding sphees thanks to Huygens' pinciple. Refeences N. H. Fletche and T. D. Rossing, The physics of musical instuments, Spinge-Velag, 99. G. B. Folland, Intoduction to patial dieential equations, Pinceton, 976. F. John, Patial dieential equations, Spinge-Velag, 98. P. M. Mose and H. Feshbach, Methods of theoetical physics, McGaw-Hill, 953. Lod Rayleigh, The theoy of sound, vols., Dove, 945. c999 8 Febuay 00: Kathyn Haiman and Nick Tefethen

12 . ( k d ( k φ( d whee each β k j is independent of φ. k j0 β k j j+ dj φ d j ( 3. β k (k. Poof. Use induction. Poof of Lemma 4. [ ( k v d ( k u(x;, t] d ( k d ( k u (x;, t by Lemma 5 d ( k ( d d ( k u (x;, t d d ( k ( d d [kk u + k u ] ( k [ ] d k ( k d u + u ( k ( [ ] d n k u + u d ( k d ( k u tt d ( k d ( k u tt d v tt Clealy, v(x;, 0 g(x;, v t (x;, 0 h(x; and v(x; 0, t 0. Theefoe, the lemma is poved. Now v(x;, t is a solution of the one-dimensional wave equation on the half-line with Diichlet bounday condition implies fo 0 t, the solution is given by v(x;, t [g(x; + t g(x; t ] + t+ t h(x; y dy. Recall: u(x, t lim 0 u(x;, t. 0

13 Now Theefoe, v(x;, t ( k d ( k u(x;, t d k βj k j0 j+ j u(x;, t j β0 k u(x;, t + β k u (x;, t βk k k k u(x;, t. k which implies β0 k u(x;, t v(x;, t β k u (x;, t... βk k k k u(x;, t, k u(x;, t v(x;, t β k 0 βk β k 0 u (x;, t... βk k β0 k k k u(x;, t. k Theefoe, [ v(x;, t u(x, t lim 0 β0 k v(x;, t lim 0 β0 k lim 0 β0 k β k 0 βk β k 0 u (x;, t... βk k β0 k [ g(x; t + g(x; t + [ t g(x; t + h(x; t] whee β k (k. Recall g(x; t+ t ( k ( k φ(x;. Now n k + implies k (n /, and, theefoe, g(x; t ( t n 3 ( t n ] u(x;, t k k k ] h(x; y dy φ(y ds(y. And, Theefoe, h(x; t h(x; ( t ( k ( k ψ(x;. n 3 ( t n ψ(y ds(y.

14 Theefoe, implies u(x, t γ n ( u(x, t γ n [ t g(x; t + h(x; t] ( t + γ n ( t n 3 n 3 ( t n ( t n φ(y ds(y ψ(y ds(y whee γ n 3 5 (n. Even dimensions. As in the case of n dimensions, we use the method of descent. In paticula, suppose u(x,..., x n, t is a solution of the wave equation in R n with initial data u(x,..., x n, 0 φ(x,..., x n and u t (x,..., x n, 0 ψ(x,..., x n. Then define ũ(x,..., x n+, t u(x,..., x n, t φ(x,..., x n+ φ(x,..., x n ψ(x,..., x n+ ψ(x,..., x n. Theefoe, ũ is a solution of the wave equation in R n+, whee now n + is odd. Theefoe, fom the fomula above fo the case when the dimension is odd, ou solution at the point (x, t (x,..., x n, 0, t is given by ũ(x, t ( γ n+ + γ n+ ( t ( t n n ( t n (t n φ(y ds(y ψ(y ds(y whee γ n+ 3 5 (n, and whee B(x, t is the ball in R n+ of adius t about the point x (x,..., x n, 0. Now, φ(y ds(y φ(y ds(y. (n + α(n + t n But, notice B(x, t {y n+ 0} is the gaph of the function γ(y (t y x /. And, similaly, B(x, t {y n+ 0} is the gaph of γ. Theefoe, φ(y ds(y φ(y( + γ(y / dy (n + α(n + t n (n + α(n + t n B(x,t Now ( + γ(y / t(t y x /.

15 Theefoe, φ(y ds(y (n + α(n + t n B(x,t tα(n (n + α(n + α(nt n tα(n (n + α(n + Theefoe, ou solution fomula is given by u(x, t ( γ n+ + γ n+ γ n+ ( t ( t n n ( t n ( t n [ ( ( α(n (n + α(n + t ( n ( + t n t Now γ n+ 3 5 (n and B(x,t tφ(y dy (t y x / B(x,t φ(y dy (t y x / φ(y dy. (t y x / φ(y ds(y ψ(y ds(y n ( t n ψ(y dy (t y x / φ(y dy (t y x / ]. whee Γ(n is the gamma function, α(n πn/ Γ ( n+, Theefoe, Γ(n 0 e x x n dx. γ n+ α(n (n + α(n (n Using popeties of the gamma function, namely that πn/ Γ( n+ (n + π(n+/ Γ( n+3 n (n + Γ( π/ Γ( n+. Γ(m + mγ(m and Γ(/ π /, 3

16 we can conclude that ( ( ( n + 3 n + n Γ and And, theefoe, ( n + ( n ( n Γ ( Γ (. α(n γ n+ (n + α(n + 4 (n n ( Theefoe, the solution of the wave equation in even dimensions is given by u(x, t γ n [ ( whee γ n 4 (n n. ( t + n ( t ( t n n B(x,t ( t n φ(y dy (t y x / B(x,t 7.5 Wave Equation in R n with a souce. ψ(y dy (t y x / In this section, we conside the inhomogeneous wave equation in R n. Fist, ecall Duhamel s Pinciple. If S(t is the solution opeato fo the fist-ode initial-value poblem { Ut + AU 0 U(0 Φ, then the solution of the inhomogeneous poblem { Ut + AU F U(0 Φ ] should be given by U(t S(tΦ + t 0 S(t sf (s ds. Now conside the initial-value poblem fo the wave equation in R n, u tt u f(x, t x R n u(x, 0 φ(x u t (x, 0 ψ(x. (7.5 4