Secod Order iear Partial Differetial Equatios Part IV Oe-dimesioal udamped wave equatio; D Alemert solutio of the wave equatio; damped wave equatio ad the geeral wave equatio; twodimesioal aplace equatio The secod type of secod order liear partial differetial equatios i 2 idepedet variales is the oe-dimesioal wave equatio. Together with the heat coductio equatio, they are sometimes referred to as the evolutio equatios ecause their solutios evolve, or chage, with passig time. The simplest istace of the oe-dimesioal wave equatio prolem ca e illustrated y the equatio that descries the stadig wave ehiited y the motio of a piece of udamped viratig elastic strig. 2008, 2012 Zachary S Tseg E-4-1
Udamped Oe-Dimesioal Wave Equatio: Viratios of a Elastic Strig Cosider a piece of thi fleile strig of legth, of egligile weight. Suppose the two eds of the strig are firmly secured ( clamped ) at some supports so they will ot move. Assume the set-up has o dampig. The, the vertical displacemet of the strig, 0 < <, ad at ay time t > 0, is give y the displacemet fuctio u(, t). It satisfies the homogeeous oedimesioal udamped wave equatio: a 2 u u tt Where the costat coefficiet a 2 is give y the formula a 2 T / ρ, such that a horizotal propagatio speed (also kow as phase velocity) of the wave motio, T force of tesio eerted o the strig, ρ mass desity (mass per uit legth). It is sujected to the homogeeous oudary coditios u(0, t) 0, ad u(, t) 0, t > 0. The two oudary coditios reflect that the two eds of the strig are clamped i fied positios. Therefore, they are held motioless at all time. The equatio comes with 2 iitial coditios, due to the fact that it cotais the secod partial derivative of time, u tt. The two iitial coditios are the iitial (vertical) displacemet u(, 0), ad the iitial (vertical) velocity u t (, 0) *, oth are aritrary fuctios of aloe. (Note that the strig is merely the medium for the wave, it does ot itself move horizotally, it oly virates, vertically, i place. The resultig udulatio, or the wave-like shape of the strig, is what moves horizotally.) * Velocity rate of chage of displacemet with respect to time. The other first partial derivative u represets the slope of the strig at a poit ad time t. 2008, 2012 Zachary S Tseg E-4-2
Hece, what we have is the followig iitial-oudary value prolem: (Wave equatio) a 2 u u tt, 0 < <, t > 0, (Boudary coditios) u(0, t) 0, ad u(, t) 0, (Iitial coditios) u(, 0) f (), ad u t (, 0) g(). We first let u(, t) X()T(t) ad separate the wave equatio ito two ordiary differetial equatios. Sustitutig u X T ad u tt X T ito the wave equatio, it ecomes a 2 X T X T. 2008, 2012 Zachary S Tseg E-4-3
Dividig oth sides y a 2 X T : X X T 2 a T As for the heat coductio equatio, it is customary to cosider the costat a 2 as a fuctio of t ad group it with the rest of t-terms. Isert the costat of separatio ad reak apart the equatio: X X λ X T λ 2 X a T X λx X + λx 0, T λ a 2 T T a 2 λ T T + a 2 λ T 0. The oudary coditios also separate: u(0, t) 0 X(0)T(t) 0 X(0) 0 or T(t) 0 u(, t) 0 X()T(t) 0 X() 0 or T(t) 0 As usual, i order to otai otrivial solutios, we eed to choose X(0) 0 ad X() 0 as the ew oudary coditios. The result, after separatio of variales, is the followig simultaeous system of ordiary differetial equatios, with a set of oudary coditios: X + λx 0, X(0) 0 ad X() 0, T + a 2 λ T 0. 2008, 2012 Zachary S Tseg E-4-4
The et step is to solve the eigevalue prolem X + λx 0, X(0) 0, X() 0. We have already solved this eigevalue prolem, recall. The solutios are Eigevalues: 2 2 π λ 2, 1, 2, 3, Eigefuctios: π X si, 1, 2, 3, Net, sustitute the eigevalues foud aove ito the secod equatio to fid T(t). After puttig eigevalues λ ito it, the equatio of T ecomes T π 2 2 2 + a T 2 0. It is a secod order homogeeous liear equatio with costat coefficiets. It s characteristic have a pair of purely imagiary comple cojugate roots: aπ r ± i. Thus, the solutios are simple harmoic: aπ t T ( t) A cos + B aπ t si, 1, 2, 3, Multiplyig each pair of X ad T together ad sum them up, we fid the geeral solutio of the oe-dimesioal wave equatio, with oth eds fied, to e 2008, 2012 Zachary S Tseg E-4-5
u(, t) 1 A aπ t cos + B si aπ t si π. There are two sets of (ifiitely may) aritrary coefficiets. We ca solve for them usig the two iitial coditios. Set t 0 ad apply the first iitial coditio, the iitial (vertical) displacemet of the strig u(, 0) f (), we have u(,0) 1 A 1 si ( A cos(0) + B si(0) ) π f ( ) si π Therefore, we see that the iitial displacemet f () eeds to e a Fourier sie series. Sice f () ca e a aritrary fuctio, this usually meas that we eed to epad it ito its odd periodic etesio (of period 2). The coefficiets A are the foud y the relatio A, where are the correspodig Fourier sie coefficiets of f (). That is A 2 f ( )si π d. 0 Notice that the etire sequece of the coefficiets A are determied eactly y the iitial displacemet. They are completely idepedet of the other sequece of coefficiets B, which are determied solely y the secod iitial coditio, the iitial (vertical) velocity of the strig. To fid B, we differetiate u(, t) with respect to t ad apply the iitial velocity, u t (, 0) g(). 2008, 2012 Zachary S Tseg E-4-6
u (, t) t 1 A aπ si aπ t + B aπ cos aπ t si π Set t 0 ad equate it with g(): aπ π ut (,0) B si 1 g( ). We see that g() eeds also e a Fourier sie series. Epad it ito its odd periodic etesio (period 2), if ecessary. Oce g() is writte ito a sie series, the previous equatio ecomes u (,0) t 1 B aπ si π g( ) 1 si π Compare the coefficiets of the like sie terms, we see Therefore, B aπ 2 g( )si π d. 0 B aπ 2 g( )si π d. aπ 0 As we have see, half of the particular solutio is determied y the iitial displacemet, the other half y the iitial velocity. The two halves are determied idepedet of each other. Hece, if the iitial displacemet f () 0, the all A 0 ad u(, t) cotais o sie-terms of t. If the iitial velocity g() 0, the all B 0 ad u(, t) cotais o cosie-terms of t. 2008, 2012 Zachary S Tseg E-4-7
et us take a closer look ad summarize the result for these 2 easy special cases, whe either f () or g() is zero. Special case I: Nozero iitial displacemet, zero iitial velocity: f () 0, g() 0. Sice g() 0, the B 0 for all. A 2 f ( )si π d, 1, 2, 3, 0 Therefore, u(, t) 1 A aπ t cos si π. 2008, 2012 Zachary S Tseg E-4-8
The D Alemert Solutio I 1746, Jea D Alemert produced a alterate form of solutio to the wave equatio. His solutio takes o a especially simple form i the aove case of zero iitial velocity. Use the product formula si(a) cos(b) [si(a B) + si(a + B)] / 2, the solutio aove ca e rewritte as u(, t) 1 2 1 A π ( at) π ( + at) si + si Therefore, the solutio of the udamped oe-dimesioal wave equatio with zero iitial velocity ca e alteratively epressed as u(, t) [F( at) + F( + at)] / 2. I which F() is the odd periodic etesio (period 2) of the iitial displacemet f (). A iterestig aspect of the D Alemert solutio is that it readily shows that the startig waveform give y the iitial displacemet would keep its geeral shape, ut it would also split eactly ito two halves. The two halves of the wave form travel i the opposite directios at the same fiite speed of propagatio a. This ca e see y the fact that the two halves of the wave form, i terms of, are eig traslated/moved i the opposite directio, to the right ad left, i the form of phase shifts, at the rate of distace a uits per uit time. Hece the value a is also kow as the wave s phase velocity. Jea le Rod d Alemert (1717 1783) was a Frech mathematicia ad physicist. He is perhaps est kow to calculus studets as the ivetor of the Ratio Test for covergece. 2008, 2012 Zachary S Tseg E-4-9
Furthermore, oce the wave frot has passed over a poit o the strig, the displacemet at that poit will e restored to its previous state efore the arrival of the wave. I physics, this aspect of a clearly-defied, echo-less, wave motio of a oe-dimesioal wave is called the Huyges Priciple. (The priciple also holds for solutios of a three-dimesioal wave equatio. But it is ot true for two-dimesioal waves.) Special case II: Zero iitial displacemet, ozero iitial velocity: f () 0, g() 0. Sice f () 0, the A 0 for all. B 2 g( )si π d, 1, 2, 3, aπ 0 Therefore, u(, t) 1 B si aπ t si π. 2008, 2012 Zachary S Tseg E-4-10
Eample: Solve the oe-dimesioal wave prolem 9 u u tt, 0 < < 5, t > 0, u(0, t) 0, ad u(5, t) 0, u(, 0) 4si(π) si(2π) 3si(5π), u t (, 0) 0. First ote that a 2 9 (so a 3), ad 5. The geeral solutio is, therefore, u(, t) 3π t A cos + B si si 1 5 5 5 3π t π. Sice g() 0, it must e that all B 0. We just eed to fid A. We also see that u(, 0) f () is already i the form of a Fourier sie series. Therefore, we just eed to etract the correspodig Fourier sie coefficiets: A 5 5 4, A 10 10 1, A 25 25 3, A 0, for all other, 5, 10, or 25. Hece, the particular solutio is u(, t) 4cos(3π t) si(π ) cos(6π t) si(2π ) 3cos(15π t) si(5π ). 2008, 2012 Zachary S Tseg E-4-11
We ca also solve the previous eample usig D Alemert s solutio. The prolem has zero iitial velocity ad its iitial displacemet has already ee epaded ito the required Fourier sie series, u(, 0) 4si(π) si(2π) 3si(5π) F(). Therefore, the solutio ca also e foud y usig the formula u(, t) [F( at) + F( + at)] / 2, where a 3. Thus u(, t) [ [ 4si(π( + 3t)) + 4si(π( 3t)) ] [si(2π( + 3t)) + si(2π( + 3t)) ] [3si(5π( + 3t)) + 3si(5π( + 3t)) ] ] / 2 Ideed, you could easily verify (do this as a eercise) that the solutio otaied this way is idetical to our previous aswer. Just apply the additio formula of sie fuctio ( si(α ± β) si(α)cos(β) ± cos(α)si(β) ) to each term i the aove solutio ad simplify. 2008, 2012 Zachary S Tseg E-4-12
Eample: Solve the oe-dimesioal wave prolem 9 u u tt, 0 < < 5, t > 0, u(0, t) 0, ad u(5, t) 0, u(, 0) 0, u t (, 0) 4. As i the previous eample, a 2 9 (so a 3), ad 5. Therefore, the geeral solutio remais u(, t) 3π t A cos + B si si 1 5 5 5 3π t π. Now, f () 0, cosequetly all A 0. We just eed to fid B. The iitial velocity g() 4 is a costat fuctio. It is ot a odd periodic fuctio. Therefore, we eed to epad it ito its odd periodic etesio (period T 10), the equate it with u t (, 0). I short: B 2 π 2 g( )si d a 4si π 3 π 0 80 2 2 3 π 0,, odd eve 5 0 π 5 d Therefore, u(, t) 1 80 3(2 1) 3(2 1) π t si 5 (2 1) π si 5 2 2 π. 2008, 2012 Zachary S Tseg E-4-13
The Structure of the Solutios of the Wave Equatio I additio to the fact that the costat a is the stadig wave s propagatio speed, several other oservatios ca e readily made from the solutio of the wave equatio that give isights to the ature of the solutio. To reduce the clutter, let us look at the form of the solutio whe there is o iitial velocity (whe g() 0). The solutio is u(, t) 1 A aπ t cos si π. The sie terms are fuctios of. They descried the spatial wave patters (the wavy shape of the strig that we could visually oserve), called the ormal modes, or atural modes. The frequecies of those sie waves that we could see, π /, are called the spatial frequecies of the wave. They are also kow as the wave umers. It measures the agular motio, i radias, per uit distace that the wave travels. The period of each spatial (sie) fuctio, 2/( π / ) 2 /, is the wave legth of each term. Meawhile, the cosie terms are fuctios of t, they give the vertical displacemet of the strig relative to its equilirium positio (which is just the horizotal, or the -ais). They descrie the up-ad-dow viratig motio of the strig at each poit of the strig. These temporal frequecies (the frequecies of fuctios of t; i this case, the cosies ) are the actual frequecies of oscillatig motio of vertical displacemet. Sice this is the udamped wave equatio, the motio of the strig is simple harmoic. The frequecies of the cosie terms, aπ / (measured i radias per secod), are called the atural frequecies of the strig. I a strig istrumet, they are the frequecies of the soud that we could hear. The correspodig atural periods ( 2π / atural frequecy) are, therefore, T 2 / a. For 1, the oservale spatial wave patter is that of si(π / ). The wave legth is 2, meaig the legth strig carries oly a half period of the siusoidal motio. It is the strig s first atural mode. The first atural 2008, 2012 Zachary S Tseg E-4-14
frequecy of oscillatio, aπ /, is called the fudametal frequecy of the strig. It is, give the set-up, the lowest frequecy ote the viratig strig ca produce. It is also called, i acoustics, as the first harmoic of the strig. For 2, the spatial wave patter is si(2π / ) is the secod atural mode. Its wavelegth is, which is the legth of the strig itself. The secod atural frequecy of oscillatio, 2aπ /, is also called the secod harmoic, or the first overtoe, of the strig. It is eactly twice of the strig s fudametal frequecy; hece its wavelegth ( ) is oly half as log. Acoustically, it produces a toe that is eactly oe octave higher tha the first harmoic. For 3, the third atural frequecy, 3aπ /, is also called the third harmoic, or the secod overtoe. It is 3 times larger tha the fudametal frequecy ad, at a 3:2 ratio over the secod harmoic, is situated eactly halfway etwee the adjacet octaves (at the secod ad the fourth harmoics). The fourth atural frequecy (fourth harmoic/ third overtoe), 4aπ /, is four times larger tha the fudametal frequecy ad twice of that the secod atural frequecy. The toe it produces is, therefore, eactly 2 octaves ad 1 octave higher tha those geerated y the first ad secod harmoics, respectively. Together, the sequece of all positive iteger multiples of the fudametal frequecy is called a harmoic series (ot to e cofused with that other harmoic series that you have studied i calculus). The motio of the strig is the comiatio of all its atural modes, as idicated y the terms of the ifiite series of the geeral solutio. The presece, ad magitude, of the ature modes are solely determied y the (Fourier sie series epasio of) iitial coditios. astly, otice that the wavelike ehavior of the solutio of the udamped wave equatio, quite ulike the solutio of the heat coductio equatio discussed earlier, does ot decrease i amplitude/itesity with time. It ever reaches a steady state (uless the solutio is trivial, u(, t) 0, which occurs whe f () g() 0). This is a cosequece of the fact that the udamped wave motio is a thermodyamically reversile process that eeds ot oey the secod law of Thermodyamics. 2008, 2012 Zachary S Tseg E-4-15
First atural mode (oscillates at the fudametal frequecy / 1st harmoic): Secod atural mode (oscillates at the 2d atural frequecy / 2d harmoic): Third atural mode (oscillates at the 3rd atural frequecy / 3rd harmoic): 2008, 2012 Zachary S Tseg E-4-16
Summary of Wave Equatio: Viratig Strig Prolems The vertical displacemet of a viratig strig of legth, securely clamped at oth eds, of egligile weight ad without dampig, is descried y the homogeeous udamped wave equatio iitial-oudary value prolem: a 2 u u tt, 0 < <, t > 0, u(0, t) 0, ad u(, t) 0, u(, 0) f (), ad u t (, 0) g(). The geeral solutio is u(, t) 1 A aπ t cos + B si aπ t si π. The particular solutio ca e foud y the formulas: A 2 f ( )si π d, ad 0 B 2 g( )si π d. aπ 0 The solutio waveform has a costat (horizotal) propagatio speed, i oth directios of the -ais, of a. The viratig motio has a (vertical) velocity give y u t (, t) at ay locatio 0 < < alog the strig. 2008, 2012 Zachary S Tseg E-4-17
Eercises E-4.1: 1. Solve the viratig strig prolem of the give iitial coditios. 4 u u tt, 0 < < π, t > 0, u(0, t) 0, u(π, t) 0, (a) u(, 0) 12si(2) 16si(5) + 24si(6), u t (, 0) 0. () u(, 0) 0, u t (, 0) 6. (c) u(, 0) 0, u t (, 0) 12si(2) 16si(5) + 24si(6). 2. Solve the viratig strig prolem. 100 u u tt, 0 < < 2, t > 0, u(0, t) 0, ad u(2, t) 0, u(, 0) 32si(π) + e 2 si(3π) + 25si(6π), u t (, 0) 6si(2π) 16si(5π / 2). 3. Solve the viratig strig prolem. 25 u u tt, 0 < < 1, t > 0, u(0, t) 0, ad u(2, t) 0, u(, 0) 2, u t (, 0) π. 4. Verify that the D Alemert solutio, u(, t) [F( at) + F( + at)] / 2, where F() is a odd periodic fuctio of period 2 such that F() f () o the iterval 0 < <, ideed satisfies the give iitial-oudary value prolem y checkig that it satisfies the wave equatio, oudary coditios, ad iitial coditios. 2008, 2012 Zachary S Tseg E-4-18
a 2 u u tt, 0 < <, t > 0, u(0, t) 0, u(, t) 0, u(, 0) f (), u t (, 0) 0. 5. Use the method of separatio of variales to solve the followig wave equatio prolem where the strig is rigid, ut ot fied i place, at oth eds (i.e., it is ifleile at the edpoits such that the slope of displacemet curve is always zero at oth eds, ut the two eds of the strig are allowed to freely slide i the vertical directio). a 2 u u tt, 0 < <, t > 0, u (0, t) 0, u (, t) 0, u(, 0) f (), u t (, 0) g(). 6. What is the steady-state displacemet of the strig i #5? What is limu(, t)? Are they the same? t 2008, 2012 Zachary S Tseg E-4-19
Aswers E-4.1: 1. (a) u(, t) 12cos(4t) si(2) 16cos(10t) si(5) + 24cos(12t) si(6). (c) u(, t) 3si(4t) si(2) 1.6si(10t) si(5) + 2si(12t) si(6). 5. The geeral solutio is aπ t aπ t π u(, t) A0 + B0 t+ A cos + B si cos 1 The particular solutio ca e foud y the formulas: 1 A0 f ( ) d 0, A B 2 g( )cos π d aπ 0 2 f ( ) cos π d. 0, B. 1 0 g( ) d 0, ad 6. The steady-state displacemet is the costat term of the solutio, A 0. The limit does ot eist uless u(, t) C is a costat fuctio, which happes whe f () C ad g() 0, i which case the limit is C. They are ot the same otherwise. 2008, 2012 Zachary S Tseg E-4-20
The Geeral Wave Equatio The most geeral form of the oe-dimesioal wave equatio is: a 2 u + F(, t) u tt + γ u t + k u. Where a the propagatio speed of the wave, γ the dampig costat k (eteral) restoratio factor, such as whe viratios occur i a elastic medium. F(, t) aritrary eteral forcig fuctio (If F 0 the the equatio is homogeeous, else it is ohomogeeous.) 2008, 2012 Zachary S Tseg E-4-21
The Telegraph Equatio The most well-kow eample of (a homogeeous versio of) the geeral wave equatio is the telegraph equatio. It descries the voltage u(, t) iside a piece of telegraph / trasmissio wire, whose electrical properties per uit legth are: resistace R, iductace, capacitace C, ad coductace of leakage curret G: a 2 u u tt + γ u t + k u. Where a 2 1 / C, γ G / C + R /, ad k GR / C. 2008, 2012 Zachary S Tseg E-4-22
Eample: The Oe-Dimesioal Damped Wave Equatio a 2 u u tt + γ u t, γ 0. Suppose oudary coditios remai as the same (oth eds fied): (0, t) 0, ad u(, t) 0. The equatio ca e separated as follow. First rewrite it as: a 2 X T X T + γ X T, Divide oth sides y a 2 X T, ad isert a costat of separatio: X X T + γ T a T λ 2. Rewrite it ito 2 equatios: X λ X X + λ X 0, T + γ T a 2 λ T T + γ T + a 2 λ T 0. The oudary coditios also are separated, as usual: u(0, t) 0 X(0)T(t) 0 X(0) 0 or T(t) 0 u(, t) 0 X ()T(t) 0 X() 0 or T(t) 0 As efore, settig T(t) 0 would result i the costat zero solutio oly. Therefore, we must choose the two (otrivial) coditios i terms of : X(0) 0, ad X() 0. 2008, 2012 Zachary S Tseg E-4-23
After separatio of variales, we have the system X + λx 0, X(0) 0 ad X() 0, T + γ T + α 2 λ T 0. The et step is to fid the eigevalues ad their correspodig eigefuctios of the oudary value prolem X + λx 0, X(0) 0 ad X() 0. This is a familiar prolem that we have ecoutered more tha oce previously. The eigevalues ad eigefuctios are, recall, Eigevalues: 2 2 π λ 2, 1, 2, 3, Eigefuctios: π X si, 1, 2, 3, The equatio of t, however, has differet kid of solutios depedig o the roots of its characteristic equatio. 2008, 2012 Zachary S Tseg E-4-24
(Optioal topic) Nohomogeeous Udamped Wave Equatio Prolems of partial differetial equatio that cotais a ozero forcig fuctio (which would make the equatio itself a ohomogeeous partial differetial equatio) ca sometimes e solved usig the same idea that we have used to hadle ohomogeeous oudary coditios y cosiderig the solutio i 2 parts, a steady-state part ad a trasiet part. This is possile whe the forcig fuctio is idepedet of time t, which the could e used to determie the steady-state solutio. The trasiet solutio would the satisfy a certai homogeeous equatio. The 2 parts are thus solved separately ad their solutios are added together to give the fial result. et us illustrate this idea with a simple eample: whe the strig s weight is o loger egligile. Eample: A fleile strig of legth has its two eds firmly secured. Assume there is o dampig. Suppose the strig has a weight desity of 1 Newto per meter. That is, it is suject to, uiformly across its legth, a costat force of F(, t) 1 uit per uit legth due to its ow weight. et u(, t) e the vertical displacemet of the strig, 0 < <, ad at ay time t > 0. It satisfies the ohomogeeous oe-dimesioal udamped wave equatio: a 2 u + 1 u tt. The usual oudary coditios u(0, t) 0, ad u(, t) 0, apply. Plus the iitial coditios u(, 0) f () ad u t (, 0) g(). Sice the forcig fuctio is idepedet of time t, its effect is to impart, permaetly, a displacemet o the strig that depeds oly o the locatio (the effect is suject to the oudary coditios, thus might chage with ). That is, the effect is to itroduce a ozero 2008, 2012 Zachary S Tseg E-4-25
steady-state displacemet, v(). Hece, we ca rewrite the solutio u(, t) as: u(, t) v() + w(, t). By settig t to e a costat ad rewrite the equatio ad the oudary coditios to e depedet of oly, the steady-state solutio v() must satisfy: a 2 v + 1 0, v(0) 0, v() 0. Rewrite the equatio as v 1 / a 2, ad itegrate twice, we get 1 v ( ) C + C 2a 2 + 2 1 2. Apply the oudary coditios to fid C 1 / 2a 2 ad C 2 0: 1 2 ( ) + 2. 2a 2a v 2 Commet: Thus, the sag of a wire or cale due to its ow weight ca e see as a maifestatio of the steady-solutio of the wave equatio. The sag is also paraolic, rather tha siusoidal, as oe might have reasoaly assumed, i ature. We ca the sutract out v() from the equatio, oudary coditios, ad the iitial coditios (try this as a eercise), the trasiet solutio w(, t) must satisfy: a 2 w w tt, 0 < <, t > 0, w(0, t) 0, w(, t) 0, w(, 0) f () v(), w t (, 0) g(). 2008, 2012 Zachary S Tseg E-4-26
The prolem is ow trasformed to the homogeeous prolem we have already solved. The solutio is just w(, t) 1 A aπ t cos + B si aπ t si π. Comiig the steady-state ad trasiet solutios, the geeral solutio is foud to e u(, t) v( ) + w(, t) 1 2 aπ t aπ t + + A cos + si 2 2 B si 2a 2a 1 π The coefficiets ca e calculated ad the particular solutio determied y usig the formulas: A 2 ( f ( ) v( ) ) si π d, ad 0 B 2 g( )si π d. aπ 0 Note: Sice the velocity u t (, t) v t () + w t (, t) 0 + w t (, t) w t (, t). The iitial velocity does ot eed ay adjustmet, as u t (, 0) w t (, 0) g(). Commet: We ca clearly see that, eve though a ozero steady-state solutio eists, the displacemet of the strig still will ot coverge to it as t. 2008, 2012 Zachary S Tseg E-4-27
The aplace Equatio / Potetial Equatio The last type of the secod order liear partial differetial equatio i 2 idepedet variales is the two-dimesioal aplace equatio, also called the potetial equatio. Ulike the other equatios we have see, a solutio of the aplace equatio is always a steady-state (i.e. time-idepedet) solutio. Ideed, the variale t is ot eve preset i the aplace equatio. The aplace equatio descries systems that are i a state of equilirium whose ehavior does ot chage with time. Some applicatios of the aplace equatio are fidig the potetial fuctio of a oject acted upo y a gravitatioal / electric / magetic field, fidig the steady-state temperature distriutio of the (2- or 3-dimesioal) heat coductio equatio, ad the steady-state flow of a ideal fluid (where the flow velocity forms a vector field that has zero curl ad zero divergece). Sice the time variale is ot preset i the aplace equatio, ay prolem of the aplace equatio will ot, therefore, have ay iitial coditio. A aplace equatio prolem has oly oudary coditios. et u(, y) e the potetial fuctio at a poit (, y), the it is govered y the two-dimesioal aplace equatio u + u yy 0. Ay real-valued fuctio havig cotiuous first ad secod partial derivatives that satisfies the two-dimesioal aplace equatio is called a harmoic fuctio. Similarly, suppose u(, y, z) is the potetial fuctio at a poit (, y, z), the it is govered y the three-dimesioal aplace equatio u + u yy + u zz 0. 2008, 2012 Zachary S Tseg E-4-28
Commet: The oe-dimesioal aplace equatio is rather dull. It is merely u 0, where u is a fuctio of aloe. It is ot a partial differetial equatio, ut rather a simple itegratio prolem of u 0. (What is its solutio? Where have we see it just very recetly?) The oudary coditios that accompay a 2-dimesioal aplace equatio descrie the coditios o the oudary curve that ecloses the 2- dimesioal regio i questio. While those accompay a 3-dimesioal aplace equatio descrie the coditios o the oudary surface that ecloses the 3-dimesioal spatial regio i questio. 2008, 2012 Zachary S Tseg E-4-29
The Relatioships amog aplace, Heat, ad Wave Equatios (Optioal topic) Now let us take a step ack ad see the igger picture: how the homogeeous heat coductio ad wave equatios are structured, ad how they are related to the aplace equatio of the same spatial dimesio. Suppose u(, y) is a fuctio of two variales, the epressio u + u yy is called the aplacia of u. It is ofte deoted y 2 u u + u yy. Similarly, for a three-variale fuctio u(, y, z), the 3-dimesioal aplacia is the 2 u u + u yy + u zz. (As we have just oted, i the oe-variale case, the aplaia of u(), degeerates ito 2 u u.) The homogeeous heat coductio equatios of 1-, 2-, ad 3- spatial dimesio ca the e epressed i terms of the aplacias as: α 2 2 u u t, where α 2 is the thermo diffusivity costat of the coductig material. Thus, the homogeeous heat coductio equatios of 1-, 2-, ad 3- dimesio are, respectively, α 2 u u t α 2 (u + u yy ) u t α 2 (u + u yy + u zz ) u t 2008, 2012 Zachary S Tseg E-4-30
As well, the homogeeous wave equatios of 1-, 2-, ad 3- spatial dimesio ca the e similarly epressed i terms of the aplacias as: a 2 2 u u tt, where the costat a is the propagatio velocity of the wave motio. Thus, the homogeeous wave equatios of 1-, 2-, ad 3-dimesio are, respectively, a 2 u u tt a 2 (u + u yy ) u tt a 2 (u + u yy + u zz ) u tt Now let us cosider the steady-state solutios of these heat coductio ad wave equatios. I each case, the steady-state solutio, eig idepedet of time, must have all zero as its partial derivatives with respect to t. Therefore, i every istace, the steady-state solutio ca e foud y settig, respectively, u t or u tt to zero i the heat coductio or the wave equatios ad solve the resultig equatio. That is, the steady-state solutio of a heat coductio equatio satisfies α 2 2 u 0, ad the steady-state solutio of a wave equatio satisfies a 2 2 u 0. Eve the electromagetic waves are descried y this equatio. It ca e easily show y vector calculus that ay electric field E ad magetic field B satisfyig the Mawell s Equatios will also satisfy the 3- dimesioal wave equatio, with propagatio speed a c 299792 km/s, the speed of light i vacuum. 2008, 2012 Zachary S Tseg E-4-31
I all cases, we ca divide out the (always positive) coefficiet α 2 or a 2 from the equatios, ad otai a uiversal equatio: 2 u 0. This uiversal equatio that all the steady-state solutios of heat coductio ad wave equatios have to satisfy is the aplace / potetial equatio! Cosequetly, the 1-, 2-, ad 3-dimesioal aplace equatios are, respectively, u 0, u + u yy 0, u + u yy + u zz 0. Therefore, the aplace equatio, amog other applicatios, is used to solve the steady-state solutio of the other two types of equatios. Ad all solutios of a aplace equatio are steady-state solutios. To aswer the earlier questio, we have had see ad used the oe-dimesioal aplace equatio (which, with oly oe idepedet variale,, is a very simple ordiary differetial equatio, u 0, ad is ot a PDE) whe we were tryig to fid the steady-state solutio of the oe-dimesioal homogeeous heat coductio equatio earlier. 2008, 2012 Zachary S Tseg E-4-32
aplace Equatio for a rectagular regio Cosider a rectagular regio of legth a ad width. Suppose the top, ottom, ad left sides order free-space; while eyod the right side there lies a source of heat/gravity/magetic flu, whose stregth is give y f (y). The potetial fuctio at ay poit (, y) withi this rectagular regio, u(, y), is the descried y the oudary value prolem: (2-dim. aplace eq.) u + u yy 0, 0 < < a, 0 < y <, (Boudary coditios) u(, 0) 0, ad u(, ) 0, u(0, y) 0, ad u(a, y) f (y). The separatio of variales proceeds similarly. A slight differece here is that Y(y) is used i the place of T(t). et u(, y) X()Y(y) ad sustitutig u X Y ad u yy X Y ito the wave equatio, it ecomes Dividig oth sides y X Y : X Y + X Y 0, X Y X Y. X Y X Y Now that the idepedet variales are separated to the two sides, we ca isert the costat of separatio. Ulike the previous istaces, it is more coveiet to deote the costat as positive λ istead. X X Y λ Y 2008, 2012 Zachary S Tseg E-4-33
X X λ X λx X λx 0, Y Y λ Y λ Y Y + λ Y 0. The oudary coditios also separate: u(, 0) 0 X()Y(0) 0 X() 0 or Y(0) 0 u(, ) 0 X()Y() 0 X() 0 or Y() 0 u(0, y) 0 X(0)Y(y) 0 X(0) 0 or Y(y) 0 u(a, y) f (y) X(a)Y(y) f (y) [caot e simplified further] As usual, i order to otai otrivial solutios, we eed to igore the costat zero fuctio i the solutio sets aove, ad istead choose Y(0) 0, Y() 0, ad X(0) 0 as the ew oudary coditios. The fourth oudary coditio, however, caot e simplified this way. So we shall leave it as-is. (Do t worry. It will play a useful role later.) The result, after separatio of variales, is the followig simultaeous system of ordiary differetial equatios, with a set of oudary coditios: X λx 0, X(0) 0, Y + λ Y 0, Y(0) 0 ad Y() 0. Plus the fourth oudary coditio, u(a, y) f (y). The et step is to solve the eigevalue prolem. Notice that there is aother slight differece. Namely that this time it is the equatio of Y that gives rise to the two-poit oudary value prolem which we eed to solve. 2008, 2012 Zachary S Tseg E-4-34
Y + λy 0, Y(0) 0, Y() 0. However, ecept for the fact that the variale is y ad the fuctio is Y, rather tha ad X, respectively, we have already see this prolem efore (more tha oce, as a matter of fact; here the costat ). The eigevalues of this prolem are 2 λ σ 2 2 π 2, 1, 2, 3, Their correspodig eigefuctios are Y π y si, 1, 2, 3, Oce we have foud the eigevalues, sustitute λ ito the equatio of. We have the equatio, together with oe oudary coditio: 2 2 π X X 2 0, X(0) 0. 2 2 2 π Its characteristic equatio, r 0 2, has real roots Hece, the geeral solutio for the equatio of is r π ±. X π π C1 e + C2 e. The sigle oudary coditio gives X(0) 0 C 1 + C 2 C 2 C 1. 2008, 2012 Zachary S Tseg E-4-35
2008, 2012 Zachary S Tseg E-4-36 Therefore, for 1, 2, 3,, e e C X π π. Because of the idetity for the hyperolic sie fuctio 2 sih θ θ θ e e, the previous epressio is ofte rewritte i terms of hyperolic sie: K X π sih, 1, 2, 3, The coefficiets satisfy the relatio: K 2C. Comiig the solutios of the two equatios, we get the set of solutios that satisfies the two-dimesioal aplace equatio, give the specified oudary coditios: y K y Y X y u π π si sih ) ( ) ( ), (, 1, 2, 3, The geeral solutio, as usual, is just the liear comiatio of all the aove, liearly idepedet, fuctios u (, y). That is, y K y u π π si sih ), ( 1.
This solutio, of course, is specific to the set of oudary coditios u(, 0) 0, ad u(, ) 0, u(0, y) 0, ad u(a, y) f (y). To fid the particular solutio, we will use the fourth oudary coditio, amely, u(a, y) f (y). u( a, y) 1 K sih aπ si π y f ( y) We have see this story efore. There is othig really ew here. The summatio aove is a sie series whose Fourier sie coefficiets are K sih(aπ / ). Therefore, the aove relatio says that the last oudary coditio, f (y), must either e a odd periodic fuctio (period 2), or it eeds to e epaded ito oe. Oce we have f (y) as a Fourier sie series, the coefficiets K of the particular solutio ca the e computed: K sih aπ 2 0 f ( y)si π y dy Therefore, K 2 π y f ( y)si dy aπ aπ sih sih. 0 2008, 2012 Zachary S Tseg E-4-37
(Optioal topic) aplace Equatio i Polar Coordiates The steady-state solutio of the two-dimesioal heat coductio or wave equatio withi a circular regio (the iterior of a circular disc of radius k, that is, o the regio r < k) i polar coordiates, u(r, θ), is descried y the polar versio of the two-dimesioal aplace equatio u 1 u r 1 rr + r + u 0 2 θθ. r The oudary coditio, i this set-up, specifyig the coditio o the circular oudary of the disc, i.e., o the curve r k, is give i the form u(k, θ) f (θ), where f is a fuctio defied o the iterval [0, 2π). Note that there is oly oe set of oudary coditio, prescried o a circle. This will cause a slight complicatio. Furthermore, the ature of the coordiate system implies that u ad f must e periodic fuctios of θ, of period 2π. Namely, u(r, θ) u(r, θ + 2π), ad f (θ) f (θ + 2π). By lettig u(r, θ) R(r)Θ(θ), the equatio ecomes 1 1 R Θ+ R Θ+ RΘ 0 2. r r Which ca the e separated to otai R + rr Θ λ R Θ r 2. This equatio aove ca e rewritte ito two ordiary differetial equatios: r 2 R + rr λr 0, Θ + λθ 0. 2008, 2012 Zachary S Tseg E-4-38
The eigevalues are ot foud y straight forward computatio. Rather, they are foud y a little deductive reasoig. Based solely o the fact that Θ must e a periodic fuctio of period 2π, we ca coclude that λ 0 ad λ 2, 1, 2, 3, are the eigevalues. The correspodig eigefuctios are Θ 0 1 ad Θ A cos θ + B si θ. The equatio of r is a Euler equatio (the solutio of which is outside of the scope of this course). The geeral solutio of the aplace equatio i polar coordiates is u( r, A0 + si 2 ( A cos θ+ B θ) θ ) r. 1 Applyig the oudary coditio u(k, θ) f (θ), we see that A u( k, θ ) + 2 ( A k cos θ+ B k si θ) f ( ) 0 θ 1. Sice f (θ) is a periodic fuctio of period 2π, it would already have a suitale Fourier series represetatio. Namely, f ( a0 ) + si 2 1 ( a cos θ+ θ) θ. Hece, A 0 a 0, A a / k, ad B / k, 1, 2, 3 For a prolem o the uit circle, whose radius k 1, the coefficiets A ad B are eactly idetical to, respectively, the Fourier coefficiets a ad of the oudary coditio f (θ). 2008, 2012 Zachary S Tseg E-4-39
(Optioal topic) Udamped Wave Equatio i Polar Coordiates The viratig motio of a elastic memrae that is circular i shape ca e descried y the two-dimesioal wave equatio i polar coordiates: u rr + (1 / r) u r + (1 / r 2 ) u θθ a 2 u tt. The solutio is u(r, θ, t), a fuctio of 3 idepedet variales that descries the vertical displacemet of each poit (r, θ) of the memrae at ay time t. 2008, 2012 Zachary S Tseg E-4-40