The Wave Equation ♦ The method of characteristics ♦ Inclusion of boundary conditions ♦ Traveling waves and stationary waves TRANSVERSE OSCILLATIONS OF AN ELASTIC STRING T uniform linear density ρ y 2 θ 2 small displacements y θ T 1 1 • T1 cos θ1 = T2 cos θ2 x for small θ, cos θ ≃ 1 ⇒ T1 = T2 = T ∂2y • ρ δx = T sin θ − T sin θ ∂t2 2 1 ∂y ∂2y ∂y ∂y sin θ ≃ tan θ ≃ =⇒ ρ δx = T ( ) − ( ) ∂x ∂t2 ∂x 2 ∂x 1 (∂2y/∂x2)δx+... | {z } ∂2y ∂2y Thus ρ = T ∂t2 ∂x2 ∂2y 1 ∂2y T i.e., = , c2 ≡ wave equation ∂x2 c2 ∂t2 ρ 2nd-order linear PDEs A(x, y)uxx +2B(x, y)uxy +C(x, y)uyy +D(x, y)ux +E(x, y)uy +F (x, y)u = R(x, y) 2 • B − AC < 0 elliptic. Ex.: Laplace eqn. uxx + uyy = 0 2 • B − AC = 0 parabolic. Ex.: diffusion eqn. ut − αuxx = 0 2 2 • B − AC > 0 hyperbolic. Ex.: wave eqn. utt − c uxx = 0 ♦ General solutions of PDEs depend on arbitrary functions (analogous to solutions of ODEs depending on arbitrary constants) −→ boundary conditions to determine such functions • Cauchy boundary conditions: assign function u and normal derivative ∂u/∂n on given curve γ in xy plane (relevant to hyperbolic PDEs) !2 y 1 !2 y D’Alembert’s solution 2 = 2 2 !x c !t Change variables u = x ! ct v = x + ct ! !u ! !v ! ! ! !x = !x !u + !x !v = !u + !v ! !u ! !v ! ! ! " c c !t = !t !u + !t !v = !u + !v 2 2 2 !2 y 1 !2 y ! ! % 1 ! ! % ! ! % = " + y = (c + c y = ( y !x2 c2 !t 2 $ u v$ c2 $ u v$ $ u v$ " ! ! % " ! ! % " ! ! % "2 y ! = 0 f, g arbitrary functions "u"v General solution y() u,v = f() u + g() v i.e. y() x,t = f() x ! ct + g() x + ct • The curves in the xt plane x − ct = const. x + ct = const. are called the characteristics of the wave equation. t x−ct=const. x x+ct=const. CHARACTERISTICS Consider the PDE Aytt + 2Bytx + Cyxx + Dyt + Eyx + F y = R A B Q = matrix of 2nd − order coefficients B C • Characteristics of the above PDE are defined as the curves χ(t, x) = const. such that their normal n is rotated by 90◦ by Q or is annihilated by Q, i.e., n · Q n = 0. • Since n ∝ ∇χ, for characteristics ∇χ · Q ∇χ = 0. 1 −1/c2 0 ⊲ Example. Wave eqn. : − ytt + yxx = 0 ⇒ Q = c2 0 1 The curves χ∓(t, x)= x ∓ ct =const. are the characteristics of the wave equation because ∓c −1/c2 0 ∓c ±1/c ∇χ∓ = ⇒ ∇χ · Q ∇χ =( ∓c 1) =( ∓c 1) = 0 1 0 1 1 1 • The condition ∇χ · Q ∇χ = 0 implies, using ∇χ =(χt,χx), that A B χt ( χt χx ) = 0 , B C χx 2 2 i.e., Aχt + 2Bχtχx + Cχx = 0 . ′ Expressing the derivatives in terms of x (t)= −χt/χx, A[x′(t)]2 − 2Bx′(t)+ C = 0 . ♠ Hyperbolic eqns. (B2 − AC > 0) have 2 families of characteristics ♠ Parabolic eqns. (B2 − AC = 0) have 1 (Q ∇χ = 0) ♠ Elliptic eqns. (B2 − AC < 0) have none Uses of characteristics • Characteristics χ∓(t, x)= const. can be used to solve hyperbolic equations by means of the transformation of variables u = χ−(t, x) v = χ+(t, x) ⊲ Example: D’Alembert solution of the wave equation • Characteristics serve to analyze whether boundary value problems for PDEs are well posed. ⊲ Example: Cauchy conditions on curve γ well-defined provided γ is not a characteristic [Cauchy-Kovalevska theorem] Cauchy boundary conditions and characteristics Consider the PDE Aytt + 2Bytx + Cyxx = H(yt, yx, y, t, x) • Cauchy conditions: Suppose y and the normal derivative yn are assigned on the curve γ specified by G(t, x) = 0 ⊲ The normal and tangential directions to γ are n ∝∇G =(Gt,Gx), τ ∝ (−Gx,Gt). ⊲ Given y on γ, we can compute tangential derivative yτ . From yn and yτ we can get yt and yx. • Can we determine higher derivatives as well? ∂ ytt yt = τ · ∇ yt ∝ ( −Gx Gt ) = −Gxytt + Gtyxt ∂τ yxt ∂ ytx yx = τ · ∇ yx ∝ ( −Gx Gt ) = −Gxytx + Gtyxx ∂τ yxx ⇒ 3 linear equations in ytt, ytx, yxx, with unique solution if det =6 0 A 2B C det −Gx Gt 0 =6 0 − 0 Gx Gt 2 2 ⇒ AGt + 2BGtGx + CGx =6 0 That is, Cauchy conditions on curve γ are well-defined provided γ is not a characteristic .#$%$'%().(+(,$,.,/+ !2 !2 y 1 y ! , = " + + 2 = 2 2 y() x t f() x ct g() x ct !x c !t f and g are determined by the initial conditions: Suppose at time t = 0, the wave has an initial displacement U(x) and an initial velocity V (x) y() x,0 = f() x + g() x = U() x x y() x,0 1 = "cf#() x + cg#() x = V() x $ f() x " g() x = " V() x' dx ' t c % b x 1 1 f() x = U() x ! V() x' dx ' 2 2c " b x 1 1 g() x = U() x + V() x' dx ' 2 2c ! b " x+ct x!ct $ x+ct 1 1 ( 1 1 y() x,t = "U() x ! ct + U() x + ct $ + V() x dx ! V() x dx = "U() x ! ct + U() x + ct $ + V() x dx 2 # % 2c " " ( 2 # % 2c " # b b %( x!ct !69$ &4($5(0,$(-(.&/$.(10&-23/&0$4(5-/&1(7(-0$.(/(&5(4$8.,7$.(508$$ V(x) = 0 ' x+ct % 1 1 1 " $ y x,t = "U x ct + U x + ct ( + V x dx , y() x,t = U() x ! ct + U() x + ct () # ()! ()) () 2 % 2 2c $ , ( x!ct ' U() x y x,t = f x ! ct + g x + ct () () () What is the form of f (x), g(x)? y() x,t = f() x ! ct + g() x + ct If time dependence is cos ()!t the full ()x,t dependence is given by y(x,t)=Acos ()kx ! !t + Bcos ()kx " !t ! ""$%&"%'( ' • Speed of wave c = k 2" 1 y(x,t) ! = Frequency f = = • ! 2# t " 2" 2 ( , ) ! • Wavelength ! = y x t = k k x k is "wavenumber" We can write the equation of a travelling wave in a number of analogous forms: Velocity Wavelength Period Angular frequency Asin () kx"" t " / k 2 / k 2 /" " Asin k() x" vt v 2 / k 2 / vk vk &,()x t $/ % $ % 2 /% Asin,- 2 $" * 0./$ % . v $ $ / v 2 v / $ Asin0&, 2 () x" vt / $ . N.B. Can include phase most easily by putting = &, y()() x, t Re0 A exp 0&, i kx"" t . where A is complex. N.B.2 Sometimes more convenient to switch x and t, i.e. y()() x, t= A sin " t" kx This is still a travelling wave moving to the right. For non-sinusoidal wave moving to right with speed v, can always write as f() x! vt . y() t )#$%%#'.)-#,,-) y = Asin()kx ! "t + Asin()kx + "t 2 sin cos " = A kx t y() t + !t 1-)#$%%#'.)-#,,1) .%',)/,%,'#00.) sin ! 2# sin 2# y = A ()kx t + 1 + A ()kx + t + 2 2 sin # # cos # ! # = A ()kx + 1 + 2 ()t + 1 2 !$"$%('( *&",$-. -"0- † !" !.E = 0 !"10%%%3()%).",-.()401%%)(2"3%5) !" !.B = 0 !" !" B !" E # = t !" 0 !" 1 E !" B = !" !" !" c2 t 2 " " E = # E + () .E !" !" !" B " # B 1 2 E !" # = ! = ! t t c2 t 2 !" !" 1 "2 E !2 E = c2 "t 2 !" !" EM plane wave E = E()z !" !" † !" !2 E 1 !2 E !E = = 0 " 0 ! 2 2 ! 2 ! Ez = z c t z 1".(6%1(%)0"6%) !$" %&' ($)* sin Ex = A ()kx ! t sin Ey = B ()kx ! t + # E x E y.