The Wave Equation the Method of Characteristics Inclusion Of

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+ct=const. CHARACTERISTICS

Consider the PDE Aytt + 2Bytx + Cyxx + Dyt + Eyx + F y = R

AB Q = matrix of 2nd − order coeﬃcients BC

• 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

AB 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 y 1 !2 y = ! y() x,t = f() x " ct + g() x + ct !x2 c2 !t 2

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$$5(0'$(+(1!)$.$"0!+&3)!.$#(/-)!"$*$+0$.$)$!/$#$%.,*$.$/08$$ 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 $" * (.%$ ! + v $ $ / v 2 $v / Asin(.&, 2 $ () x" vt /

N.B. Can include phase most easily by putting = &, y()() x, t Re(. A exp (.&, 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

0()!*&%!'.)-!,"1)

.&'")#"%"'!$$.)

sin ! 2# sin 2# y = A ()kx t + 1 + A ()kx + t + 2 2 sin # # cos # ! # = A ()kx + 1 + 2 ()t + 1 2 !$"!*('% #&"+!-. -")- † !" !.E = 0 !!10#%%3+)#).!,'&+)4$*##)+(!"#5) !" !.B = 0 !" !" B !" E # = t !" 0 !" 1 E !" B = c2 t !" !" !" " " E = # 2 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 *!&+/#*+#)0!/#) !$" %!& '$#*

sin Ex = A ()kx ! t sin Ey = B ()kx ! t + #

E x E y