Wave Equations and Waves Flashcards
General Function of a Wave in One Dimension
moving to the right
y = f (x-vt+𝛿)
General Function of a Wave in One Dimension
moving to the left
y = f(x+vt+𝛿)
The Wave Equation in One Dimension
∂²y/∂x² = 1/v² ∂²y/∂t²
- where v is the speed of the propagation of the wave
- any function of the form y=f(x-vt+𝛿) will satisfy the 1D wave equation
General Form of a Harmonic Wave
y = A sin(kx - ωt + 𝛿)
-where ω=2πf , k=2π/λ , v=ω/k
Proof that a harmonic wave satisfies the 1D wave equation
y = Asin(kx - ωt + 𝛿) = Asin( k(x - ω/k t + 𝛿/k) = f(x-vt+𝛿)
The Laplacian
∇²V = |∇.|∇ V = ∂²V/∂x² + ∂²V/∂y² + ∂²V/∂z²
-where V is a scalar
Vector Form of the Laplacian
∇²|E = ∇²Ex ^i + ∇²Ey ^j + ∇²Ez ^k
The Wave Equation in Three Dimensions
∇² |E = 1/v² * ∂²|E/∂t²
General Equation of a Plane Wave
|E = Eo ^i sin ( |k.|r - ωt + 𝛿)
- where |E is a plane wave travelling in the |k direction
- its a plane wave because if you change your position |r by moving perpendicular to |k then |k.|r does not change, so in a plane perpendicular to |k all the values for |E are the same
Compare the Wave Equations in 3D and 1D
- the 3D wave equation is similar to the 1D wave equation, but:
- -now a vector whose size oscillates and,
- -to move from one wavelength you have to change |k.|r by 2π as this makes the sine function go through one oscillation, therefore creating one wavelength
A Vector Identity
|∇ x (|∇ x |E) = |∇ (|∇ . |E) - ∇² |E
Maxwell’s Equations in a Vacuum
-for |E and |B in a vacuum, there can be no conduction current density in a vacuum as there are no charges to conduct with:
|∇ x |B = μoεo * ∂|E/∂t
and
|∇ . |E = 0
Electric Field and Waves
-sub in to the R.H.S of the vector identity
|∇ x (|∇ x |E) = |∇ x (-∂|B/∂t) from Faraday’s Law
-switch the order of differentiation:
= -∂/∂t (|∇ x |B)
-sub in Ampere-Maxwell Law:
= - ∂/∂t (μoεo * ∂|E/∂t)
= - μoεo * ∂²|E/∂t²
-using the L.H.S. of the vector identity and Gauss’s Law for electric fields in a vacuum:
|∇ (|∇ . |E) - ∇² |E = |∇ (0) - ∇² |E
-equate L.H.S. and R.H.S.:
- ∇² |E = - ∂/∂t (μoεo * ∂|E/∂t)
∇² |E = μoεo * ∂²|E/∂t²
-this is the same form as the wave equation:
∇² |E = 1/v² * ∂²|E/∂t²
-so we can equate
1/v² = μoεo
-this gives v = 2.9979 x 10^8 m/s , the speed of light
-electric fields propagate through space at the speed of light
Magnetic Field and Waves
-sub in to the R.H.S of the vector identity
|∇ x (|∇ x |B) = |∇ x (μoεo * ∂|E/∂t)
-switch the order of differentiation:
= μoεo ∂/∂t (|∇x|E)
-sub in using Faraday’s Law:
= μoεo ∂/∂t (- ∂|B/∂t)
= - μoεo * ∂²|B/∂t²
-using the R.H.S of the vector identity and Gauss’s Law for magnetic fields:
|∇ (|∇ . |B) - ∇² |B = |∇ (0) - ∇² |B = - ∇² |B
-equate the L.H.S and R.H.S of the vector identity:
- μoεo * ∂²|B/∂t² = - ∇² |B
μoεo * ∂²|B/∂t² = ∇² |B
-this is the same form as the wave equation:
∇² |B = 1/v² * ∂²|B/∂t²
-so we can equate
1/v² = μoεo
-this gives v = 2.9979 x 10^8 m/s , the speed of light
-magnetic fields propagate through space at the speed of light
Mawell - Light and Magnetism
“the agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.”
What is a wave?
-a disturbance that propagates in time and space
What is an electromagnetic wave?
-for electromagnetic waves, a time dependent magnetic field induces a time dependent electric field that then induces a time dependent magnetic field and so on
-electromagnetic waves are therefore self proagating
-in a vacuum there are no sources or sinks of |E and |B so:
|∇.|E = 0 and |∇.|B = 0
-which means that there are no beginnings or ends to our field lines
-these electromagnetic waves therefore propagate indefinitely through free space
Representing Plane Waves Using Complex Numbers
Description
-a plane wave is represented by:
|E = |Eo cos (|k . |r - ωt + 𝛿) where c=ω/k
-a useful way to represent a plane wave is using complex numbers, where e^iθ = cosθ + isinθ
|E = |Eo exp (i (|k . |r - ωt + 𝛿))
-the real part of this is what we observe and measure, the imaginary part just makes the maths easier
Representing Plane Waves Using Complex Numbers
∂/∂t
-differentiate |E
∂|E/∂t = (-iω) |Eo exp (i (|k . |r - ωt + 𝛿)) = -iω |E
∂²|E/∂t² = (-iω)(-iω) |Eo exp (i (|k . |r - ωt + 𝛿)) = -ω² |E
-therefore for our complex plane wave:
∂/∂t = -iω
Representing Plane Waves Using Complex Numbers
|∇
-evaluate the dot product in the plane wave equation in terms of components:
|k . |r = (kx,ky,kz) . (x,y,z) = kxx + kyy + kzz
so,
|E = Eo exp(i (kxx + kyy + kzz - ωt + 𝛿))
-find |∇E for each component of |E:
∂Ex/∂x = i kx |E
∂Ey/∂y = i ky |E
∂Ez/∂z = i kz |E
-therefore del is represented by:
|∇ = i |k , where |k is the wave vector (not z)
Representing Plane Waves Using Complex Numbers
Divergence
-since |∇ = i |k
|∇ . |E = i |k . |E
Representing Plane Waves Using Complex Numbers
Curl
-since |∇ = i |k
|∇ x |E = i |k x |E
Orientation of |E and |B relative to |k
-in free space (vacuum) we know that |∇ . |E = 0
so |∇ . |E = i |k . |E = 0
-this means that |E is perpendicular to the direction of propagation |k
-also, in free space |∇ . |B = 0
so |∇ . |B = i |k . |B = 0
-this means that |B is also perpendicular to the direction of propagation |k
-but we can’t tell from this the relative orientations of |E and |B
Relative Orientations of |E and |B
-from Faraday’s Law:
|∇ x |E = - ∂|B/∂t , using complex notation
i |k x |E = iω |B, rewrite |k = k* ^k and cancel i
k* ^k x |E = ω |B , divide by k
^k x |E = ω/k |B = c |B
-from Ampere-Maxwell Law in free space
|∇ x |B = 1/c² ∂|E/∂t , using complex notation:
i |k x |B = -iω|E / c², rewrite |k = k* ^k and cancel i
k* ^k x |B = -ω/c² |E , divide by k
^k x |B = -ω/kc² |E = - 1/c |E
-using both equations:
^k x |B = - 1/c |E
-so |E is perpendicular to both ^k and |B
-also taking the modulus over both sides:
| ^k x |B | = | - 1/c |E |
-as ^k and |B are orthogonal and ^k is a unit vector:
B = E / c
Proving that |E and |B in Electromagnetic Waves are in Phase
-using faraday's law: ∫ |E . d|l = - d/dt ∫ |B . ^n dA -calculate the circulation of E (L.H.S.) -equate with R.H.S. ∂Ey/∂x = - ∂Bz/∂t -let Ey = Eyo sin (kx-ωt) ∂Bz/∂t = - ∂Ey/∂x = - Eyo * k cos(kx-ωt) -integrate w.r.t. t to find Bz: Bz = Eyo*k/ω sin (kx-ωt) + C = Eyo/c sin (kx-ωt) + C -constant of integration, C, is zero since if there was no oscillating field, there would be no induced magnetic field in free space, Eyo=0 => Bz = 0 Bz = Eyo/c sin (kx-ωt) -B and E have the same k and ω, so must be in phase
Poynting Vector
Definition
|S = (|E x |B) / μo
- the vector S is in the direction of propagation of the wave
- its size is the rate of energy flow across a unit area placed perpendicular to the flow direction
