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An introduction to waves, the wave equation, and phase velocity. It covers the concept of waves, propagating waves, the one-dimensional wave equation, and the solution to the wave equation. The document also discusses complex numbers, plane waves, and boundary conditions.
What you will learn
Typology: Study notes
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What is a wave?
Forward [ f ( x -v t )] and backward [ f ( x +v t )] propagating waves
The one-dimensional wave equation
Wavelength, frequency, period, etc.
Phase velocity Complex numbers
Plane waves and laser beams Boundary conditions
Div, grad, curl, etc., and the 3D Wave equation
Source: Trebino, Georgia Tech
A wave is anything that moves. To displace any function f ( x ) to the right, just change its argument from x to x-a , where a is a positive number. If we let a = v t , where v is positive and t is time, then the displacement will increase with time. So represents a rightward, or forward, propagating wave. Similarly, represents a leftward, or backward, propagating wave. v will be the velocity of the wave.
f ( x - v t )
f ( x + v t )
Proof that f ( x ± v t ) solves the wave equation
Now, use the chain rule:
So ⇒ and ⇒
Substituting into the wave equation:
u x
f f u x u x
∂ ∂ ∂
f f u t u t
∂ ∂ ∂
f f x u
2 2 2
2 2 2 2
f f x u
2 2 2 2 2 2 2 2 2 2 2
We’ll use cosine- and sine-wave solutions:
or
where:
2 2
με
E x t ( , ) = B cos[ ( k x ± v )] t + C sin[ ( k x ±v )] t
E x t ( , ) = B cos( kx ± ω t ) + C sin( kx ±ω t )
1 v k
ω
με
= =
kx ± ( v) k t
where E is the light electric field
The speed of light in vacuum, usually called “c”, is 3 x 10 10 cm/s.
Definitions: Amplitude and Absolute phase
E ( x,t ) = A cos[( k x – ω t ) – θ ]
A = Amplitude
Spatial quantities:
Temporal quantities:
The phase is everything inside the cosine.
E(x,t ) = A cos( ϕ), where ϕ = k x – ω t – θ
ϕ = ϕ( x,y,z,t ) and is not a constant, like θ!
In terms of the phase, ω = – ∂ϕ /∂ t
k = ∂ϕ /∂ x And
This formula is useful when the wave is really complicated.
A 0.4 m B 4 m C 8 m D 16 m
Animation: http://www.youtube.com/watch?v=3mclp9QmCGs
exp( i ϕ) = cos( ϕ) + i sin( ϕ)
so the point, P = A cos(ϕ) + i A sin(ϕ), can be written:
P = A exp( i ϕ)
where
ϕ = Phase
Proof of Euler's Formula
Use Taylor Series:
2 3 4
2 4 3
exp( ) 1 ... 1! 2! 3! 4!
1 ... ... 2! 4! 1! 3! cos( ) sin( )
i i i
i
i
= + − − + +
= (^) − + + (^) + (^) − + = +
2 3
exp( i ϕ ) = cos( ϕ ) + i sin( ϕ )
2 3 4
2 4 6 8
3 5 7 9
exp( ) 1 ... 1! 2! 3! 4!
cos( ) 1 ... 2! 4! 6! 8!
sin( ) ... 1! 3! 5! 7! 9!
x x x x x
x x x x x
x x x x x x
= + + + + +
= − + − + +
= − + − + +
into exp( x ), then:
Any complex number, z , can be written:
z = Re{ z } + i Im{ z } So Re{ z } = 1/2 ( z + z* ) and Im{ z } = 1/2 i ( z – z* )
where z * is the complex conjugate of z ( i → – i )
The "magnitude," | z |, of a complex number is:
| z |^2 = z z* = Re{ z }^2 + Im{ z }^2
A^2 = Re{ z }^2 + Im{ z }^2
tan(ϕ) = Im{ z } / Re{ z }
[ ]
[ ]
We can let the amplitude be complex:
where we've separated the constant stuff from the rapidly changing stuff.
The resulting "complex amplitude" is:
So:
ω θ
θ ω
E 0 (^) = A exp( − i θ) ←(note the " ~ ")
How do you know if E 0 is real or complex?
Sometimes people use the "~", but not always. So always assume it's complex.
As written, this entire field is complex!
Complex numbers simplify waves!
This isn't so obvious using trigonometric functions, but it's easy with complex exponentials:
1 2 3 1 2 3
ω ω ω ω
where all initial phases are lumped into E 1 , E 2 , and E 3.
Adding waves of the same frequency, but different initial phase, yields a wave of the same frequency.
