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Waves, Wave Equation, and Phase Velocity: An Introduction, Study notes of Calculus

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

  • What are boundary conditions in waves?
  • What is a wave?
  • What is the solution to the wave equation?
  • What are complex numbers?
  • What is the one-dimensional wave equation?

Typology: Study notes

2021/2022

Uploaded on 09/12/2022

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Waves, the Wave Equation, and Phase
Velocity
What is a wave?
Forward [f(x-vt)] and
backward [f(x+vt)]
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
f(x)
f(x-3)
f(x-2)
f(x-1)
x
0 1 2 3
Source: Trebino, Georgia Tech
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12
pf13
pf14
pf15
pf16
pf17
pf18
pf19
pf1a
pf1b
pf1c
pf1d
pf1e
pf1f
pf20

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Waves, the Wave Equation, and Phase

Velocity

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

f ( x )

f ( x- 3)

f ( x- 2)

f ( x- 1)

0 1 2 3^ x

Source: Trebino, Georgia Tech

What is a wave?

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 )

f ( x- 3)

f ( x- 2)

f ( x- 1)

0 1 2 3^ x

f ( x - v t )

f ( x + v t )

Proof that f ( x ± v t ) solves the wave equation

Write f ( x ± v t ) as f ( u ), where u = x ± v t. So and

Now, use the chain rule:

So ⇒ and ⇒

Substituting into the wave equation:

u x

v

u

t

f f u x u x

∂ ∂ ∂

∂ ∂ ∂

f f u t u t

∂ ∂ ∂

∂ ∂ ∂

f f x u

v
f f
t u

2 2 2

2 v 2
f f
t u

2 2 2 2

f f x u

2 2 2 2 2 2 2 2 2 2 2

v 0
v v
f f f f
x t u u
∂ ∂ ∂ ^ ∂ 

The 1D wave equation for light waves

We’ll use cosine- and sine-wave solutions:

or

where:

2 2

E E

x t

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

θ = Absolute phase (or initial phase)

kx

Definitions

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

  • ∂ϕ /∂ t v = ––––––– ∂ϕ /∂ x

The Phase of a Wave

This formula is useful when the wave is really complicated.

Tacoma Narrows Bridge

  1. The animation shows the Tacoma Narrows Bridge shortly before its collapse. What is its frequency? A .1 Hz B .25 Hz C .50 Hz D 1 Hz
  2. The distance between the bridge towers (nodes) was about 860 meters and there was also a midway node. What was the wavelength of the standing torsional wave? A 1720 m B 860 m C 430 m D There is no way to tell.
  3. What is the amplitude?

A 0.4 m B 4 m C 8 m D 16 m

Animation: http://www.youtube.com/watch?v=3mclp9QmCGs

Euler's Formula

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

A = Amplitude

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

x x x
f x = f + f + f + f +

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

= + + + + +

= − + − + +

= − + − + +

If we substitute x = i ϕ

into exp( x ), then:

More complex number theorems

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

To convert z into polar form, A exp( i ϕ):

A^2 = Re{ z }^2 + Im{ z }^2

tan(ϕ) = Im{ z } / Re{ z }

We can also differentiate exp( ikx ) as if

the argument were real.

[ ]

[ ]

exp( ) exp( )

cos( ) sin( ) sin( ) cos( )

sin( ) cos( )

1/ sin( ) cos( )

d

ikx ik ikx

dx

d

kx i kx k kx ik kx

dx

ik kx kx

i

i i ik i kx kx

Proof :

But , so :

Waves using complex amplitudes

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:

, exp

, exp( ) exp

E x t A i kx t

E x t A i i k x t

ω θ

θ ω

E  0 (^) = A exp( − i θ) ←(note the " ~ ")

E ( x t , ) = E 0 exp i kx ( − ω t )

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

( , ) exp ( ) exp ( ) exp ( )

( ) exp ( )

E tot x t E i kx t E i kx t E i kx t

E E E i kx t

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.