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Waves Notes

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Waves and Geometrical Optics 2014

Contents 1 Wave Motion

2

2 The wave equation

2.1 Harmonic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 Wave velocities: which is which? . . . . . . . . . . . . . . . . . . . . . .

3 4 5

3 Waves on a string

3.1 Energetics of waves on a string . . . . . . . . . . . . . . . . . . . . . . .

5 6

4 Waves in 3D and Plane Waves

7

5 Reflection and Transmission 8

5.1 Power reflection coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 10 6 Optics: an Introduction

11

7 Geometrical Optics 13

7.1 The sign convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7.2 A simple lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1

1

1

WAVE MOTION

2

Wave Motion

In this section we are going to study the general mathematics of wave motion. The reason to do so is very simple: a number of very interesting physical systems exhibit wave motion. One of the simplest ways to demonstrate wave motion is shake up and down a rope
fixed at the other end. Crests and troughs move down the rope, making up a travelling wave which transmits energy and phase. A wave formed in this way is nothing more than a collection of many harmonic oscillators (the material points of the string), with a phase difference in space (on top of the phase evolution over time of the harmonic oscillator) related to their physical separation. This inherent similarity between wave motion and the simple harmonic oscillator means that a lot of the Mathematics you learn about in the Oscillations part of the course is going to come in handy here!
If the medium is limited in extent, for example if the rope was replaced by a violin string fixed at both ends, the waves travelling on the string would be reflected at both ends. The vibration of the string would then be the combination of the forward and backward waves and standing waves would be formed.

Figure 1: A simple sketch of a sinusoidal wave pattern at one instant in time. Before embarking on the study of the mathematics of wave motion, let me first walk you though some of the interesting Physics that you will be able to tackle.

1. Mechanical waves, like water waves, waves on strings, sound waves, seismic waves etc. They are all governed by Newton's Laws (hence mechanical) and propagate through a medium, whose particles perform Simple Harmonic Motion.

2. Electromagnetic waves, which represent the oscillation in time and space of electric and magnetic fields. There are no particles moving, so electromagnetic radiation does not need a medium to propagate. We will study how the wave equation emerges from the Laws of Electromagnetism (Maxwell's Equations) later this year.

3. Quantum (or Matter) waves. In Quantum Mechanics every particle exhibits wave motion, hence elementary particles like the electron and the proton can be shown to behave like wave on the appropriate distance (or energy) scales. In the non-relativistic regime, the dynamics of particles in Quantum Mechanics is governed by the Schr¨dinger equation. o Waves on strings are transverse waves: the displacements or oscillations in the medium are transverse (i.e. perpendicular) to the direction of wave propagation.

2

THE WAVE EQUATION

3

When the oscillations are parallel to the direction of wave propagation the waves are called longitudinal. Sound waves are longitudinal waves; a gas can sustain only longitudinal waves because transverse waves require a shear force to maintain them. Both transverse and longitudinal waves can travel in a solid. Electromagnetic waves are transverse (the oscillating electric field is always perpendicular to the direction of travel.)

2

The wave equation

There is no reason to restrict ourselves to sinusoidal waves (like the one in figure 1). In general terms, a wave can be defined as a disturbance in a field ψ(r, t) (a field is a physical property defined at every point of space and time), such that:

1. its phase propagates at a constant velocity v.

2. its functional form does not change in shape over time. Mathematically the two conditions above mean that a wave needs to be a function f that depends only on x ± vt, (in particular, the plus sign indicates a wave travelling in the -ve x direction, and a negative sign a wave moving towards to +ve x direction)
Ψ = f (x ± vt)

(1)

It is interesting to see what is the differential equation obeyed by this family of solutions1 . The resulting differential equation will inevitably involve derivatives in time and space, since the solution depends on both of them as independent variables, i.e. it will be a partial differential equation. It is universally known as the (non-dispersive) Wave equation. To get there, let us start by (restricting ourselves to one dimension) defining the phase of the wave φ = x ± vt and taking derivatives with respect to it df ∂φ
df
∂Ψ
=
=

∂x dφ ∂x dφ
∂Ψ
df ∂φ
df
=
= ±v

∂t dφ ∂t dφ

∂2Ψ
d2 f
=
2
∂x dφ2 2
∂ Ψ
d2 f
= v2 2
∂t2 dφ

(2) (3)

Hence comparing the second order partial derivatives we get the (1D non-dispersive) wave equation
∂2Ψ
1 ∂2Ψ
= 2 2 . 2
∂x v ∂t

(4)

Note that comparing the first order partial derivatives we get to the "travelling wave" equation, relating the first order derivatives
∂Ψ
1 ∂Ψ

.
∂x v ∂t

(5)

While the wave equation above is satisfied by both the forward and backward propagating waves, the travelling wave equation is only satisfied by waves propagating in 1 yeah,

we're going from the solution back to the differential equation, rather crazy it seems. Of course I could have started by telling you: this is the wave equation, learn it and show that it has solutions of this type.

2

THE WAVE EQUATION

4

either direction, depending on the sign in front of the RHS. Its use lies in the relation it proves between the wave velocity v and the partial derivatives of the field Ψ2 . v=±

2.1

∂Ψ/∂t
.
∂Ψ/∂x

(6)

Harmonic Solutions

It is very important to note that the wave equation is linear, because it contains only linear powers of Ψ. This means that the principle of superposition applies: if Ψ1 and
Ψ2 are solutions, a1 Ψ1 + a2 Ψ2 is also a solution. As a consequence, since Fourier Theory allows us to write any function as a sum of sines and cosines, it is generally convenient to consider harmonic solutions of the type Ψ = Re Aeik(vt±x) , where k is a constant (with the dimensions of L−1 called the wavenumber). It is conventional to write this solution in terms of the angular frequency of oscillation ω = kv, leading to the standard expression for a harmonic wave
Ψ = Re Aei(±kx−ωt)

(7)

Obviously the -ve sign in front of ω is conventional, but the -ve sign is the universally agreed standard in Quantum Mechanics, hence we will use it from now on. Let us take a moment to revise all the quantities we have now defined with relation to wave motion:

• The angular frequency ω is related to the frequency f and period T by 2π
(8) T The period is the time needed for the phase of the wave to advance by 2π.
ω = 2πf =

• The wavelength is defined as the difference in space between two points that have a phase difference of 2π. It is related to the wavenumber by
λ=


k

(9)

• The phase velocity (i.e. the velocity at which the phase of the wave advances) is related to the wavelength and frequency by
ω
k

(10)

v = fλ

(11)

v=
or equivalently

2 Although it seems we just proved that v = ± dx , this is an incorrect conclusion, since the chain dt rule can be used with partial derivatives only when both derivatives are taken keeping the same variable constant. In this section, however, all the partial derivatives with respect to time are at constant x and all the partial derivatives wrt x are a constant t. Hence the chain rule as such is generally inapplicable. However you might want to have a go at using the reciprocity relation to find a cleaner expression for the phase velocity.

3

WAVES ON A STRING

2.2

5

Wave velocities: which is which?

In the context of wave motion there are three important velocities to be considered, (the third one is only quoted here for completeness and is different from the second one only for dispersive waves).

1. The particle velocity ∂ψ , representing the speed of the field or material particle
∂t that is oscillating at each point in space.
∂Ψ/∂t

2. The phase velocity v = ω = ± ∂Ψ/∂x , which represents the velocity at which k the phase advances. Note the fact that even in 1D the general solution to the wave equation always contains a forward and a backwards propagating wave, leading the the appearance of the ± sign in the definition above.

3. The group velocity v = ∂ω , which represents the velocity at which energy is
∂k propagated by the wave. We will come back to this later on. In the following sections we will be making use of the analogy we have drawn between waves and a set of harmonic oscillators. One such example is in the definition of wave impedance Z driving force
∂ψ
∂ψ
→F =Z
→ F = Zv (12) particle velocity
∂t
∂x We can therefore make direct use of the equation relating power to impedance and (particle) velocity (see the Oscillation part of the course) as Z=

P =

3

1 dΨ 2
|
| Re(Z) 2 dt

(13)

Waves on a string

This section is dedicated to the study of the simplest example of waves I could think of: waves on a string. The good news is that this is very simple! The bad news is that we will not be studying explicitly any other types of waves in this section, because they are too complicated (however keep an eye for electromagnetic waves in the Electromagnetism part of the course!).

Figure 2: A simple sketch of the analysis of wave motion of a stretched string. To study the dynamics of the string, consider a small length of string between points x and x + dx and resolve the forces vertically and horizontally, defining T (x) as the tension, and µ(x) the mass per unit length of the string.

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