9.1.1 The Wave Equation What is a "wave?"

Chapter 9. Electromagnetic Waves

9.1 Waves in One Dimension

9.1.1 The Wave Equation What is a "wave?"

Let's start with the simple case: fixed shape, constant speed:

How would you represent such a string object mathematically?

f (z, t) represents the displacement of the string at the point z, at time t.

Given the initial shape of the string,

The displacement at point z, at the later time t, is the same as

the displacement a distance vt to the left (i.e. at z - vt), back at time t = 0:

 It represents a wave of fixed shape traveling in the +z direction at speed v.

(Mechanical or Classical) Waves Equation

 (Mechanical or Classical) Waves Equation = Equation of Motion governed by Newton’s second law!

Consider a stretched string under tension T.

The net transverse force on the segment between z and (z + z) is

If the distortion of the string is not too great, sin ~ tan.

If the mass per unit length is , Newton's second law says

Why does a stretched string support wave motion?

 Actually, it follows from Newton's second law.

where v (which, as we'll soon see, represents the speed of propagation) is

 (classical) Wave Equation

because it admits as solutions all functions of the form

Waves Equation

: All functions that depend on the variables z and t in the special combination u = z - vt, represent waves propagating in the z direction with speed v.

Therefore, the Wave Equation admits as solutions all functions of the form

For example, if A and b are constants (with the appropriate units),

All represent waves

All do not represent waves

(Mechanical or Classical) Waves Equation

Problem 9.1 By explicit differentiation, check that the functions f₁, f₂, and f₃ satisfy the wave equation.

Show that f₄ and f₅ do not.

(Mechanical or Classical) Waves Equation

Problem 9.2 Show that the standing wave f(z, t) = A sin(kz) cos(kvt) satisfies the wave equation,

and express it as the sum of a wave traveling to the left and a wave traveling to the right.

By using the trig identity

the sum of a wave traveling to the left and a wave traveling to the right.

9.1.2 Sinusoidal Waves

(i) Terminology:

(At time t = 0)

A is the amplitude of the wave

it is positive, and represents the maximum displacement from equilibrium.

The argument of the cosine,  = k(z-vt) +  is called the phase

 is the phase constant

Obviously, you can add any integer multiple of 2to  without changing f (z, t)

Ordinarily, one uses a value in the range 0 ≤  ≤ 2p

k is the wave number

it is related to the wavelength  by the equation 

One full cycle in a time period   = kvT = 

Frequency v (number of oscillations per unit time) 

Angular frequency , the number of radians swept out per unit time 

A sinusoidal waves in terms of k and 

Sinusoidal Waves

(ii) Complex notation

In view of Euler's formula,

Complex wave function:

 Complex amplitude

 The actual wave function is the real part:

 The advantage of the complex notation is that exponentials are much easier to manipulate than sines and cosines.

Example 9.1

In particular, if they have the same frequency and wave number, you just add the (complex) amplitudes.

Then take the real part 

The combined wave still has the same frequency and wavelength.

Without using the complex notation, you will find yourself looking up trig identities and slogging through nasty algebra.

Sinusoidal Waves

(iii) Linear combinations of sinusoidal waves

Any wave can be expressed as a linear combination of sinusoidal ones:

k includes negative values

This does not mean that  and  are negative: wavelength and frequency are always positive.

k to run through negative values in order to represent waves going in both directions.

(Fourier transformation)

Note the point that any wave can be written as a linear combination of sinusoidal waves,

 therefore if you know how sinusoidal waves behave, you know in principle how any wave behaves.

So from now on, we shall confine our attention to sinusoidal waves.

9.1.3 Boundary conditions: Reflection and Transmission

What happens depends a lot on how the string is attached at the end.

 that is, how the wave propagation depends on the specific boundary conditions.

Suppose, for instance, that the string is simply tied onto a second string with different mass at z =0.

The incident wave 

The reflected wave 

The reflected wave 

All parts of the system are oscillating at the same frequency  (a frequency determined by the person who is shaking the string)

( Assume the wave velocities v₁and v₂are different)

Since the wave velocities are different in the two strings,

 the wavelengths and wave numbers are also different:

Boundary conditions: Reflection and Transmission

For a sinusoidal incident wave, then, the net disturbance of the string is:

At the join (z = 0),

the displacement and slope just slightly to the left (z = 0^-) must equal those slightly to the right (z = 0⁺), or else there would be a break between the two strings.

Boundary conditions: Reflection and Transmission

These boundary conditions apply directly to the real wave function f(z, t).

But since the imaginary part differs from the real part only in the replacement of cosine by sine,

it follows that the complex wave function obeys the same rules:

The real amplitudes and phases, then, are related by

If the second string is lighter than the first, all three waves have the same phase angle

If the second string is heavier than the first, the reflected 'Nave is out of phase by 180^o

In particular, if the second string is infinitely massive,

Boundary conditions:

Problem 9.5

It gives rise to a reflected wave, h_R(z + v₁t), and a transmitted wave, g_T(z - v₂t).

By imposing the boundary conditions, find h_Rand g_T.

9.1.4 Polarization

Transverse Waves: the displacement is perpendicular to the direction of propagation Longitudinal Waves: the displacement is along the direction of propagation

Vertical polarization:

Horizontal polarization:

Any other Polarization:

Polarization Vector:

Any transverse wave can be considered a superposition of two waves: one horizontally polarized, the other vertically:

Transverse waves occur in two independent states of polarization