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 Harmonic waves Wavelength, frequency, period, etc. Phase velocity Complex numbers Plane waves and laser beams f(x)f(x) f(x-3) f(x-2) f(x-1) x

Introduction Wave propagation in a medium is described mathematically in terms of a wave equation; this is a differential equation relating the dynamics and static of small displacements of the medium, and whose solution may be a propagating disturbance. Once an equation has been set up, to call it a wave equation we require it to have propagation solutions. This means that if we supply initial conditions in the form of a disturbance which is centered around some given position at time zero, then we shall find a disturbance of similar type centered around a new position at a later time.

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) f(x-3) f(x-2) f(x-1) x f(x - v t) f(x + v t)

The non-dispersive wave equation in one dimension The most important elementary wave equation in one dimension can be derived from the requirement that any solution: Propagates in either direction (± x) at a constant velocity v, Does not change with time, when referred to a center which moving at this velocity.

The solution to the one-dimensional wave equation where f (u) can be any twice-differentiable function. The wave equation has the simple solution: Is called the phase of the wave.

Proof that f (x ± vt) solves the wave equation Write f (x ± vt) as f (u), where u = x ± vt. So and Now, use the chain rule: So  and 

The wave equation

The one-dimensional wave equation We’ll derive the wave equation from Maxwell’s equations. Here it is in its one-dimensional form for scalar (i.e., non-vector) functions, f : Light waves (actually the electric fields of light waves) will be a solution to this equation. And v will be the velocity of light.

The 1D wave equation for light waves We’ll use cosine- and sine-wave solutions or where: where E is the light electric field The speed of light in vacuum, usually called “c”, is 3 x cm/s.

A simpler equation for a harmonic wave: E(x,t) = A cos[(kx –  t) –  ] Use the trigonometric identity: cos(z–y) = cos(z) cos(y) + sin(z) sin(y) where z = k  x –   t and y =  to obtain: E(x,t) = A cos(kx –  t) cos(  ) + A sin(kx –  t) sin(  ) which is the same result as before, as long as: A cos(  ) = B and A sin(  ) = C For simplicity, we’ll just use the forward- propagating wave.

Definitions: Amplitude and Absolute phase E(x,t) = A cos[(k x –  t ) –  ] A = Amplitude  = Absolute phase (or initial phase) 

Definitions Spatial quantities: Temporal quantities:

The Phase Velocity How fast is the wave traveling? Velocity is a reference distance divided by a reference time. The phase velocity is the wavelength / period: v = /  In terms of the k-vector, k = 2  /, and the angular frequency,  = 2  / , this is: v =  / k

Human wave A typical human wave has a phase velocity of about 20 seats per second.

The Phase of a Wave This formula is useful when the wave is really complicated. The phase is everything inside the cosine. E(t) = A cos(  ), where  = k x –  t –   =  (x,y,z,t) and is not a constant, like  ! In terms of the phase,  = –  /  t k =  /  x And –  /  t v = –––––––  /  x