Waves, the Wave Equation, and Phase Velocity

1. f(x) f(x-2) f(x-1) f(x-3) x 0 1 2 3 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

2. 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.

3. f(x) f(x-2) f(x-1) f(x-3) x 0 1 2 3 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. vwill be the velocity of the wave. f(x - v t) f(x + v t)

4. 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.

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

6. 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 Þ and Þ

7. The wave equation

8. 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.

9. The 1D wave equation for light waves where E is the light electric field We’ll use cosine- and sine-wave solutions   or where: The speed of light in vacuum, usually called “c”, is 3 x 1010 cm/s.

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

11. p Definitions: Amplitude and Absolute phase E(x,t) = A cos[(k x – w t ) – q ] A = Amplitude q = Absolute phase (or initial phase)

12. Definitions Spatial quantities: Temporal quantities:

13. 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 = l / t In terms of the k-vector, k = 2p / l, and the angular frequency, w = 2p / t, this is: v = w / k

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

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