Making Waves Wednesday October 16, 2002

One dimensional wave equation Example Disturbance on a string y yo T = tension (N) Small amplitude pulse x µ = kg/m Particles move vertically only – i.e. transverse to direction of wave This is a transverse wave – particles move perpendicular to 11/29/2018

One dimensional wave One can show that the disturbance (y = vertical co-ordinate of particle in the string ) obeys the equation, and 11/29/2018

Wave Equation The equation, is an example of a one-dimensional wave equation. It describes propagation in a non-dispersive media i.e. velocity of propagation is the same for all frequencies 11/29/2018

Wave Equation Solutions are of the form x x These solutions can be verified by direct substitution. 11/29/2018

Solution to wave equation Let Then 11/29/2018

Solution to the wave equation 11/29/2018

Solution to the wave equation Hence and similarly for g (x + vt) 11/29/2018

Interpretation Represents a disturbance traveling along the positive x-axis With velocity v – to see this let: be a disturbance at x2 and time t2 11/29/2018

Interpretation Now at t1 = t2 – Δt we can write, f (x2 – vt2) = f (x2 – v(t1+ Δt)) = f ( x2- vΔt – vt1) = f ( x1- vt1) i.e. the disturbance at x2 , t2 is exactly the same as it was at x1 = x2 - v Δt, a time earlier. 11/29/2018

Interpretation y yo x v x1 vΔt x2 11/29/2018

The Nature of f and g The functions f, g are completely general thus far They represent a huge variety of possibilities – some of which cannot be written down in analytical form i.e. there is no analytical description for many types of irregular disturbances 11/29/2018

Harmonic functions Almost any type of disturbance can, however, be described approximately by a suitable combination of sines and cosines (i.e. in a Fourier series expansion) Thus if we let f or g be a harmonic function (sin or cos), then we can provide an excellent description of almost any arbitrary disturbance with a suitable combination of sin and cos 11/29/2018

Harmonic Functions Let’s begin by discussing harmonic functions with a single frequency Harmonic – a given particle excecutes single harmonic motion (SHM) Now if y = magnitude of disturbance, and the wave propagates in the positive x -direction 11/29/2018

Harmonic Functions Here the angular frequency ω = 2πf ; f = frequency Фo = phase angle which is determined by initial condition (t=0) Note that we could also write in terms of sin, with a different value of Фo. 11/29/2018

Harmonic Waves y At a particular position xo: yo t T T = period – time required for one complete cycle of particle motion yo = amplitude of wave – maximum displacement 11/29/2018

Harmonic Waves y At a particular time to: yo x λ λ = wavelength – distance traveled in one period = vT = v/f Or, λf = v – true for all monochromatic waves 11/29/2018

Harmonic motion λf = v ω = vk ; v = ω/k Definitions (a) Wavenumber k = 2π/λ ω = vk ; v = ω/k 11/29/2018

Harmonic motion λf = v Definitions (a) Phase of wave The phase of a harmonic wave is simply the argument of the sinusoidal function where Фo = initial phase angle 11/29/2018

Harmonic Waves Order of x and t is unimportant in description of waves kx ± ωt can be written ωt ± kx Both describe the same wave – just different phase But sign is important kx – ωt and ωt – kx both describe +ive x dir’n kx + ωt and ωt + kx both describe -ive x dir’n 11/29/2018