Waves 1 1
Waves 1 1. N coupled oscillators towards the continuous limit. Stretched string and the wave equation 3. The d Alembert solution 4. Sinusoidal waves, wave characteristics and notation
T 1 T N coupled oscillators Consider fleible elastic string to which are attached N particles of mass m, each a distance l apart. The string is fied at each end. Small transverse displacements are applied transverse oscillations p1 p 0 1 3 p 1 p 1 l p N N 1 3
N coupled oscillators: special cases Let s first consider the special cases N=1 and N= 4
N coupled oscillators: general case Now let s tr and find solution for a general value N 5
N coupled oscillators: the solution T 1 T p1 p 0 1 3 p 1 p 1 l p N N 1 Displacement for mass p when oscillating in mode n and angular frequenc: pn pn ( Cn sin cos( nt n) N 1 n n sin 0 ( N 1) 0 T / ml Although the value of n can go beond N, this just generates duplicate solutions, i.e. there are N normal modes in total. 6
N coupled oscillators: modes for N=5 Look at each mode for N=5, with snapshot taken at t=0 n=1 n= n=5 n=3 n=4 Note how the displacement of ever particle falls on a sine curve! 7
N coupled oscillators: N ver large Let s eplore the scenario where N is ver large, which starts to approimate case of a real, continuous, string 8
Sstem of springs and N masses: longitudinal oscillations k m m k k m k m k u p Let u p be displacement from equilibrium position of mass p 9
Stretched string Consider a segment of string of linear densit ρ stretched under tension T T T, small 10
Stretched string and wave equation Will show that the displacements on a stretched string obe t T which is the wave equation 1 t c T c with 11
Jean-Baptiste le Rond d Alembert 1717-1783 Lived in Paris Mathematician and phsicist Also a music theorist and co-editor with Diderot of a famous encclopaedia 1
d Alembert solution of wave equation We will show how the wave equation can be solved to ield solutions of form: (, f ( c g( c Here f and g are an functions of (-c & (+c, determined b initial conditions. We will then interpret this solution. 13
Interpretation of D Alembert solution (, f ( c Focus on =0 and consider situations at t=0 and t=δt (0,0) ( 0, c t Wave moves to right with speed c 14
Interpretation of D Alembert solution (, g( c Focus on =0 and consider situations at t=0 and t=δt (0,0) ( 0, c t Wave moves to left with speed c 15
d Alembert s solution with boundar conditions Eample: rectangular wave of length a released from rest 1 (, U ( c U( c t 0 t a / c U() a a a a t a / c t 3a / c a a a a 16
Sinusoidal waves A ver common functional dependence for f and g... (, f ( c g( c...is sinusoidal. In this case it is usual to write: (, Acos( k Bcos( k with k and ω (and A and B) constants or Asin(k-ω... etc (choice doesn t matter, unless we are comparing one wave with another and then relative phases become importan speed of wave c / k 1/ T frequenc where ω is angular frequenc f / k / wavelength where k is the wave-number (or wave-vector if also used to indicate direction of wave) 17
Notation choices Sinusoidal solution (, Acos( k (writing here, for compactness, onl the forward-going solution) Using the relationships between k,ω, λ & c this can be epressed in man forms (, Acos[ k( c] (, Acos( t k) Also note that sometimes it is convenient to write Changes nothing (for cosine, triviall so, & practicall not even for sine function, as overall sign can be absorbed in constan & still describes forward-going wave. A ver frequent approach is to use comple notation (we alread made use of this when analsing normal modes, and ou will have seen it in circuit analsis) (, Re Aep[ i( k] or (, Im Aep[ i( k] if its important to pick out sine function. Note that often the Re or Im is implicit, and it gets omitted in discussion. 18
Phase differences Often important to specif phase shifts. Onl meaningful to do so when we are comparing one wave to another. 1(, Acos( k (, Acos( kt ) wave wave 1 In this eample wave leads wave 1 b π/, i.e. φ=-π/ Can be epressed with comple notation (, Re Aep[ i( kt )] Nicer still to subsume phase into amplitude (, Re Aep[ i( k] with π/ A A ep( i) k=0 ωt 19