Lecture 26: Standing Waves

Interference

when two or more waves pass through the same re-

gion at the same time

Superposition:

when two or more waves overlap, the total dis-

placement is the sum of each wave function

basic idea behind standing wave on a string:

boundary conditions at the edges of the string limit the number of possible

waveforms

function that rep-

resents the standing wave pattern:

yn = Bn sin (npi/L x ) cos (wt)

to get function that rep-

resents the standing wave pattern need :

superposition principle, boundary conditons

n definition

normal mode

when do normal modes occur?

all particles move sinusoidally with same frequency (fn)

what is a harmonic?

fn = n f1 the frequency all particles are travlling at sinusoidally for a normal mode

funcamental frequency

f1 = vn/2L = 1/2L sqrt(T/u) u=linear density

wavelength of each normal mode formula:

wavelength n = 2L /n = 2piv/w

explanation for wavelength formula

string clamped, sin function crosses points, can only fit intervals of half wavelengths

difference from normal modes to ideal spring

ideal spring = only one normal mode and fundamental frequency

nodes defintion

points on standing wave that never move

antinodes defintion

points with maximum amplitude, y = extrenum values

when L decrease: f1 ___, T____

increase, inc

Travelling Wave on String definiton

some waveform moving through space as a function

of time

Travellling wave solution to equation of motion:

y=A cos (kx +- wt + o)

General form of solution for Standing Wave

y = A(x) cos (wt + o)

equation for SHM of some function A(x)

A²A(x)/Ax² = -w²/v² A(x),

General solution for A(x)

A(x) = A cos (w/v x) + B sin (w/v x)

Boundary conditions (initial conditions) : x=0 and x=L, ampltide at these points:

A(0) = 0 = A, A(L) = 0 = A cos (w/v L) + B sin (w/v L) , no motion at these points

equation relating our arguments of 0 = B sin (ω

v L) and sin (nπ) = 0

w/vL = npi

condition for allowed angular frequencies

w = vnpi/L

what does cos (wt) in the function do

changes relative position of each of the harmonics

to sketch each harmonic plot_____

An(x) = Bn sin (n pi / L x)

Period (T) formula

Tn = 2pi/wn = 2L/vn

wavenumber formula

k = 2pi/wavelength, npi/L