Standing Waves

Physics 202Professor Vogel (Professor Carkner’s notes, ed)Lecture 5

Equation of a Standing Wave

The string oscillates with timeThe amplitude varies with position

yr = [2ym sin kx] cos te.g. at places where sin kx = 0 the

amplitude is always 0 (a node)

Nodes and AntinodesConsider different values of x (where n is an

integer)For kx = n, sin kx = 0 and y = 0Node:

x=n (/2)Nodes occur every 1/2 wavelength

For kx=(n+½), sin kx = 1 and y=2ym

Antinode: x=(n+½) (/2)

Antinodes also occur every 1/2 wavelength, but at a spot 1/4 wavelength before and after the nodes

Resonance Frequency When do you get resonance?

The reflected wave must be in phase with the incoming wave at both ends

Since you are folding the wave on to itself If both ends are fixed you have to have

a node at both ends You need an integer number of half

wavelengths to fit on the string (length = L)

n½=L In order to produce standing waves

through resonance the wavelength must satisfy:

= 2L/n where n = 1,2,3,4,5 …

Resonance?

Under what conditions will you have resonance?Must satisfy = 2L/n

n is the number of loops on a string fractions of n don’t work

v = ()½ = fChanging, , , or f will change Can find new in terms of old and see if

it is an integer fraction or multiple

HarmonicsWe can express the resonance condition in

terms of the frequency (v=f or f=v/) f=(nv/2L)

For a string of a certain length that will have waves of a certain velocity, this is the frequency you need to use to get strong standing wavesRemember v depends only on and

The number n is called the harmonic numbern=1 is the first harmonic, n=2 is the second etc.

For cases that do not correspond to the harmonics the amplitude of the resultant wave is very low (destructive interference)

Generating Musical Frequencies

Many devices are designed to produce standing wavese.g., Musical instruments

Frequency corresponds to notee.g., Middle A = 440 Hz

Can produce different f by changing v

Tightening a string

Changing LUsing a fret

Superposition

When 2 waves overlap each other they add algebraically

yr = y1 +y2

Traveling waves only add up as they overlap and then continue onSuperposition does not effect the velocity or

the shape of the waves after overlapWaves can pass right through each other

with no lasting effect

Pulse Collision

InterferenceConsider 2 waves of equal wavelength, amplitude

and speed traveling down a stringThe waves may be offset by a phase constant

y1 = ym sin (kx - t) y2 = ym sin (kx - t +)

From the principle of superposition the resulting wave yr is the sum of y1 and y2

yr = ymr sin (kx - t +½)What is ymr (the resulting amplitude)?

Is it greater or less than ym?

Interference and Phase

The amplitude of the resultant wave (ymr) depends on the phase constant of the initial waves

ymr = 2 ym cos (½)The phase constant can be

expressed in degrees, radians or cyclesExample: 180 degrees = radians =

0.5 cycles

Resultant Equation

Combining Waves

Types of InterferenceConstructive Interference -- when the resultant

has a larger amplitude than the originalsFully constructive -- = 0 and ymr = 2ym

No offset or offset by a full wavelengthThe two peaks re-enforce each other

Destructive Interference -- when the resultant has a smaller amplitude than the originalsFully destructive -- = and ymr = 0Offset by 1/2 wavelengthPeak and trough cancel out

Standing WavesConsider 2 waves traveling on the same string in

opposite directionsThe two waves will interfere, but if the input waves

do not change, the resultant wave will be constantThe sum of the 2 waves is a standing wave, it does not

move in the x directionNodes -- places with no displacement of the string

(string does not move)Antinodes -- places where the amplitude is a

maximum (only place where string has max or min displacement)The positions of the nodes and antinodes do not change,

unlike a traveling wave

Standing Wave Amplitudes