Standing Waves

Physics 202Professor Lee

CarknerLecture 7

PAL #6 Wave Energy How do you find linear density?

v = f = (/)½ or = /f22

Get frequency from function generator f =

Get wavelength by measuring on string =

Get tension from hanging weights hanging mass is 225g so = mg = (.225)(9.8) =

2.205 N =

Velocity = f v = (22.54)(1.5) =

Exam #1 Friday About 1/3 multiple choice

Study notes Study Quizdom questions Look at textbook “Checkpoint”

questions About 2/3 problems

Study PAL’s and SuperPALS Study old homework Do new practice homework questions

Try to do this with just equation sheet

Need (real) calculator and pencil

Standing Waves

The two waves will interfere, but if the input waves do not change, the resultant wave will be constant

Nodes --

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

Equation of a Standing Wave

If the two waves have equations of the form:

Then the sum is:

The amplitude varies with position e.g. at places where sin kx = 0 the amplitude

is always 0 (a node)

Nodes and Antinodes

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

x=n (/2)

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 Condition

Standing waves occur due to resonance

When do you get resonance?

You must have: An integer number of “loops”

Since each “loop” is half a wavelength

= 2L/n where n = 1,2,3,4,5 …

Resonance?

Under what conditions will you have resonance? Solve for n, must be integer

v = ()½ = f Can find new in terms of old and see if

it is an integer fraction or multiple

Harmonics

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 waves

The number n is called the harmonic number

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

Generating Harmonics Many devices are designed to produce

standing waves e.g.,

Frequency corresponds to note e.g.,

Can produce different f by changing v

Changing L

Next Time

Test #1 For Monday, December 5

Read 17.1-17.4 and do webassign homework

What kind of string propagates waves the fastest?

a) Heavy and tightb) Heavy and loosec) Light and loosed) Light and tighte) We can’t know wave speed without

knowing the input frequency

How would you modify the wave generator to input the maximum amount of energy?

a) Increase frequency, increase amplitude

b) Increase frequency, decrease amplitudec) Decrease frequency, increase amplituded) Decrease frequency, decrease amplitude e) Input energy is independent of frequency

and amplitude

What kind of string transmits energy the fastest?

a) Heavy and tightb) Heavy and loosec) Light and loosed) Light and tighte) All strings transmit energy at the same

rate

Consider a wave traveling along a string that can be combined with three otherwise identical waves with phase shifts of 0.5, 1.0, and 1.9 radians. Rank the resulting wave by amplitude, largest first.

a) 0.5, 1.0, and 1.9 b) 1.9, 1.0, 0.5c) 1.0, 0.5, 1.9d) 1.9, 0.5, 1.0e) 0.5, 1.9, 1.0