When two identical waves move toward each other on a string, they form a standing wave with the same wavelength and double the amplitude. Standing waves have nodes where the string does not move and antinodes where it moves the most. If the string's length is shortened, the fundamental frequency increases and wavelength decreases, while the wave speed remains constant according to the wave equation.

1. Lesson #4Lesson #4
Standing Waves:
Standing Waves on a String
Nelson Reference Pages:Nelson Reference Pages:
422 - 425 , 455 - 457422 - 425 , 455 - 457

2. Standing Waves
 Q. What happensWhat happens
when two continuouswhen two continuous
and identical wavesand identical waves
move towards eachmove towards each
other?other?
 A. The two wavesThe two waves
form a standing wave.form a standing wave.
This wave has theThis wave has the
samesame λλ, twice the, twice the
amplitude. Theamplitude. The
crests and troughscrests and troughs
alternate positionsalternate positions
and nodes appear notand nodes appear not
to move.to move.
D O M D O M
D O M
N o d e
A n t in o d e

3. Terms Defined and Demos
 Nodes are where destructive interference occursNodes are where destructive interference occurs
and antinodes are where constructiveand antinodes are where constructive
interference occurs.interference occurs.
 Consecutive nodes and antinodes are separatedConsecutive nodes and antinodes are separated
byby ½ λ..
 Loops extend from one node to the next. For aLoops extend from one node to the next. For a
standing wave that starts and ends with a node,standing wave that starts and ends with a node,
the number of loopsthe number of loops equals the number ofthe number of
antinodes.antinodes.
 Wave machine demonstration of standing wavesWave machine demonstration of standing waves
and video.and video.
 http://www.youtube.com/watch?v=no7ZPPqtZEg&feature=relatedhttp://www.youtube.com/watch?v=no7ZPPqtZEg&feature=related

4. Standing Wave on a String
 Consider a string that isConsider a string that is
fixed at each end (similarfixed at each end (similar
to a stringed musicalto a stringed musical
instrument). If the string isinstrument). If the string is
plucked it will most likelyplucked it will most likely
vibrate at itsvibrate at its lowest
natural frequency (called(called
fundamental frequency).fundamental frequency).
Here,Here, λ = __= __ L
 TheThe second lowest natural
frequency (1frequency (1stst
overtone) isovertone) is
shown on the right.shown on the right.
Here,Here, λ = __= __ L
L
C a l l e d f u n d a m e n t a l
f r e q u e n c y o r f i r s t
h a r m o n i c .
L
C a lle d f ir s t o v e r t o n e o r
s e c o n d h a r m o n i c .

5.  Similarly, the 3Similarly, the 3rdrd
lowest natural frequency could be drawnlowest natural frequency could be drawn
by _______ _______ _______. Here,by _______ _______ _______. Here, λ = __= __ L
 The standing wave on the string is created by wavesThe standing wave on the string is created by waves
being continuously reflected from the two ends of thebeing continuously reflected from the two ends of the
string.string.
 If we apply the fundamental wave equation to the threeIf we apply the fundamental wave equation to the three
lowest natural frequencies, both ___________ andlowest natural frequencies, both ___________ and
__________ change but ___________ remains constant.__________ change but ___________ remains constant.
L
C a l l e d s e c o n d o v e r t o n e
o r t h i r d h a r m o n i c .

6.  Q. What happens toWhat happens to
the following quantitiesthe following quantities
if the string length isif the string length is
made shorter (tensionmade shorter (tension
does not change) bydoes not change) by
using a clamp ?using a clamp ?
1.1. Fundament FrequencyFundament Frequency
(f(f11) ___________) ___________
2.2. λλ11 ________________________
3.3. Speed of the waveSpeed of the wave
________________________
The relationship we canThe relationship we can
use is: ___________use is: ___________
L 1
T h e s t r in g is n o w c l a m p e d
m a k in g i t s h o r t e r .
C
L
A
M
P
L 2

7. Practice Questions
Long video (not shown in class) using a simulation program
which goes over the same lesson topics.
http://www.youtube.com/watch?v=mh3o8gUu4AEhttp://www.youtube.com/watch?v=mh3o8gUu4AE
Nelson TB:
Page 425 #1 – 2 (you may wish to use the equations givenPage 425 #1 – 2 (you may wish to use the equations given
in the TB), 3in the TB), 3
Page 426 #1, 2, 4, 5Page 426 #1, 2, 4, 5
Page 457 #1, 2Page 457 #1, 2
Page 460 #4, 5a,bPage 460 #4, 5a,b
 HandoutHandout