• Waves are closely related to oscillations
• We’ll mainly deal with sinusoidal waves.
- Water waves: Water molecules oscillate in a circle
- Sound waves: Air molecules oscillate back and forth
- Stadium waves: People move up and down
- Electromagnetic wave: (in Physics 114)
Chapter 18: Superposition and Standing Waves
Reading assignment: review for test
Homework : (due Monday, Nov. 28, 2005):
Problems: Q3, Q12, 7, 8, 13, 31, 34, 35, 47
Standing waves
)sin(1 tkxAy )sin(2 tkxAy
tkxAyR cos)sin(2 The resultant wave is a standing wave:
Now we are considering two sinusoidal waves (same A, k and ) that travel in the same medium, but in the opposite direction.
A standing wave is a an oscillating pattern with a stationary outline. It has nodes and antinodes.
tkxAyR cos)sin(2 A standing wave is a an oscillating pattern with a stationary outline. It has nodes and antinodes.
Standing waves
The nodes occur when sin(kx) = 0
Thus, kx = …
Nodes:
... 3, 2, 0,1, n 2
,.....2
3,,
2,0
nx
nodeantinode antinode
tkxAyR cos)sin(2 A standing wave is a an oscillating pattern with a stationary outline. It has nodes and antinodes.
Standing waves
The antinodes occur when sin(kx) = 1
Thus, kx = …
Antinodes:
... 5, 3, 1, n 4
.....4
5,
4
3,
4
nx
nodeantinode antinode
nodeantinode antinode
Standing waves
The distance between nodes is /2.
The distance between antinodes is /2
The distance between nodes and antinodes is /4
Two waves traveling in opposite directions produce a standing wave. The individual wave functions are:
)0.20.3
sin()4(1 ts
xcm
cmy )0.20.3
sin()4(2 ts
xcm
cmy
(a) What is the amplitude of a particle located at x = 2.3 cm.(b) Find the position of the nodes and antinodes.(c) What is the amplitude of a particle located at an antinode?
Black board example 17.4
Standing waves in a string fixed at both ends.
Normal modes of a string
... 3, 2, 1,n 2
n
Ln
... 3, 2, 1,n 2
L
vn
vf
nn
Wavelength:
Frequency:
1
... 3, 2, 1,n T
2
fnf
L
nf
n
n
Tv :Using
frequency lfundamenta thecalled is T
2
11 Lf
Standing waves in a string fixed at both ends.
f1 is called the fundamental frequency
The higher frequencies fn are integer
multiples of the fundamental frequency
These normal modes are called harmonics.
f1 is the first harmonic, f2 is the second
harmonic and so on…
String instruments:
When playing string instruments, standing waves (harmonics) are excited in the strings by plucking (guitar), bowing cello) or striking (piano) them.
A violin string has a length of 0.350 m and is tuned to concert G with fG = 392 Hz.
(a) Calculate the speed of the wave on the string.
(b) Where should the violinist press her finger down to play an A (fA = 440 Hz).
(c) Why are some violins so expensive (Stradivarius : $ 1.5 M)?
Black board example 17.5
Harmonics in a String
• In a string, the overtone pitches are– two times the fundamental frequency (octave)– three times the fundamental frequency– etc.
• These integer multiples are called harmonics
• Bowing or plucking a string tends to excite a mixture of fundamental and harmonic vibrations, giving character to the sound
notes
E5
A4
D4
G3
Music and Resonance:Primary and secondary oscillators
String Instruments Wind Instruments
Air column
body Mouthpiecestrings
Connecting primary (strings) and secondary (body) oscillators
Producing Sound• Thin objects don’t project sound well
– Air flows around objects– Compression and rarefaction is minimal
• Surfaces project sound much better– Air can’t flow around surfaces easily– Compression and rarefaction is substantial
• Many instruments use surfaces for sound
Violin Harmonics
Viola Harmonics
Computer Tomography scan of a Nicolo Amati Violin (1654)
Notes and their fundamental frequency