• 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