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 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.

Standing waves The nodes occur when sin(kx) = 0 Thus, kx =  … Nodes: node antinode

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: node antinode

node 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: (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 Wavelength : Frequency :

Standing waves in a string fixed at both ends. f 1 is called the fundamental frequency The higher frequencies f n are integer multiples of the fundamental frequency These normal modes are called harmonics. f 1 is the first harmonic, f 2 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 m and is tuned to concert G with f G = 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 (f A = 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 E 5 A 4 D 4 G 3

Music and Resonance: Primary and secondary oscillators String Instruments Wind Instruments Air column body Mouthpiece strings

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