Combining Waves interference § 14.7 1

Principle of Superposition Where waves meet, the displacement is the sum of the displacements from the individual waves. 3 Run at: W1 = 0.2; k1 = 0.2; ampl = 15 integral multiples (half, third, quarter) of lambda: multiply W1, k1 by 2, 3, 4 add same-lambda wave with negative amplitude Beats: slightly vary w2 and k2 together from wave 1 values Standing waves (use phase velocity of 0.5: k = 2*w) result –3

Interference Constructive: Sum of waves has increased amplitude Destructive: Sum of waves has decreased amplitude Two-wave simulation

Interference Patterns Interference of similar wavelengths 4

Patterns Positions of constructive and destructive interference destructive: nodes constructive: antinodes Ripple tank simulator http://www.falstad.com/ripple/

waves that don’t actually travel Standing Waves waves that don’t actually travel § 14.8 6

Standing Waves Sum of waves of equal amplitude and wavelength traveling in opposite directions Half-wavelength divides exactly into the available space Wave pattern has locations of minimum and maximum variation (nodes and antinodes) (standing longitudinal waves) Run at: w1 = 0.2; k1 = 0.2; ampl = 15 integral multiples (half, third, quarter) of lambda: multiply w1, k1 by 2, 3, 4 add same-lambda wave with negative amplitude Beats: slightly vary w2 and k2 together from wave 1 values (0.22 and 0.22; 0.21 and 0.21, etc.) Standing waves (use w of about 0.2; try w = 0.2, k = 0.1) 7

standing waves generalized Normal modes standing waves generalized 8

Modes Objects have characteristic frequencies at which standing waves are sustained Lowest frequency = fundamental Higher frequencies = overtones Sustained motion is a combination of normal modes 9

Vibrational Modes: Clamped String Insert Figure 15.3 from class text Source: Griffith, The Physics of Everyday Phenomena, Figure 15.13 10

Combinations of Harmonics Characteristic sounds arise from combining particular harmonics in specific ratios Fourier analysis suimulation flute oboe saxophone Simulation

“Closed” and “Open” Tube Modes Source: Halliday, Resnick, and Walker, Fundamentals of Physics, 2003, p 419.

Sequence of Harmonics Western musical scale and harmonies are based on overtone series (sound files) Sound files: overtones of open tube or clamped string

Circular membrane standing waves 2-D Standing Waves Nodes are lines or curves Circular membrane standing waves edge node only diameter node circular node Source: Dan Russel’s page Higher frequency  more nodes

Aside Electron orbitals in atoms and molecules are 3-D standing waves All particles have wave natures Orbitals are interference patterns that persist (don’t cancel over time) Stationary states are like harmonics

Resonance Boundary conditions determine nodal positions For uniform media, resonant wavelengths and frequencies have simple relationships Clamped strings http://www.surendranath.org/Applets/Waves/Harmonics/HarmonicsApplet.html Air cylinders http://www.physics.smu.edu/~olness/www/05fall1320/applet/pipe-waves.html More complex media are more interesting http://paws.kettering.edu/~drussell/Demos.html 16

coincidence of similar frequencies Beats coincidence of similar frequencies § 14.9 17

Beats Waves of similar frequency combine to give alternating times of constructive and destructive interference Distinctive “waa-waa” sound with beat frequency equal to the difference in frequency of the component waves fbeat = |f1 – f2| (Why?)

Beats Sound files Ripple tank simulator http://www.falstad.com/ripple/