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Standing Waves

Standing Waves. Physics 202 Professor Vogel (Professor Carkner’s notes, ed) Lecture 5. Equation of a Standing Wave. The string oscillates with time The amplitude varies with position y r = [2y m sin kx] cos w t e.g. at places where sin kx = 0 the amplitude is always 0 (a node).

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Standing Waves

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  1. Standing Waves Physics 202 Professor Vogel (Professor Carkner’s notes, ed) Lecture 5

  2. Equation of a Standing Wave • The string oscillates with time • The amplitude varies with position • yr = [2ym sin kx] cos wt • e.g. at places where sin kx = 0 the amplitude is always 0 (a node)

  3. Nodes and Antinodes • Consider different values of x (where n is an integer) • For kx = np, sin kx = 0 and y = 0 • Node: • x=n (l/2) • Nodes occur every 1/2 wavelength • For kx=(n+½)p, sin kx = 1 and y=2ym • Antinode: • x=(n+½) (l/2) • Antinodes also occur every 1/2 wavelength, but at a spot 1/4 wavelength before and after the nodes

  4. Resonance Frequency • When do you get resonance? • The reflected wave must be in phase with the incoming wave at both ends • Since you are folding the wave on to itself • If both ends are fixed you have to have a node at both ends • You need an integer number of half wavelengths to fit on the string (length = L) • n½l=L • In order to produce standing waves through resonance the wavelength must satisfy: l = 2L/n where n = 1,2,3,4,5 …

  5. Resonance? • Under what conditions will you have resonance? • Must satisfy l = 2L/n • n is the number of loops on a string • fractions of n don’t work • v = (t/m)½ = lf • Changing, m, t, or f will change l • Can find new l in terms of old l and see if it is an integer fraction or multiple

  6. Harmonics • We can express the resonance condition in terms of the frequency (v=fl or f=v/l) • f=(nv/2L) • For a string of a certain length that will have waves of a certain velocity, this is the frequency you need to use to get strong standing waves • Remember v depends only on t and m • The number n is called the harmonic number • n=1 is the first harmonic, n=2 is the second etc. • For cases that do not correspond to the harmonics the amplitude of the resultant wave is very low (destructive interference)

  7. Generating Musical Frequencies • Many devices are designed to produce standing waves • e.g., Musical instruments • Frequency corresponds to note • e.g., Middle A = 440 Hz • Can produce different f by • changing v • Tightening a string • Changing L • Using a fret

  8. Superposition • When 2 waves overlap each other they add algebraically • yr = y1 +y2 • Traveling waves only add up as they overlap and then continue on • Superposition does not effect the velocity or the shape of the waves after overlap • Waves can pass right through each other with no lasting effect

  9. Pulse Collision

  10. Interference • Consider 2 waves of equal wavelength, amplitude and speed traveling down a string • The waves may be offset by a phase constant f • y1 = ym sin (kx - wt) • y2 = ym sin (kx - wt +f) • From the principle of superposition the resulting wave yr is the sum of y1 and y2 • yr = ymr sin (kx - wt +½f) • What is ymr (the resulting amplitude)? • Is it greater or less than ym?

  11. Interference and Phase • The amplitude of the resultant wave (ymr) depends on the phase constant of the initial waves • ymr = 2 ym cos (½f) • The phase constant can be expressed in degrees, radians or cycles • Example: 180 degrees = p radians = 0.5 cycles

  12. Resultant Equation

  13. Combining Waves

  14. Types of Interference • Constructive Interference -- when the resultant has a larger amplitude than the originals • Fully constructive -- f = 0 and ymr = 2ym • No offset or offset by a full wavelength • The two peaks re-enforce each other • Destructive Interference -- when the resultant has a smaller amplitude than the originals • Fully destructive -- f = p and ymr = 0 • Offset by 1/2 wavelength • Peak and trough cancel out

  15. Standing Waves • Consider 2 waves traveling on the same string in opposite directions • The two waves will interfere, but if the input waves do not change, the resultant wave will be constant • The sum of the 2 waves is a standing wave, it does not move in the x direction • Nodes -- places with no displacement of the string (string does not move) • Antinodes -- places where the amplitude is a maximum (only place where string has max or min displacement) • The positions of the nodes and antinodes do not change, unlike a traveling wave

  16. Standing Wave Amplitudes

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