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 y r = [2y m sin kx] cos  t 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 = n , sin kx = 0 and y = 0 Node: x=n ( /2) Nodes occur every 1/2 wavelength For kx=(n+½) , sin kx = 1 and y=2y m Antinode: x=(n+½) ( /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) n½ =L In order to produce standing waves through resonance the wavelength must satisfy: = 2L/n where n = 1,2,3,4,5 …

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

6. Harmonics We can express the resonance condition in terms of the frequency (v=f or f=v/ ) 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  and  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 y r = y 1 +y 2 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  y 1 = y m sin (kx -  t) y 2 = y m sin (kx -  t +  ) From the principle of superposition the resulting wave y r is the sum of y 1 and y 2 y r = y mr sin (kx -  t +½  ) What is y mr (the resulting amplitude)? Is it greater or less than y m ?

11. Interference and Phase The amplitude of the resultant wave (y mr ) depends on the phase constant of the initial waves y mr = 2 y m cos (½  ) The phase constant can be expressed in degrees, radians or cycles Example: 180 degrees =  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 --  = 0 and y mr = 2y m 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 --  =  and y mr = 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