PHYS 012A Lecture Notes - Lecture 4: Standing Wave, Wave Function, Square Wave
12 Jun 2018
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Chapter 18: Superposition and Standing Waves 9/13/17
Waves vs. Particles
Waves are very different from particles
- Particles have zero size. Waves have a characteristic size – their wavelength
- Multiple particles must exist at different locations. Multiple waves can combine at one point
in the same medium – they can be present at the same location.
Quantization
- When waves are combined in systems with boundary conditions, only certain allowed
frequencies can exist.
o We say the frequencies are quantized.
o Quantized is at the heat or quantum mechanics.
- Quantization can be used to understand the behavior of the wide array of musical instruments
that are based on string and air columns.
Superposition Principle
- Waves can be combined in the same location in space.
- Use the superposition principles (linear waves obey this principle).
- If two or more traveling waves are moving through a medium, the resultant value of the wave
function at any point is the algebraic sum of the values of the wave functions of the individual
waves.
Superposition and Interference
- Two traveling waves can pass through each other without being destroyed or altered.
- The combination of separate waves in the same region of space to produce a resultant wave
is called interference – waves pass through each other.
- Two pulses traveling towards each other – the waves overlap – the wave function is the sum
of the individual wave functions.
- When crest meets crest the resultant wave has a larger amplitude than the original waves.
o They continue moving in their original directions and remain unchanged after
separation – constructive interference (a type of superposition).
o Constructive interference occurs when the displacements caused by the two pulses
are in the same direction.
- When two pulses traveling in opposite directions meet, and their displacements are inverted
with respect to each other. This is destructive interference.
o Destructive interference occurs when the displacements caused by the two pulses are
in opposite directions. The amplitude of the resultant pulse is less than both
individual pulses.
- Superposition of Sinusoidal Waves
o The waves differ only in phase:
▪ y1 = A sin (kx - wt)
▪ y2 = A sin (kx - wt + f)
▪ y = y1+y2 = 2A cos (f /2) sin (kx - wt + f /2)
o The resultant wave has the same frequency and wavelength as the original waves.
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find more resources at oneclass.com
o The amplitude of the resultant wave is 2A cos (f / 2) .
o The phase of the resultant wave is f / 2.
Sinusoidal Waves with Constructive Interference (phi = npi and n is even)
- Amplitude of the resultant wave is 2A. The aplitudes add if they ae’t eual.
Sinusoidal Waves with Destructive (phi = npi and n is odd)
- Amplitude of the resultant wave is 0.
Sinusoidal Waves, General Interference
- Interference is neither constructive nor destructive. Wave functions still add and the
amplitude would be between 0 and 2A.
Interference in Sound Waves
- The distance along any path from speaker to receiver is called the path length, r.
- A phase difference may arise between two waves generated by the same source when they
travel along paths of unequal lengths.
Standing Waves
- They interfere according the superposition principle.
- Wave function of the resultant standing wave will be y = (2A sin kx) cos wt.
o There is no kx – wt term, and therefore it is not a traveling wave.
- At the node -> destructively added – where a point of zero amplitude occurs
- At the anti-node-> constructively added – where there is a point of maximum
displacement,2A.
- Half a wavelength between nodes, half a wavelength between antinodes, and a quarter
wavelength between a node and antinode.
- Peak and valley is destructive interference, but peak and peak is constructive interference.
Notes on Amplitudes
- The amplitude of the individual waves, A
- The amplitude of the simple harmonic motion of the elements in the medium, 2Asinkx
o A given element in the standing wave vibrates within the constraints
- The amplitude of the standing wave, 2A.
Standing Waves in a String
- Set up by a continuous superposition of waves incident on and reflected from the ends.
- Ends of the strings must be nodes – must be fixed with zero displacement.
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
Waves have a characteristic size their wavelength. Multiple particles must exist at different locations. Multiple waves can combine at one point in the same medium they can be present at the same location. When waves are combined in systems with boundary conditions, only certain allowed frequencies can exist: we say the frequencies are quantized, quantized is at the heat or quantum mechanics. Quantization can be used to understand the behavior of the wide array of musical instruments that are based on string and air columns. Waves can be combined in the same location in space. Use the superposition principles (linear waves obey this principle). If two or more traveling waves are moving through a medium, the resultant value of the wave function at any point is the algebraic sum of the values of the wave functions of the individual waves. Two traveling waves can pass through each other without being destroyed or altered.
