04-01-2013, 10:38 AM
Superposition and Standing Waves
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Superposition 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
Waves that obey the superposition principle are linear waves
For mechanical waves, linear waves have amplitudes much smaller than their wavelengths
Superposition and Interference
Two traveling waves can pass through each other without being destroyed or altered
A consequence of the superposition principle
The combination of separate waves in the same region of space to produce a resultant wave is called interference
Superposition Example
Two pulses are traveling in opposite directions
The wave function of the pulse moving to the right is y1 and for the one moving to the left is y2
The pulses have the same speed but different shapes
The displacement of the elements is positive for both
Types of Interference
Constructive interference occurs when the displacements caused by the two pulses are in the same direction
The amplitude of the resultant pulse is greater than either individual pulse
Destructive interference occurs when the displacements caused by the two pulses are in opposite directions
The amplitude of the resultant pulse is less than either individual pulse
Destructive Interference Example
Two pulses traveling in opposite directions
Their displacements are inverted with respect to each other
When they overlap, their displacements partially cancel each other
Use the active figure to vary the pulses and observe the interference patterns
Sinusoidal Waves, Summary of Interference
Constructive interference occurs when f = np where n is an even integer (including 0)
Amplitude of the resultant is 2A
Destructive interference occurs when f = np where n is an odd integer
Amplitude is 0
General interference occurs when 0 < f < np
Amplitude is 0 < Aresultant < 2A
Standing Waves
Assume two waves with the same amplitude, frequency and wavelength, traveling in opposite directions in a medium
y1 = A sin (kx – wt) and y2 = A sin (kx + wt)
They interfere according to the superposition principle
Note on Amplitudes
There are three types of amplitudes used in describing waves
The amplitude of the individual waves, A
The amplitude of the simple harmonic motion of the elements in the medium,
2A sin kx
The amplitude of the standing wave, 2A
A given element in a standing wave vibrates within the constraints of the envelope function 2A sin kx, where x is the position of the element in the medium
Standing Waves, Particle Motion
Every element in the medium oscillates in simple harmonic motion with the same frequency, w
However, the amplitude of the simple harmonic motion depends on the location of the element within the medium
Notes on Quantization
The situation where only certain frequencies of oscillations are allowed is called quantization
It is a common occurrence when waves are subject to boundary conditions
It is a central feature of quantum physics
With no boundary conditions, there will be no quantization
Waves on a String, Harmonic Series
The fundamental frequency corresponds to n = 1
It is the lowest frequency, ƒ1
The frequencies of the remaining natural modes are integer multiples of the fundamental frequency
ƒn = nƒ1
Frequencies of normal modes that exhibit this relationship form a harmonic series
The normal modes are called harmonics
Resonance
A system is capable of oscillating in one or more normal modes
If a periodic force is applied to such a system, the amplitude of the resulting motion is greatest when the frequency of the applied force is equal to one of the natural frequencies of the system
[attachment=46404]
Superposition 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
Waves that obey the superposition principle are linear waves
For mechanical waves, linear waves have amplitudes much smaller than their wavelengths
Superposition and Interference
Two traveling waves can pass through each other without being destroyed or altered
A consequence of the superposition principle
The combination of separate waves in the same region of space to produce a resultant wave is called interference
Superposition Example
Two pulses are traveling in opposite directions
The wave function of the pulse moving to the right is y1 and for the one moving to the left is y2
The pulses have the same speed but different shapes
The displacement of the elements is positive for both
Types of Interference
Constructive interference occurs when the displacements caused by the two pulses are in the same direction
The amplitude of the resultant pulse is greater than either individual pulse
Destructive interference occurs when the displacements caused by the two pulses are in opposite directions
The amplitude of the resultant pulse is less than either individual pulse
Destructive Interference Example
Two pulses traveling in opposite directions
Their displacements are inverted with respect to each other
When they overlap, their displacements partially cancel each other
Use the active figure to vary the pulses and observe the interference patterns
Sinusoidal Waves, Summary of Interference
Constructive interference occurs when f = np where n is an even integer (including 0)
Amplitude of the resultant is 2A
Destructive interference occurs when f = np where n is an odd integer
Amplitude is 0
General interference occurs when 0 < f < np
Amplitude is 0 < Aresultant < 2A
Standing Waves
Assume two waves with the same amplitude, frequency and wavelength, traveling in opposite directions in a medium
y1 = A sin (kx – wt) and y2 = A sin (kx + wt)
They interfere according to the superposition principle
Note on Amplitudes
There are three types of amplitudes used in describing waves
The amplitude of the individual waves, A
The amplitude of the simple harmonic motion of the elements in the medium,
2A sin kx
The amplitude of the standing wave, 2A
A given element in a standing wave vibrates within the constraints of the envelope function 2A sin kx, where x is the position of the element in the medium
Standing Waves, Particle Motion
Every element in the medium oscillates in simple harmonic motion with the same frequency, w
However, the amplitude of the simple harmonic motion depends on the location of the element within the medium
Notes on Quantization
The situation where only certain frequencies of oscillations are allowed is called quantization
It is a common occurrence when waves are subject to boundary conditions
It is a central feature of quantum physics
With no boundary conditions, there will be no quantization
Waves on a String, Harmonic Series
The fundamental frequency corresponds to n = 1
It is the lowest frequency, ƒ1
The frequencies of the remaining natural modes are integer multiples of the fundamental frequency
ƒn = nƒ1
Frequencies of normal modes that exhibit this relationship form a harmonic series
The normal modes are called harmonics
Resonance
A system is capable of oscillating in one or more normal modes
If a periodic force is applied to such a system, the amplitude of the resulting motion is greatest when the frequency of the applied force is equal to one of the natural frequencies of the system