Standing Waves
What is a Standing Wave?
- A vibrational pattern created within a medium when the vibrational frequency of the source causes reflected waves from one end of the medium to interfere with the incident waves from the source
- Waves vibrate up and down but stays in the
same place
- Created by waves with the same frequency, lambda and amplitude travelling in the opposite directions and interfering
- Formed as the result of the perfectly timed
interference of two waves passing through the same medium
What is Resonance?
- A physical property of an object
- All physical objects resonate
- Some have plain, uniform resonance patterns
- Others have complicated resonance patterns
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Standing Waves
The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms resonant standing waves.
If a string is tied between two fixed supports, pulled tightly and sharply plucked at one end, a pulse will travel from one end of the string to the other.
Standing waves on a string are a result of travelling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference and the antinodes (places of max amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. A wave has both a frequency and a wavelength that are related by the equation.
If a string is tied between two fixed supports, pulled tightly and sharply plucked at one end, a pulse will travel from one end of the string to the other.
Standing waves on a string are a result of travelling waves interfering both destructively and constructively. The nodes (places of zero amplitude) are due to destructive interference and the antinodes (places of max amplitude) are due to constructive interference. When a standing wave appears, the nodes and antinodes are fixed in place. A wave has both a frequency and a wavelength that are related by the equation.
Equations
Below are the two equations used, the first equation is to find the length of the wave where as the second equation is used to figure our the velocity. Notes that both equations can be rearranged to solves for another variable.
L → Length
λ → Lambda *remember 1 lambda is made up of two loops!!*
n → number of harmonic (frequency)
v → velocity
f → frequency (measured in Hertz → Hz)
L → Length
λ → Lambda *remember 1 lambda is made up of two loops!!*
n → number of harmonic (frequency)
v → velocity
f → frequency (measured in Hertz → Hz)
Standing Waves and Harmonics (frequency)
Looking at the diagram to the left, we can identify key properties:
- Wave is NOT a full wave; in order to have a full wave, there should be two antinodes (loops).
- Thus, this is only a half lambda
- This diagram only has two nodes (the point where the wave is at rest)
Looking at the diagram to the right, we can identify key properties:
- Wave is a full wave; as mentioned in the above diagram, this wave has TWO LOOPS, two loops makes 1 lambda.
- Instead of two nodes, this wave has three nodes and two antinodes
Looking at the diagram to the left, we can identify key properties:
To calculate the wavelength, the formula is slightly different compared to the formula for a full wavelength.
The formula to calculate wavelengths differ from each other as it depends on the number of frequency.
- This wave has 3 antinodes which mean it is a little more than a full wave; this wave has a full wave and a half of a wave.
- Instead of three nodes, this wave has has four nodes
To calculate the wavelength, the formula is slightly different compared to the formula for a full wavelength.
The formula to calculate wavelengths differ from each other as it depends on the number of frequency.
Relationship of Frequency and Wavelength
For example, higher note (pitch) has high frequency, you can't see or feel higher pitch notes whereas low notes have lower frequency which you can see as well as feel.
In the video below, you can visually see that as the frequency gets higher, there are more loops created, at times it goes really fast that you can't see the wave; but as the frequency get lower, there are less loops created and the string is more visible. It seems as if there are two strings but there is actually only one string but since the string moves fast, it looks like there are two strings.
In the video below, you can visually see that as the frequency gets higher, there are more loops created, at times it goes really fast that you can't see the wave; but as the frequency get lower, there are less loops created and the string is more visible. It seems as if there are two strings but there is actually only one string but since the string moves fast, it looks like there are two strings.
- If the frequency is high then the wave length is smaller.
- If the frequency is low then the wave length is longer.
For reference, look at the diagram to the right
- With higher frequency, the wave lengths are shorter as there are more loops created.
- With lower frequency, the wave length are longer because there aren't many loops created.
NOTE: The total length of both waves are the same!
Changing the medium changes the wave speed.
Examples
- Vertically shaking a jump rope that is bounded to a surface
- Ocean or pool waves hitting a wall and bouncing back in the opposite direction
- Touching a ringing tuning fork to an instrument string or just plucking a string
- When you pluck the string you put energy into an elastic medium, and this energy travels through the medium as a transverse pulse