4.3.2 Explain the formation of standing waves in one dimension
When two waves of the same frequency and wavelength travel in opposite directions a standing wave is created. They can also be created when a single wave is reflected off a fixed boundary (string reflecting with one end of the string attached to the wall). It is called a standing wave because it does not appear to move. The peaks become troughs (anti - nodes) and vice versa, but the points with zero displacement (nodes) stay in the same place, i.e. they do not travel.
4.3.3 Compare standing waves and traveling waves
- While traveling waves transmit energy a standing wave does not. However there is energy associated with standing waves.
- Any given point on a traveling wave will have amplitudes ranging from the minimum to the maximum amplitude, i.e. all point can attain any amplitude. Where as a given point on a standing wave will have amplitudes ranging from a max to min, but not necessarily the max and min of the wave…
- The wavelength of a traveling wave is the physical distance from a peak to the next peak or from a trough to the next trough. The wavelength of a standing wave is twice the distance between nodes or twice the distance between antinodes.
4.3.4 Explain the concept of resonance and state the conditions necessary for resonance to occur
Every object has a “natural frequency” this can be thought of as a frequency that he object “likes to vibrate” at. At the natural frequency of an object the energy of the vibration tends to stay in the object and build up or collect in the object. If an object is forced to vibrate at its natural frequency we call this resonance. Resonance can be observed when you sit in a swing and rock your legs back and forth to generate motion. Sometimes people sing in the shower and they think they sound good… This is because the shower stall has several resonant frequencies, the sound waves build up and sound amplified. The amplified waves make the voice sound richer and just plain better.
4.3.5 Describe the fundamental and higher resonant modes in strings and open and closed pipes
Musical instruments depend on standing waves for their characteristic sounds. Each instrument has different natural frequencies. When a string is plucked or “bowed” the string resonates in such a way as to create a standing wave. If air is blown across the opening of a pipe a standing wave is set up inside of the pipe and a sound is generated.
Strings: Each end of the string is fixed and therefore a node with an antinode in the middle. Since the wavelength is twice the distance between the nodes, the longest wavelength is twice the length of the string. This is known as the fundamental mode or fundamental frequency.
Fundamental Frequency
The string must have a node at both ends of the string, but it could also have a node in the middle of the string. 3 nodes and 2 antinodes generates a wavelength equal to the length of the string.
Second Harmonic Frequency
Third Harmonic Frequency
We can also generate a standing wave with 4 nodes and 5 nodes and so on… These other modes or frequencies are called higher resonant modes or higher resonant frequencies. In Theory the string can vibrate at an infinite number of frequencies or wavelengths. In general we can describe the possible wavelengths of a string with the equation:
(1)Where L is the length of the string and n is the mode. If n = 2 we would say the string is vibrating in the second harmonic, if n =3 third harmonic and so on.
Closed Pipes: If you blow across the opening of a pipe that is closed at one end a sound is generated. The sound is generated by a standing wave. Just like with a string there is a fundamental frequency and higher resonant modes. However with a closed pipe the requirement is that there is a node at the closed end and an antinode at the open end. Why? Sound is generated by moving air, at the closed end the air can not move, i.e. there is zero displacement… At the open end of the pipe there is maximum displacement…
Fundamental Frequency
Second Harmonic Frequency
For a closed pipe the wavelength of the fundamental frequency is 4 times the length of the pipe. The second resonant mode has a wavelength of 4/3 times the length of the pipe. In general we can find the wavelength by:
(2)Open Pipe: Just like with a closed pipe an open pipe can support a standing wave. The boundary conditions are that there must be antinodes at both ends of the pipe.
The fundamental frequency occurs when there is two antinodes and one node. The wavelength is 2L. The second fundamental frequency has nodes, and three antinodes.
Fundamental Frequency
Second resonant mode/frequency
Hello :)
Are nodes and antinodes confused in this explanation?
The peaks become troughs (nodes) and vice versa, but the points with zero displacement (antinodes) stay in the same place, i.e. they do not travel.
Yep, typo? Not sure how that got messed up. I think I fixed it.
Thanks for providing all these notes :D
Had practise paper 1 and 2 today and did quite well.
Very helpful website!!
super useful… thx!
In the open pipe section there is actually a closed pipe illustration instead of an open pipe illustration for the fundamental node
A life saver… thank you.
can you explain the formation of standing waves in an open pipe?
is it a consequence of resonance?
as for the closed pipe, i can understand that the formation is due to the incident wave being reflected at the antinode. But i cant seem to apply this concept to the explanation for open pipe..
In the past I used a now semi-defunct program called Crocodile Physics to show how this works. Its a bit tough to understand a visualization helps a lot.
Essentially a pulse or wave front moves through the open pipe compressing the air as it goes. When the pulse or wave front gets to the other end of the pipe there is a "collision" with the lower pressure air. This causes a new "wave source" (think Huygens Principle) and effectively causes a minor reflection back through the tube. This starts a chain reaction of bouncing back and forth. A good visualization can be found at:
http://www.phys.unsw.edu.au/jw/flutes.v.clarinets.html#time
Hope that helps I know it is not a perfect or even 100% complete answer.
I am having following queries:
(1)For formation of standing waves is reflection at both ends (boundary conditions) a must?
(2) Since during superposition of two waves the original waveform is not lost, so if reflection at second boundary is not must then is it possible that the reflected wave beyond the region of superposition goes ahead in its original reflected form?
yes or no & why if the answer is no ?
the value of the transverse velocity increases when the linear mass density is increased
There's a mistake in this. Under closed pipes, the second diagram is the THIRD harmonic frequency not the second.
Why? a quarter (1/4) of a wave is added to make the next harmonic frequency in a pip with only one end open.
That's not a mistake.
You cannot produce a sound on a closed pipe with the second harmonic frequency as you need a node at the closed end and an antinode at the open end.
Post preview:
Close preview