A sound-pressure disturbance, shown in red, enters the vocal tract at the glottis and propagates towards the lips.
This input wave has been chosen to have a sinusoidal form, with the length of the complete wave equal to 4 x vocal tract length.
Thus, when the intial value of 1 reaches the lips the value entering at the glottis is zero and the red curve forms 1/4
of a complete sinusoid.
On reaching the lips the wave is reflected WITH A CHANGE OF POLARITY (positive becomes negative).
This is a crucial feature of what happens at the open end of the tube.
The reflected wave is shown in green.
One way of thinking of this is that the pressure at the mouth opening has to be zero, since here the vocal tract is connected to a virtually infinite volume at atmospheric pressure (zero relative pressure).
Thus the sum of all waves just reaching the lips, and of those that have just been reflected has to be zero.
At this stage of the animation there is only one wave reaching the lips from the glottis (in red) and one reflected wave (in green). However, from now on the animation will show the SUM of all waves passing through the vocal tract in black. Thus the black curve is zero at the lip end of the tube and will remain so.
Let us now turn our attention to the black curve (the sum of all reflections).
Over the first few milliseconds of the animation the black curve gets bigger and bigger until it reaches an amplitude of about 4. Notice also that as the reflections go to and fro through the vocal tract they decline in amplitude. This decay will happen in any realistic physical system due to loss of energy to the surroundings, and has the advantage here of making it easier to see multiple reflections.
The peak value of 4 in the black curve thus represents a state where energy input at the glottis is balanced by energy loss in the decaying reflections.
Look at the animation from a timepoint where the black curve is close to its maximum and also fairly smooth through the superposition of multiple reflections. e.g. from about 4ms.
How long does it take the black curve to go through one complete cycle, e.g. from its peak positive value, via the peak negative value back to the peak positive value?
You should come up with a value close to 2ms.
Thus the "standing" wave is an oscillation with a frequency of 1000/2 = 500Hz.
It is referred to as "standing" because the locations of zero pressure and maximum/minimum pressure in the tube do not change.
Notice that this period duration of 2ms is also the time required for four passes of the reflected wave through the vocal tract (follow the blue circle at the leading edge of the reflected wave). This must be so because the sound pressure disturbance (e.g. at the blue circle) moves at the speed of sound. With c=34000cm/s 2ms is the time required to travel 4x17 = 64cm. Thus we see here the link between frequency and wavelength captured in the formula f = c/lambda.
The following animation shows the next standing wave in a tube closed at one end and open at one end.
Based on these first two examples we cannot exclude the possibility that a standing-wave will develop regardless of the nature
of the input. The next example illustrates that this is not the case:
This point can be emphasized by considering the following two animations:
The first one shows the original quarter-wave animation, but with a slower rate of decay in the amplitude of the reflections (the green and magenta lines are closer together than in the original animation).