If we have let us say fixed air column of length 'L', in a open-closed column problem, lamba is equal to 4*L/(2n-1).
n = number of nodes / anti nodes in air column
How does 'n' changes with respect to the temperature? Is the "number of nodes / anti nodes" at room temperature equal to "number of nodes/anti nodes" in same air column at let us say T =1200C? or do they increase or decrease with temperature?
As speed = frequency * wavelenght
and speed and temperature have reciprocal relationship, that means
increase Temperature will increase speed, which will result in increase of frequency but is there any relationship that after certain frequency(or temperature indirectly) the quarter standing waves turn into 3 quarters ?
Best Answer
The standing waves you introduced are models for how air moves as it resonates. In fact, each is called a mode (of oscillation). In the continuum approximation of air being an infinitely divisible, continuous fluid, you need infinitely many of them to simultaneously model arbitrary resonating movement.
In practice, you can hope to neglect all but low numbered modes. How low depends on what criteria you use. The way you phrased your question, I must guess because you did not elaborate on what purpose you intend to serve.
You can certainly stop when you reach such short wavelengths that the air does not support: If they get too close to the typical intermolecular distance, the motion will essentially be heat, not sound. As you approach that limit, you turn your sound (oscillatory motion) into unordered motion or a temperature increase. That happens sooner as air gets less dense and the distance between molecules larger: Your relevant number of standing waves goes down as temperature increases. In practice, other effects may cause higher modes to become uninteresting, and perhaps too dissipative, much earlier, such as interaction with a perhaps frequency-dependently dissipative tube wall.
If it is sound that you are after, there will likely be another more relevant effect: We cannot hear sound above a certain frequency. Since the speed of sound varies only slightly with temperature, that may mean the number of interesting standing waves does not significantly chance with temperature.