In a previous presentation, we introduced wave parameters and standing waves along strings. Here we will extend these ideas to sound waves traveling in air.

*infrasound*frequencies, but elephants can send out and receive these type of sounds over great distances to communicate with each other. Likewise if a source vibrates quickly enough, it will produce sounds with frequencies higher than 20,000 Hz--we humans can't hear these

*ultrasound*frequencies, but bats can send out and receive these type of sounds, typically for navigating and preying on insects.)

*absolute*temperature (in kelvin, not in Celsius!), and at 273 K (0° C) would have a speed of 331 m/s, and at "room temperature" (according to some) of 293 K (20° C) sound waves would have a speed of:

*v*= (

__331__m/s)·sqrt((

__293__K)/(

__273__K)) =

__342__.91026079 m/s,

or to three significant figures, 343 m/s. (Some may define room temperature as high as 25° C, but apparently physicists like it a little cooler than some other people.) Essentially sound travels faster through warmer air, and slower through cooler air (neglecting changes in pressure). These sound waves are fast, but bullets and jets can certainly move faster than these speeds.

(This is a review of our previous discussion of one-dimensional rope and string waves.) Note the hierarchy of these wave parameters. Since the wave speed is determined by properties of the material it travels through (independent of the source), and the frequency is determined by the source (independent of the medium), these are said to be independent wave parameters. In contrast, the wavelength of the wave is dependent on both the independent speed and frequency parameters. Algebraically there is nothing wrong with expressing this relation as

*v*= λ

*f*and

*f*=

*v*/λ, as long as you recognize that the dependency of λ doesn't change.

Now we'll briefly extend our previous discussion on standing waves on strings to sound standing waves in pipes.

*antinode*in the middle, where air molecules slosh back-and-forth, while there are

*nodes*at either end, where air molecules are "trapped" because they cannot move into (against) the closed ends.

In contrast, the open-open pipe has antinodes at either end, as air molecules are free to slosh back-and-forth, being exposed to the atmosphere, while there is a node in the middle, where air molecules are "trapped" because of the motion at either end.

Because the distance between consecutive nodes and antinodes in these pipes is proportional to the wavelength of the standing wave, the fundamental frequency would be the same for either closed-closed or open-open pipes.

For a closed-closed or open-open symmetric pipe, the frequency at which a source of sound waves would resonate within this pipe are multiples of the fundamental frequency

*f*

_{1}, which depends on the wave speed

*v*of sound (343 m/s for "room temperature"), and the length of the pipe.

*antinode*, where air molecules exposed to the atmosphere slosh back-and-forth, while the closed end has a

*node*, where air molecules are "trapped" because they cannot move into (against) the closed ends.

(Because the distance between consecutive nodes and antinodes in these pipes is proportional to the wavelength of the standing wave, the fundamental frequency would be the same for either closed-open or open-closed pipes. Note for a given length of pipe, the asymmetric systems have a

*longer*(bigger value) node-antinode spacing than the symmetric systems, and thus an asymmetric system would have a

*lower*(smaller value) fundamental frequency (in this animation, approximately 2 Hz) than the symmetric system.)

For a closed-open or open-closed asymmetric pipes, the fundamental frequency is lower than a comparable symmetric pipe (note the factor of 4 instead of 2 in the denominator), and resonant frequencies are all

*odd*multiples of the fundamental frequency (rather than integer multiples). How very, very...odd. But this follows from the asymmetry of closed-open and open-closed pipes, such that even standing wave frequency multiples are not allowed.

*but only at multiples of the fundamental frequency*in order to resonate the air along the pipe. Higher resonant frequency notes can be accomplished either by changing the length

*L*of the trombone by shortening the slide... (Video link: "How to play the Trombone B Flat Major Scale.")

*L*is constant), and buzzing your lips to match higher resonant frequency notes. (Video link: "Trombone B Flat 7 Octaves Attempt.") This takes a fair amount of manual and lip coordination, so keep that in mind the next time you are subjected to a novice trombone player practicing the scales.

## No comments:

Post a Comment