# Acoustics

## Vibrations Of An Air Column

When air is blown across the entrance to an organ pipe, it causes the air in the pipe to vibrate, so that there are alternate small increases and decreases of the density of the air (condensations and rarefactions). These alternate in space, with the distance between successive condensations (or rarefactions) being the wavelength; they alternate in time, with the frequency of the vibration. One major difference here is that the string vibrates transversely (perpendicular to the length of the string), while the air vibrates longitudinally (in the direction of the column of air). If the pipe is open at both ends, then the density of the air at the ends must be the same as that of the air outside the pipe, while the density inside the pipe can vary above or below that value. Again, as for the vibrations of the string, the density of the air in the pipe can be analyzed into a fundamental and overtones. If the density of the air vibrating in the fundamental mode (of the open pipe) is plotted across the pipe length, the graph is as in Figure 4.

The "zero value" at the ends denotes the fact that the density at the ends of the pipe must be the same as outside the pipe (the ambient density), while inside the pipe the density varies above and below that value with the frequency of the fundamental, with a maximum (and minimum) at the center. The density plot for the fundamental looks just like that for the fundamental of the vibrating stretched string (Figure 1). In the same manner, plots of the density for the various overtones would look like those of the string overtones. The frequency of the fundamental can be calculated from the fact that the velocity, which is analogous to that found for vibrations of the string, is the velocity with which sound travels in the air, usually denoted by c. Since the wavelength of the fundamental is twice the pipe length, its frequency is (c/2)l, where l is the length of the organ pipe. (While the discussion here is in terms of the density variations in the air, these are accompanied by small variations in the air pressure, and small motions of the air itself. At places of increased density the pressure is increased; where the pressure is changing rapidly, the air motion is greatest.) When a musician blows into the mouthpiece of a clarinet, the air rushing past the reed causes it to vibrate which then causes the column of air in the clarinet to vibrate in a manner similar to, but more complicated than, the motion of the organ pipe. These vibrations (as for all vibrations) can also be analyzed into harmonics. By opening and closing the keyholes in the clarinet, different harmonics of the clarinet are made to grow louder or softer causing different tones to be heard.