ABSTRACT 
General expressions for the velocity potential, particle velocity, instan- 
taneous sound pressure, mean-square sound pressure, and sound intensity for 
the sound generated by turbulence are derived, starting from Lighthill’s funda- 
mental relations for the density fluctuations in the medium outside a turbulent 
region. The dipole radiation from the turbulent boundary layer of a rigid bound- 
ary, the quadrupole radiation from isotropic turbulence, and the lateral quadrupole 
radiation from turbulence in the presence of a large mean shear are discussed. 
The frequency spectra and directional patterns of the sound observed at points 
in the sound field that are much less than a wavelength from multipole sound 
sources are shown to differ significantly from the frequency spectra and direc- 
tional patterns of the sound observed at points in the sound field that are many 
wavelengths from the multipole sound sources. 
INTRODUCTION 
Significant progress has been made toward an understanding of the subject of turbulence 
noise since the publication of Lighthill’s fundamental relations! for the density fluctuations 
in the medium outside a turbulent region. The expressions defining the radiated or far-sound 
field of a turbulent region in terms of multipole fields have been extensively discussed by 
Lighthill,? Proudman,® Phillips,* and others.°-® The purpose of this report is to present 
expressions which also explicitly define the near-sound field at distances either not large 
compared with a wavelength of the radiated sound or not large compared with the geometric 
extent of the turbulent region. It is hoped that these expressions will be useful in interpreting 
noise data taken close to turbulent jets and boundary layers. 
Acoustic multipoles are also associated with phenomena other than turbulence. The 
general expressions for the sound fields of multipoles, of course, are also applicable to these 
sound fields. For instance, the near-sound fields produced by rotating propellers® and by 
surface disturbances!° have been investigated and analyzed using acoustic dipole theory. 
Because the far-sound field is simpler and usually of more interest, measurements in 
the near-sound field of a turbulent region have usually been avoided. Some reports on the 
sound from turbulent jets, however, do contain data on the sound pressures in the near 
field.11—18 In general, the data indicate an enhancement of the low-frequency portion of the 
sound spectra and a different sound pressure-distance dependence than has been observed 
for the far-sound field. In the future, careful measurements of the noise close to turbulent 
boundary layers may alsoshow near-field effects.1? Since observations in the near-sound 
field are often necessary or convenient, it is advantageous to have means of interpreting the 
data. This report is intended to facilitate the making of corrections to existing and future 
data and the planning of new experiments. 
Dee ferences are listed on page 33. 
