For a simple monopole the pressure at a distance r is,
- p(r,t) = pressure at a distance r and at time time [Pa]
- ρ0 = density of air [kgm-3]
- c = speed of sound in air [ms-1]
- ω = frequency [rads-1]
- t = time [s]
- k = wavenumber [m-1]
- r = distance from centre of source [m]
- Q = source strength [m3s-1]
- The source strength, Q is the product of the surface area and the normal surface velocity of the monopole.
- A pulsating sphere is a simple monopole.
The intensity at a distance r is given by
- λ = wavelength [m]
- I(r) = sound intensity at distance r [Wm-2]
- Integrating the intensity over a sphere centred at the source gives the radiated power
- Π = sound power [W]
- The simple monopole may be approximated by a loudspeaker with the rear closed off by a box, where the dimensions of the box in any direction are small compared to the wavelength.
- A simple extension to this is to consider the source as being baffled in which case the pressure increases by a factor of two in the half space occupied by the source. This condition may be considered for a source above a rigid ground plane or for an element that is vibrating within an infinite baffle. This simple approximation allows a more complex source to be constructed from a number of discrete monopoles.
This can be considered as two equal strength monopoles that are out of phase and a small distance, d apart (such that kd<<1). There is no net introduction of fluid by a dipole. As one source exhales, the other source inhales and the fluid surrounding the dipole simply sloshes back and forth between the sources. It is the net force on the fluid which causes energy to be radiated in the form of sound waves.
- The far-field expression for the pressure radiated by an acoustic dipole may be written as:
- The sound power radiated by a dipole is given by:
- An open-back speaker that radiates sound equally front and rear will form a dipole. The front and rear waves are out of phase and cancellation will occur when the wavelengths are long enough to wrap around. One way to stop this is to use a large, wide baffle or to enclose the driver creating a monopole.
This can be considered as four monopoles with two out of phase with the other two. They are either arranged in a line with alternating phase or at the vertices of a cube with opposite corners in phase. In the case of the quadrupole, there is no net flux of fluid and no net force on the fluid. It is the fluctuating stress on the fluid that generates the sound waves. However, since fluids donít support shear stresses well, quadrupoles are poor radiators of sound.
- Lateral Quadrupole
- The quadrupole arranged at the vertices of a square. The far-field sound pressure amplitude produced by a lateral quadrupole may be written as
- Longitudinal Quadrupole
- The quadrupole arranged in a line with alternating phase. The far-field sound pressure amplitude produced by a longitudinal quadrupole may be written as
- Relative radiation efficiency of a dipole is:
- Relative radiation efficiency of a quadrupole is:
- Relative efficiency calculated for c=343ms-1, d=0.005m and D=0.005m.
A continuous line acoustic source such as a vibrating string. This may be simplified mathematically as a number of acoustic monopoles in a line of the correct amplitude and phase.
See also: Acoustics, Reference Sound Source, Sound Source, Source.
Subjects: Architectural Acoustics Noise & Vibration