Near field and far field (acoustics)

When a sound wave propagates, a distinction is made between the near field and the far field, two distance ranges from the signal source with very different physical properties.

The near field describes the immediate area around the sound source, which is characterized by an uneven change between locations with constructive and destructive interference. In contrast, the far field refers to an area that is far away from the sound source. Interference effects only play a subordinate role here for the structure of the sound field . In particular, in fluid media in the far field, pressure and velocity are in phase, so that a real-valued wave impedance can be defined in the fluid . The distance at which the near field changes into the far field depends on the sound wavelength and the size of the transducer. To distinguish between near and far field, only the direct sound generated by the sound source, i.e. the radiation properties of the transducer and the properties of the propagation medium, is considered. Near field and far field are not room-dependent, but are dependent on the sound source. Reflections from objects in the room, such as the diffuse field generated by the walls or standing waves , are not taken into account. ${\displaystyle {\underline {Z}}=\rho \cdot c}$

Monopole radiator

The boundary between the near and far field is to be shown using the example of a zero order radiator as a breathing sphere (or pulsating sphere) when it is radiated into space. The wave equation for a spherical wave provides the solution for the sound pressure : ${\displaystyle p}$

${\displaystyle p={\frac {p_{0}}{kr}}\cdot e^{ikr}}$

The following applies to the speed of sound : ${\displaystyle v}$

${\displaystyle v={\frac {p_{0}}{Z_{0}\cdot kr}}\left(1+{\frac {1}{ikr}}\right)\cdot e^{ikr}}$

The following applies to the sound impedance ${\displaystyle Z}$

${\displaystyle Z=Z_{0}{\frac {ikr}{1+ikr}}}$

Mean

• p ₀ = pressure amplitude
• k = circular wave number
• r = distance of the measuring point from the spherical emitter
• Z ₀ = characteristic acoustic impedance
• i is the imaginary unit with the property ${\displaystyle i^{2}=-1}$

Far field

The far field is characterized by the condition

${\displaystyle k\cdot r\gg 1}$

whereby the formula for the speed of sound can be simplified

${\displaystyle v={\frac {p_{0}}{Z_{0}\cdot kr}}\cdot e^{ikr}}$

In the far field, there is no phase difference between sound pressure and sound velocity , which can be seen if the (measurable) real components of Euler's formula are given for both sound field sizes :

${\displaystyle p\sim {\frac {1}{r}}\cdot \cos(kr)}$

${\displaystyle v\sim {\frac {1}{r}}\cdot \cos(kr)}$

${\displaystyle Z=Z_{0}}$

Near field

The near field is characterized by the condition

${\displaystyle k\cdot r\ll 1}$

whereby the formula for the speed of sound can be simplified

${\displaystyle v={\frac {p_{0}}{Z_{0}\cdot kr}}\cdot {\frac {1}{ikr}}\cdot e^{ikr}}$

In the near field, sound pressure and sound velocity differ in two essential points:

• The speed of sound decreases quadratically with increasing distance and
• between and there is a phase difference of 90 °, which can be seen if the (measurable) real components of Euler's formula are given for both sound field sizes: ${\displaystyle p}$ ${\displaystyle v}$

${\displaystyle p\sim {\frac {1}{r}}\cdot \cos(kr)}$

${\displaystyle v\sim {\color {red}{\frac {1}{r^{2}}}}\cdot \sin(kr)}$

${\displaystyle Z\approx Z_{0}\cdot ikr}$

The sound pressure leads the sound velocity by a maximum of 90 °. At the transmitter location (in the immediate vicinity of the spherical transmitter) only reactive power occurs and the sound impedance is (almost) purely imaginary. This means that the monopole radiator gives off energy to the environment in certain time segments and then absorbs almost exactly the same amount of energy again (it "breathes"). The small difference is emitted and can be measured in the far field.

Transition area

The transition between near and far field is continuous. The residual phase angle between sound pressure and sound velocity is decisive for the characterization . This changes continuously between 90 ° in the immediate vicinity of the monopole radiator to 0 ° at a distance of many wavelengths. The limit is often drawn at r ≈ λ because the brackets in the formula for the sound velocity then indicate the value ${\displaystyle \phi }$

${\displaystyle \left(1+{\frac {1}{2\pi i}}\right)\approx (1-0{,}16i)\approx 1}$

Has.

Extended radiator

Harmonic sound field of an unfocused 4 MHz ultrasonic transducer with the transducer diameter D = 10 mm in water. The amplitudes of the sound pressures are displayed in log. Presentation.

Dipole radiators are rarely used in acoustics as in high-frequency technology, the dimensions of which correspond approximately to the wavelength. One reason for this is the emission of a very broad frequency spectrum in the case of sound frequencies, and in the case of ultrasound the manageable dimensions of the sounder. The following example shows the harmonic sound field of an unfocused 4 MHz ultrasonic transducer with the transducer diameter D = 10 mm in water with a sound propagation speed due to simulation calculations . The specified sound field solves the boundary value problem for the radiation from a reverberant boundary surface z = 0 into the half-space. The radiation can be described by uniformly covering the active transducer surface with monopole sources. ${\displaystyle c=1500\,\mathrm {m} /\mathrm {s} }$

The transition from the near field to the far field takes place at the sound pressure maximum furthest away from the transducer on the acoustic axis. The distance between the ultrasonic transducer and the last sound pressure maximum on the acoustic axis is called the near field length

${\displaystyle N={\frac {D^{2}-\lambda ^{2}}{4\lambda }}}$

designated. In the example given, it is . ${\displaystyle N=67\,\mathrm {mm} }$

The figure shows pronounced interference structures with an irregular change between locations of constructive and destructive interference in the near field. The far field has a regular structure with a steady decrease in sound pressure with increasing distance from the transducer.

Definition of terms

The sound terms near field and far field are not always clearly distinguished from the expressions direct field or free field and diffuse field . The near and far field characterize the sound source itself, while the direct field (free field) and diffuse field are determined by the room acoustic properties of the surrounding area.

See also

• Sound field size
• Spherical wave
• Room sound
• Near field and far field (antennas)

Individual evidence

1. ↑ Acoustic waves and fields, German Society for Acoustics eV (PDF file; 1016 kB)
2. ↑ Elfgard Kühnicke: Elastic waves in layered solid-state systems - modeling using integral transformation methods - simulation calculations for ultrasonic applications . TIMUG, Bonn 2001, ISBN 3-934244-01-7 .
3. ↑ Josef Krautkrämer, Herbert Krautkrämer: Materials testing with ultrasound . 5., completely redesigned. Edition. Springer, Berlin 1986, ISBN 3-540-15754-9 .