# Spherical wave

Two-dimensional representation of a spherical wave

The spherical wave is regularly and evenly from a source in all directions in strictly concentric wave fronts propagating wave (z. B .: sound wave , light wave ).

Such a spherical wave front only arises under the assumption of highly idealized conditions, e.g. B.

If the starting point of a wave ( transmitter ) is to be regarded as point-shaped, the wave propagates in a homogeneous, isotropic medium as a spherical wave, i.e. H. The surfaces of the same phases are spherical surfaces concentric to the transmitter and equidistant from one another.

It is characteristic of spherical waves that all field and energy quantities are constant on concentric shells around the center of excitation of the transmitter, while with plane waves these are constant in planes that are perpendicular to the direction of propagation of the wave movement. As the distance from the transmitter increases, the spherical waves become more and more similar to plane wave fronts.

A harmonic spherical wave of frequency and wave number can be represented analytically as ${\ displaystyle \ omega}$${\ displaystyle k}$

${\ displaystyle u (r, t) = u_ {0} {\ frac {\ exp \ left (\ mathrm {i} (\ omega t-kr) \ right)} {kr}}}$.

The energy of a spherical wave is distributed over ever larger areas, i.e. H. the energy density or power density decreases with the reciprocal square of the distance 1 / r 2 . This is also known as the square law of energy distance . In other words: if the distance to the transmitter is doubled , the power density is reduced to a quarter of the original value by quadrupling the spherical area. ${\ displaystyle r}$

## Acoustic spherical wave (sound)

### Sound energy quantities

The sound intensity (i.e. the surface power density of the sound) decreases as the sound energy quantity proportionally with the square of the distance from the transmitter: ${\ displaystyle I}$

${\ displaystyle I = {\ frac {P _ {\ mathrm {ak}}} {A}} \ sim {\ frac {1} {r ^ {2}}} \ quad \ mathrm {or} \ quad I \ sim {\ frac {I_ {0}} {r ^ {2}}}}$

d. H. their size value is quartered for each doubling of the distance:

${\ displaystyle \ Leftrightarrow {\ frac {I_ {2}} {I_ {1}}} = \ left ({\ frac {r_ {1}} {r_ {2}}} \ right) ^ {2}}$

This is due to the fact that the total sound power emitted by the sound source remains constant in the theoretical model on the envelope surfaces around the spherical sound source , i.e. This means that it is independent of the transmitter distance: ${\ displaystyle P _ {\ mathrm {ak}}}$

${\ displaystyle P _ {\ mathrm {ak}} = {\ text {const}}. \ neq f (r)}$

while it penetrates constantly increasing spherical surfaces that increase with the square of the distance: ${\ displaystyle A}$

${\ displaystyle A = 4 \ cdot \ pi \ cdot r ^ {2}}$

The above Decrease in sound intensity and also in sound pressure level to a quarter can each be expressed as a decrease of 6  dB .

### Sound field sizes

Similar to electromagnetic spherical waves, a distinction is also made between sound spherical waves

• a near field ${\ displaystyle \ left (r <2 \ cdot \ lambda \ right)}$
• a far field ${\ displaystyle \ left (r> 2 \ cdot \ lambda \ right)}$

With

• ${\ displaystyle r}$ the distance from the measuring point to the transmitter
• ${\ displaystyle \ lambda}$the wavelength .

#### Far field

The sound field variables sound (alternating) pressure  p , sound velocity  v and sound  displacement ξ also decrease in the far field , i.e. H. their size values ​​halve for each doubling of the distance: ${\ displaystyle {\ tfrac {1} {r}}}$

${\ displaystyle p \ sim {\ frac {1} {r}} \ quad \ Leftrightarrow \ quad {\ frac {p_ {2}} {p_ {1}}} = {\ frac {r_ {1}} {r_ {2}}}}$
${\ displaystyle v \ sim {\ frac {1} {r}} \ quad \ Leftrightarrow \ quad {\ frac {v_ {2}} {v_ {1}}} = {\ frac {r_ {1}} {r_ {2}}}}$
${\ displaystyle \ xi \ sim {\ frac {1} {r}} \ quad \ Leftrightarrow \ quad {\ frac {\ xi _ {2}} {\ xi _ {1}}} = {\ frac {r_ { 1}} {r_ {2}}}}$

(The speed of sound  v merely represents the changing speed of the particles . It should not be confused with the speed of sound  c , with which the sound energy propagates.)

#### Near field

In the near field, the sound velocity and the sound deflection decrease, while the sound pressure decreases . ${\ displaystyle {\ frac {1} {r ^ {2}}}}$${\ displaystyle {\ frac {1} {r}}}$

The 1 / r 2 drop in the velocity in the near field of a spherical sound wave is essentially caused by the blind velocity 'v, which occurs in addition to the active component -v. In the case of sound radiation in the near field, in addition to the actual (active) sound energy, there is also a reactive energy component that is created by the mass of the medium oscillating with it. This is understood to be the air mass that is pushed back and forth in the immediate vicinity of the sound source " watt los" without being compressed. As a result of this mass effect of the accompanying air, which cannot be neglected, a phase shift occurs between the sound velocity and the sound pressure , which is characteristic of the magnitude of the reactive energy (see the web link).

In a flat sound field , the speed consists only of its active component; there is no blind component there.