Akustik/ Schallgeschwindigkeit
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Die Schallgeschwindigkeit c (von lat. celeritas, "Geschwindigkeit") variiert abhängig vom Medium, das die Schallwellen durchdringen. Es werden gewöhnlich die Eigenschaften der Substanzen angegeben (z. B. siehe den Artikel über Natrium). Die Schallgeschwindigkeit sollte nicht mit der Lichtgeschwindigkeit, die ebenfalls mit c angegeben wird und 
beträgt.
Normalerweise meint man mit der Schallgeschwindigkeit die entsprechende Geschwindigkeit in Luft. Abgesehen vom Medium gibt es auch Unterschiede in Abhängigkeit von den atmosphärischen Bedingungen; Am bedeutensten ist die Temperatur T. Die Luftfäuchtigkeit hat ebenfalls einen geringen Einfluss auf die Schallgeschwindigkeit, der Luftdruck (p) ist dagegen nicht von Bedeutung. Der Schall bewegt sich langsamer mit steigender Höhe über NN, was auf die Änderungen von Luftdruck und Temperatur zurückzuführen ist. Näherungsweise kann die Schallgeschwindigkeit in Metern pro Sekunde wie folgt berechnet werden:
wobei 
(Theta) die Temperatur in °C ist.
Inhaltsverzeichnis |
[Bearbeiten] Details
Eine genauere Berechnung erhält man mit
wobei
- R (287.05 J/(kg·K) für Luft) die Gaskonstante für Luft ist: Die universelle Gas-Konstante R, welche in J/(mol·K) angegeben wird, wird durch die molare Masse von Luft dividiert)
- κ (Kappa) der adiabatische Index (1,402 bei Luft) ist gelegentlich auch als γ angegeben
- T die absolute Temperatur in Kelvin ist:
.
In der Standard-Atmosphäre:
T0 = 273,15 K (= 0 °C = 32 °F), womit wir auf 331,5 m/s kommen (= 1087,6 ft/s = 1193 km/h = 741,5 mph = 643,9 Knoten).
T20 = 293,15 K (= 20 °C = 68 °F), womit wir auf 343,4 m/s kommen (= 1126,6 ft/s = 1236 km/h = 768,2 mph = 667,1 Knoten).
T25 = 298,15 K (= 25 °C = 77 °F), womit wir auf 346,3 m/s kommen (= 1136,2 ft/s = 1246 km/h = 774,7 mph = 672,7 Knoten).
Wenn wir von einem idealen Gas ausgehen, hängt die Schallgeschwindigkeit c nur von der Temperatur ab, nicht vom Druck. Luft ist ein beinahe ideales Gas. Die Lufttemperatur variiert mit der Höhe über NN, womit wir bei Nutzung der Standard-Atmosphäre die folgenden Bedingungen haben - die eigentlichen Bedingungen können abweichen. Any qualification of the speed of sound being "at sea level" is also irrelevant.
| Höhe über NN | Temperatur | m/s | km/h | mph | Knoten |
| 0 (NN) | 15 °C (59 °F) | 340 | 1225 | 761 | 661 |
| 11000 m–20000 m (Die Flughöhe von Durchschnitts-Jets, und erste Überschallflieger) |
-57 °C (-70 °F) | 295 | 1062 | 660 | 573 |
| 29000 m (Flug mit X-43A) | -48 °C (-53 °F) | 301 | 1083 | 673 | 585 |
In einem nicht-streuenden Medium ist die Schallgeschwindigkeit unabhängig von der Frequenz, somit sind Energie-Transport-Geschwindigkeit und die der Schallverbreitung gleich. For audio sound range air is a non-dispersive medium. We should also note that air contains CO2 which is a dispersive medium and it introduces dispersion to air at ultrasound frequencies (28KHz).
In a Dispersive Medium – Sound speed is a function of frequency. The spatial and temporal distribution of a propagating disturbance will continually change. Each frequency component propagates at each its own phase speed, while the energy of the disturbance propagates at the group velocity. Water is an example of a dispersive medium.
In general, the speed of sound c is given by
where
- C is a coefficient of stiffness
- ρ is the density
Thus the speed of sound increases with the stiffness of the material, and decreases with the density.
In a fluid the only non-zero stiffness is to volumetric deformation (a fluid does not sustain shear forces).
Hence the speed of sound in a fluid is given by
where
- K is the adiabatic bulk modulus
For a gas, K is approximately given by
where
- κ is the adiabatic index, sometimes called γ.
- p is the pressure.
Thus, for a gas the speed of sound can be calculated using:
which using the ideal gas law is identical to:
(Newton famously considered the speed of sound before most of the development of thermodynamics and so incorrectly used isothermal calculations instead of adiabatic. His result was missing the factor of κ but was otherwise correct.)
In a solid, there is a non-zero stiffness both for volumetric and shear deformations. Hence, in a solid it is possible to generate sound waves with different velocities dependent on the deformation mode.
In a solid rod (with thickness much smaller than the wavelength) the speed of sound is given by:
where
- E is Young's modulus
- ρ (rho) is density
Thus in steel the speed of sound is approximately 5100 m/s.
In a solid with lateral dimensions much larger than the wavelength, the sound velocity is higher. It is found by replacing Young's modulus with the plane wave modulus, which can be expressed in terms of the Young's modulus and Poisson's ratio as:
For air, see density of air.
The speed of sound in water is of interest to those mapping the ocean floor. In saltwater, sound travels at about 1500 m/s and in freshwater 1435 m/s. These speeds vary due to pressure, depth, temperature, salinity and other factors.
For general equations of state, if classical mechanics is used, the speed of sound c is given by
where differentiation is taken with respect to adiabatic change.
If relativistic effects are important, the speed of sound S is given by:
(Note that 
is the relativisic internal energy density).
This formula differs from the classical case in that ρ has been replaced by 
.
[Bearbeiten] Speed of sound in air
| Impact of temperature | |||
|---|---|---|---|
| θ in °C | c in m/s | ρ in kg/m³ | Z in N·s/m³ |
| −10 | 325.4 | 1.341 | 436.5 |
| −5 | 328.5 | 1.316 | 432.4 |
| 0 | 331.5 | 1.293 | 428.3 |
| +5 | 334.5 | 1.269 | 424.5 |
| +10 | 337.5 | 1.247 | 420.7 |
| +15 | 340.5 | 1.225 | 417.0 |
| +20 | 343.4 | 1.204 | 413.5 |
| +25 | 346.3 | 1.184 | 410.0 |
| +30 | 349.2 | 1.164 | 406.6 |
Mach number is the ratio of the object's speed to the speed of sound in air (medium).
[Bearbeiten] Sound in solids
In solids, the velocity of sound depends on density of the material, not its temperature. Solid materials, such as steel, conduct sound much faster than air.
[Bearbeiten] Experimental methods
In air a range of different methods exist for the measurement of sound.
[Bearbeiten] Single-shot timing methods
The simplest concept is the measurement made using two microphones and a fast recording device such as a digital storage scope. This method uses the following idea.
If a sound source and two microphones are arranged in a straight line, with the sound source at one end, then the following can be measured:
- The distance between the microphones (x)
- The time delay between the signal reaching the different microphones (t)
Then v = x/t
An older method is to create a sound at one end of a field with an object that can be seen to move when it creates the sound. When the observer sees the sound-creating device act they start a stopwatch and when the observer hears the sound they stop their stopwatch. Again using v = x/t you can calculate the speed of sound. A separation of at least 200 m between the two experimental parties is required for good results with this method.
[Bearbeiten] Other methods
In these methods the time measurement has been replaced by a measurement of the inverse of time (frequency).
Kundt's tube is an example of an experiment which can be used to measure the speed of sound in a small volume, it has the advantage of being able to measure the speed of sound in any gas. This method uses a powder to make the nodes and antinodes visible to the human eye. This is an example of a compact experimental setup.
A tuning fork can be held near the mouth of a long pipe which is dipping into a barrel of water, in this system it is the case that the pipe can be brought to resonance if the length of the air column in the pipe is equal to ( {1+2n}/λ ) where n is an integer. As the antinodal point for the pipe at the open end is slightly outside the mouth of the pipe it is best to find two or more points of resonance and then measure half a wavelength between these.
Here it is the case that v = fλ
