| Acoustic Definitions, References and Decibels |
| The Decibel: A decibel, regardless of the sort of application, is always ten times the logarithmic (base ten denoted log10 or simply log) function of a number. That is:  The decibel is often used when the range of x is very broad – say from 0.0001 to 10000 – and the advantage is that the decibel “compresses” x. The table below shows some examples.
The logarithmic function, and therefore also the decibel, is only defined for dimensionless x, which means that x cannot have a unit like e.g. meters or Watt. Because of this, x is very often a fraction of two numbers with units; the nominator being the quantity of interest and the denominator being the reference.  The reference concept is very important for the proper understanding of decibels because it uniquely determines what the decibel number is referring to. An example should clarify this point. If for example “a” is the length of a stick, say a=1.3m, we cannot give the length of “a” in decibels before we choose a reference distance simply because log(a) doesn’t make sense when “a” has a unit. However, if we choose 1m as the reference we can express “a” in dB relative to 1m. That is:  Notice that the result “1.14dB re 1m” specifically points out that the chosen reference is 1m, which enable the reverse calculation back to a=1.3m from the dB. Therefore, even when the reference is left out in formulas to save time and writing, all dB numbers refer to references and any dB number only makes sense if it is clear what the reference is. The decibel and other logarithmic functions have some nice properties whereof the most important is that multiplication becomes summation. The laws of logarithmic functions are:  The last rule is often applied when the property “a” can be expressed as e.g. the square of some pressure, voltage or distance. A missing subscript k usually refers to the logarithm with base ten k=10, but it can also mean the natural logarithm ln(x) with base k=e=2.7183 |
| Underwater sound: Sound is disturbances of the medium – here water – travelling in a 3 dimensional manner as the disturbance propagate with the speed of sound. The sound is defined as a plane wave when the sound propagates in a single direction i.e. the lines for uniform phase are straight. 
 This definition is analogous to Ohm’s law for electrical circuits i.e. V=R•I and you can often think of particle velocity, acoustic impedance and sound pressure in the same way. Mechanical engineers may think of Newton’s law F=m•a as analogy. It shows that particle velocity and pressure are in phase in a plane sound wave. Acoustic intensity – power (Pa) per unit area (Aa) or energy flux - is used to describe levels of underwater sound like e.g. an echo, a whale’s call or a signal from a remote transducer. The intensity of a plane harmonic wave is:  The daily term “a high sound” refers to a sound with a high intensity. A reference intensity Iref has been defined in order to enable direct comparison of the loudness of sounds and the reference intensity used in underwater acoustics is that of a plane harmonic wave with an rms-pressure of 1µPa, which for ordinary seawater with c≈1500m/s and p ≈1000kg/m3 gives  The intensity level (IL=how high a sound is) is the intensity of the sound wave taken in decibels relative to the reference intensity of 1μPa plane wave rms-pressure (which is shortened to “re 1μPa”):  The intensity level is thus the loudness of a sound at a field point, which is different from the loudness of a source of sound because the intensity level decreases as the distance to the source increases. The intensity level of a sound is for example 200dB re 1μPa, which is the same as I=Iref •10(200dB/10)=66.7W/m2. |
| Beam patterns and Directivity The beam pattern of a transducer contain information about the transducer’s spatial response i.e. how it transmits or receives in different directions. Transducers that are very small compared to the wavelength have omni directional beams, which means that the energy is not concentrated in any particular direction. Transducers that are large compared to the wavelength have a very directive beam pattern, which means that their energy is concentrated in a specific direction. The beam width, which is the angle subtended by the points where the intensity has dropped 3dB below the maximum on-axis response, is often used as indicators of how concentrated the energy is for a specific transducer in a given cross section. The directivity index of a transmitter describes how concentrated the transmitted energy is at the maximum response point and for receivers the directivity index indicates the ability to discriminate a signal from an ambient background noise, both cases relative to an omni directional transducer.  Table 1: Approximations to far-field beam width and directivity index for various sources. Formulas for finding beam widths assume that the speed of sound is c≈1500m/s (c=λ•f). Notice that the beam width of a transducer is the same whether it is transmitting or receiving. The nearfield (or Fresnel field) of a transducer is characterized by irregularity and changes due to refraction effects leading the fact that the interference pattern (the beam) has not yet been fully formed. The Rayleigh distance r0 can approximate the nearfield extension:  Aactive is the active area of the transducer’s face. For line arrays, cylindrical arrays and the like it is often better to use Aactive=(Lmax)2 where Lmax is the longest dimension found on the active face of the transducer. The farfield (Fraunhofer field) precedes the nearfield after a transition region and is characterized by spherical spreading and regular beam patterns. |
| Underwater Sound Transmission: Sounds originating from acoustic sources are measured in intensity level, which decreases as the distance to the source is increased due to transmission loss (TL) i.e. spreading and absorption:  The formula assumes spherical spreading for the transmission loss i.e. the sound is unbounded and spreads out like it was originating from a point – the acoustic center of the source.  Spherical spreading is most common and is valid in the far field required that the source is placed far enough from any large structure. Cylindrical spreading occurs for example in shallow waters when the bottom and the surface reflects the sound and forces it to spread like a cylinder. When the sound is completely bounded (e.g. inside a pipe) it cannot spread and only absorption remains in the formula for transmission loss. The last term of the transmission loss is the attenuation, which increases very significantly with the frequency and furthermore varies with pressure, temperature, salinity and acidity. Accurate approximations are hard to come by, but the following approximation may be used:  Following Schulkin & Marsh’s approximate expressions, the special case of freshwater (S 0ppt) at room temperature (T=20ºC) and surface pressure (D=0m) gives the very simple formula for σ[dB/m] as a function of the frequency f[kHz]:  It should be noted that the effect of having saltwater (North Atlantic S 35ppt) instead of freshwater (S 0ppt) is significant. The speed of sound is a very important parameter in any echo-sounding system where a range is determined based upon the elapsed time and the speed of sound. The speed of sound can be approximated with a simple formula:  If we take the special case of freshwater (S ≈ 0ppt) at room temperature (T=20ºC) and surface pressure (D=0m) again Medvin’s formula yields:  The corresponding result for North Atlantic seawater (S ≈ 35ptt, T=20ºC, D=0m) would leave a higher speed of sound c≈ 1522m/s. The speed of sound is by definition the frequency multiplied with the wavelength:  The frequency cannot change, which implies that when the speed of sound changes the wavelength changes accordingly and this forces the sound to refract (“bend”) in order to enable the change in wavelength – see Figure 2.  Snell’s Law of refraction gives the bending angle of the sound “ray” i.e. that particular grazing angle indicating the chance in the ray propagation direction. Ray bending is only significant when the speed of sound changes and then usually only at large ranges. For more information on sound refraction and ray bending please see Urick, Robert J.: “Principles of underwater sound, 3rd edition”. McGraw-Hill Book Company, 1983 |
| Sound Levels: The source level of an acoustic source compares the intensity emitted by the acoustic source to a reference source. This of course, also enables direct comparison of acoustic sources with each other i.e. which one is the most powerful? The reference source is an omni directional source (DI=0dB) with an acoustic output power of 1W taken at the reference distance r=1m from the acoustic center. In terms of acoustic intensity the reference source has an acoustic intensity I0:  In dB relative to the reference intensity Iref this is  This is where the (to some well-known) reference level 170.8dB re 1μPa @ 1m derives from and it should be noted that the “dB re 1μPa @1m” should be understood as “the intensity level relative to the intensity of a plane wave with an rms-pressure of 1μPa taken at the reference distance 1m from the source”. Most acoustic sources have an acoustic power output different from 1W and they are not always omni-directional. To find the source level of such a more generic source we simply add (in dB) the directivity and the ratio of power output relative to 1W:  In this formula, and in many similar, it is always understood that Pa is relative to unity with the proper unit assigned i.e. 1W. So “Pa” is really an abbreviation for “Acoustic output power relative to 1W” just like “DI” is short for directivity relative to an omni-directional source. The transmit response to voltage, TRV, is defined in such a way that the source level can be calculated from:  The TRV value is, however, often measured at low power and since the electric-to-acoustic efficiency can drop significantly with increased power levels it is often best to used the TRV relation with caution. It should be emphasized that the number and term source level refers to an acoustic source, not to the level of a particular sound, and that a source level is merely a practical definition. The source level of a transmitter can be estimated (ignoring attenuation) by measuring the output voltage of a hydrophone submerged in the vicinity of the transmitting transducer, see the sketch below.  For an example, the hydrophone has a receive response RR=-190dB re 1V/μPa with an open circuit (output) voltage OCV=2.4Vrms on its terminals. This means that the intensity level at the hydrophone is IL=20log(2.4Vrms) - (-190dB re 1V/μPa)=197.6dB re 1μPa and if the distance between the hydrophone and the transmitter is r=4m the source level is calculated from SL=IL+ 20log(4m/1m)=209.6dB re 1μPa @ 1m. |
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