Marine Acoustics Formula Guide James Campbell September 15, 2016

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Introduction

This document is intended as a quick reference guide and introduction to the underwater acoustic calculations commonly used by ecologists. It is recommended that you spend some time trying to understand how the formulas are derived, as this will reduce the chance that you make errors during your analysis. The relevant matlab commands are shown in gray boxes below their respective equations.

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Decibels  10 · log10

P P0

 = dB

(1) P = Power dB = Decibel

A Decibel is a logarithmic ratio of power quantities (1). An example of a power quantity is Sound Intensity. As pressure and velocity measurements are field quantities, we convert them into power quantities by squaring them. Sound Pressure Level (dB re 1µPa):    2 p p 20 · log10 = 10 · log10 = SPL p0 p20 SPL = 20*log10(P/1) = 10*log10(P^2/1)

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(2)

Sound Velocity Level (dB re 1m/s):    2 u u = SVL 20 · log10 = 10 · log10 u0 u20

(3)

p = pressure p0 = reference pressure u = particle velocity u0 = reference particle velocity SVL = 20*log10(u/1) = 10*log10(u^2/1)

2.1

Convert between Decibels and Linear Units

The proof below shows how to convert from Decibels to Field quantities. As an example, we’re converting from SPL (dB ref 1µPa) to pascals (Pa)  p2 10 · log10 = SPL p20  20  p = SPL log10 p20 0  20 ! p log10 = SPL p0  20 p SPL 10 = p0 1  20 ! 20
1 p SPL 20 10 = p0 SPL p 10 20 = p0 

p0 · 10

SPL 20

p = 1*10^(SPL/20)

2

=p

(4)

2.2

Commonly used acoustic metrics

A brief list of commonly used acoustic measurements using the dB scale. Zero-to-peak sound pressure (µPa): pz−p = max|p| Zero-to-peak sound pressure level (dB re 1µPa): ! p2z−p SPLz−p = 10 · log10 p20

(5)

(6)

zeroToPeak = 20*log10(max(abs(p))/1) Cumulative Energy (µP a2 s): Z

tn

p(t)2 dt

(7)

t1

Sound exposure (µP a2 ·s): Z

tn

E=

p(t)2 dt

(8)

t1

To calculate the single strike sound exposure (Ess ), set t1 and tn to the occurrence of the 5th and 95th percentile of the cumulative energy. Sound exposure level (dB re 1µP a2 ·s): SEL = 10 · log10

E E0

(9)

SEL = 10*log10((sum(p^2)/dt)/1) #dt = sample period

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Calculate SPL from Power Spectrum Density (PSD) results 1. Using Parseval’s theorem, we know the area under the curve of a PSD (re dB 1µP a2 /Hz ) graph is equal to the variance of the signal (SPL). 3

Before we can calculate the area under the curve, we’ll have to convert the PSD values across all frequencies from Decibels to linear power units P. P0 · 10

PSD vector 10

PSD vector =



= Pvector

P SD1Hz Pvector =

(10)

P SD2Hz 

P1Hz

... P SDnHz

P2Hz

... PnHz

 

2. Sum the power values across all frequencies then divide the result by the sample period (seconds) to get the cumulative power normalized to a 1Hz resolution. Z

fn

Pcumulative =

Pvector (f )df

(11)

f1

f = Frequency

Pcumulative = sum(Pxx)/df #df = frequency interval 3. Convert the cumulative power to Decibel units.  10 · log10

Pcumulative P0

 = SPL

(12)

SPL = 10*log10(Pcumulative/1)

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Predicting particle velocity from pressure measurements

In the acoustic far-field, particle motion and sound pressure hold a theoretically consistent relationship as sound behaves like longitudinal wave (as opposed to a spherical shaped wave in the near-field). PFV =

4

p ρ·c

(13)

p = Measured pressure (µP a) PFV = Predicted far-field velocity (m/s) ρ = Density of water (1027 kg/m3 ) c = Speed of sound in water (1484 m/s) To convert from measured SPL to predicted far field velocity level (PFVL dB re 1nm/s), we can do the following: 1. Convert particle motion predictions to dB.  20 · log10

u u0

 = PFVL

(14)

SVL = 20*log10(u/1)

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Calculating the PSD and SPL summed across multiple recordings

5.1

Sum power values across recordings

To prevent clipping (loss of information due to a signal being louder than a sensor or microphone is able to record), you can split a relatively wide bandwidth sound (200-2000Hz for example) into multiple sounds with smaller bandwidths (200-500Hz and 500-2000Hz ). You can then analyse each of these recordings separately and sum the results to get an accurate representation of what the combined SPL would have been if the two audio tracks were played together at the same time. For this example, were going to use an intended sound of 200-2000Hz which, for purposes of avoiding clipping while recording with our vector sensor, has been divided into two smaller recordings of 200-500Hz and 5002000Hz for playback and analysis. 1. Create PSD graphs from both your analysed recordings (make sure each analysis shares a common nfft and window length!): • Recording 1: 200-500Hz playback • Recording 2: 500-2000Hz playback Recordingn =

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P1Hz

P2Hz

... PnHz



[pxx,f] = pwelch(waveForm,hamming(1024),512,1024,fs) #fs = sample rate 2. If your PSD results are in the units of dB (re p20 /Hz ), then you must first convert this into linear power units as seen in 2.1. Remember that power is equal to squared pressure (P = p2 ). p20 · 10

PSD vector 10

=P

(15)

3. Combine the summed the power values from Recording1 and Recording2 . 500 X

! Recording1 (f )

+

2000 X

! Recording2 (f )

= Psummed

(16)

500

200

Psummed = sum(pxx1) + sum(pxx2) 4. You should now have a final PSD data set that holds the power of both sounds combined together. You can now convert you results back into the Decibel scale for reporting.  SPL = 10 · log10

Psummed P0

 (17)

SPL = 20*log10(Psummed/1) For the theoretical reasons to why this works, pareval’s theorem tells us that the energy summed accross the frequency spectrum of a PSD is equal to that of the root-mean-square energy of the time domain of that signal.

6 6.1

Working with particle motion Particle Acceleration

Particle acceleration, a, can be derived from particle velocity measurements, u. The relationship between the two quantities is dependant on frequency, f , and is shown in the equation below. a = u · (2 · π · f )

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(18)

a = u * 2 * pi * f To report particle acceleration as a Decibel, use Particle Acceleration Level (PAL) with a reference value of 1µm/s.   a 10 (19) P AL = 20 · log a0 PAL = 20*log10(u * 2 * pi * f / 1)

6.2

Comparing SVL to SPL

As SPL is defined as the root mean square sound pressure in Decibel units, SVL is defined in the same way. s Z 1 n rms(u) = [u(t)]2 dt n 1   rms(u) SVL = 20 · log10 u0

(20)

SVL = 20*log10(rms(u)/1) Particle motion is a vector quantity, as opposed to scalar. To compare omni-directional pressure measurements to particle motion, we can measure the particle motion along three orthogonal directions and then combine the results. q rms (uxyz ) = rms (ux )2 + rms (uy )2 + rms (uz )2   rms (uxyz ) SVL = 20 · log10 u0

(21)

where ux , uy , and uz are vectors of the instantanious particle velocity measured along the three axes of a vector sensor. SVL = 20*log10(sqrt(rms(ux)^2 + rms(uy)^2 + rms(uz)^2)/1)

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6.3

Excess SVL

If you’d like to directly report the proportional relationship between particle motion and sound pressure, an intuitive way to do this is to comare the observed ratio of the two components and compare this to what would be expected in open water conditions. We can then report this relationship in the Decibel scale, so it can be intuitively compared to SPL and SVL measurements (SVLExcess ). We’ll do this in three steps: 1. Measure both the particle motion summed accross the three axes (uxyz ) and sound pressure (p) in your setup. 2. Calculate the predicted far field velocity (P F V ). This is the expected amount of particle motion for a given pressure measurement in open water, far field conditions. 3. Subtract the expected (P F V ) particle motion form the measured (uxyz ) and report the result as a Decibel ratio. The following equation will give the Excess SVL: 

SVLExcess SVLExcess



    uxyz PFV = 20 · log10 − 20 · log10 u0 u0 u  xyz = 20 · log10 PFV

(22)

See equations 13 & 21 for calculating the necessary quantities used in the equation. If the resulting SVLExcess is 3 dB, this means that the observed particle motion summed accross all three axes is 3 dB higher than what would be observed in a far-field, open water condition where an equivalent sound pressure has been observed. This is a useful metric of you want to compare the acoustic conditions of a tank or basin setup to the theoretical conditions of the open water. The SVLExcess can be applied to any acoustic metric, such as SVL (rms(uxyz )) and SVLz−p (max(abs(uxyz ))).

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