 # Proper water properties calculator: vertical distance travelled by sound

## 0. Brief description. Very basic theory.

The speed of sound propagation in water depends on three parameters: temperature t, hydrostatic pressure P and salinity s. And almost every water body on the Earth never has a homogenous distribution of all of them. For example, when we use echosounder to estimate the depth of a water body, sound travels down and backwards through layers of water with different speed. As we know from elementary school, the distance travelled, in this case, can be calculated as a sum of velocities ν, multiplied by equal time intervals Δt:

s = ∑ν(ti) · Δt

Where every next time interval ti+1 is calculated according to ti+1 = ti + Δt.
To make measurement more precise we have to decrease the time interval Δt as small as possible, coming to the following idea:

s = lim ∑ν(ti) · Δt, Δt→0

Which can be rewritten in a more common way:

s = ∫ν(t) dt

It means that to estimate actual distance travelled by sound in the vertical direction we need to know the speed of sound in every point from the water surface to the bottom. Which is quite unrealistic. The best that we can have is a temperature/ salinity profile - a few measurements of water salinity and temperature taken in some points on different depths. So, the best we can do is to interpolate TS-profile and calculate the sum mentioned above. Corresponding speed of sound in a point with specified t and s (and implicitly, P) calculates according to the UNESCO Equation, that can be found in a work of Chen and Millero.

## 1. Calculation

To estimate the distance travelled by sound in the vertical direction (i.e. the depth of a water body, and more precisely, the distance between echosounder's transducer and the bottom), we need to measure propagation time τ and should have the TS-profile for the current point in the water body.
Table 1. Input values
Parameter Notation Value Range Units
Measured propagation time τ 0.001 .. 16 s
Number of time intervals Nt 2 .. 99999
Geographic latitude φ -90 .. 90 °

Table 2. TS-profile parameters
Parameter Notation Value Units
Number of points Ntsp
Z coordinate step Zs m

Table 4. Calculated distance travelled by sound in the vertical direction
Notation Value Units Description
sν=1500 m s = 1500 · τ
sνmean m s = νmean · τ
sνsurf m s = νsurf · τ
s m s = ∑ν(ti) · Δt

First three values in the table 4 are given for comparison.
- sν=1500 - is the distance, estimated assuming that speed of sound is constant and equals 1500 m/s.
- sνmean - is the distance, estimated assuming constant speed of sound, calculated for mean t and s.
- sνsurf - is the distance, estimated assuming constant speed of sound, calculated for t and s on the water surface.
- s - is the distance, estimated by interpolating TS-profile and calculated the sum, discussed in the top of this document.

This calculator is made with UCNLPhysics free and open source library.

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(C) Alek Dikarev, 2020
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