Uncertainty of Speed of Sound in water, using Mackenzie's Equation

[Lloyd_Towler]

Homework Statement

Uncertainty of Speed of Sound in water, using Mackenzies Equation

Relevant Equations

c = 1448.96 + 4.591T - 5.304 x 10-2T2 + 2.374 x 10-4T3 + 1.340 (S-35) + 1.630 x 10-2D + 1.675 x 10-7D2 - 1.025 x 10-2T(S - 35) - 7.139 x 10-13TD3

T = temperature in degrees Celsius
S = salinity in parts per thousand
D = depth in metres
C (speed of sound) = 1436.30 m/s

Temperature= 7.5°C ±0.5
Salinity= 0 (no error)
Depth= 17.5cm ±0.5

I need to calculate the overall uncertainty of the value I have obtained for the speed of sound in water, using Mackenzies equation... I am not sure in what way to combine the above uncertainties. Any help would be greatly appreciated!

Lloyd

 

[gleem]

The standard approach is if c = f(x_i) =F(T,S,D) then the uncertainty in f is given by
$$
\Delta f = \sqrt{\sum_{I=1}^{n} \left (\frac{\partial f}{\partial x _{i}} \right ) ^{2}\Delta x_{i}^{2} }$$

with Δx₁ = the uncertaintly in T, Δx₂ = the uncertaintly in S, and Δx₃ = the uncertaintly in D.
The ∂f/∂x_i's are evaluated at the respecive datat points.

You know how to take a derivative, don't you?

 

[Lloyd_Towler]

Unfortunately, I don't have a clue how to take a derivative...

 

[gleem]

OK
For your equation, they are simple

Derivatives of constants are zero.

Derivatives of the form Xⁿ are ∂ Xⁿ /∂T = nX ^n-1

so for example, the ∂ T /∂T = 1 , ∂ T² /∂T = 2T , ...

See the pattern?

 

[Lloyd_Towler]

Sorry I'm not following... could you possibly give an example with a slightly simpler equation so I can try get my head round it?
Thanks for your help

 

[gleem]

Sure I can show you the general approach just substitute your own variables.

take f = aX + bY² + cZ³

using (oops I see iI didn't proof my post too well) should have been ∂ Xⁿ /∂x = nX ^n-1

Apply this to each part of f you get

∂f/∂X = ∂aX/∂x = a since Y and Z are independent of X

∂ bY² /∂Y= 2bY since X and Z are independent of Y

∂cZ³/∂z = 3cZ² since X and Y are independent of Z

Evaluate each at the data point you wish to determine the uncertainty and substitute their values along with the uncertainties of the variables into the equation for ΔX

 

[haruspex]

I strongly recommend, as a first step, writing the equation with symbols instead of all those numbers. See e.g. https://en.m.wikipedia.org/wiki/Speed_of_sound#Seawater.
This will be much easier to work with until you have the expression for the uncertainty. Only plug in the numbers then.

 

[gleem]

It looks ok except for your interpretation of the uncertainties. Use the actual uncertainty, not relative uncertainty.

The uncertainty to be used is the total value of the uncertainty in each variable so for temperature it should be the resolution plus any calibration uncertain in the instrument or accuracy stated by the manufacturer of the instrument and anything else that might affect the final value. The same for the depth. For the temperature, it should include resolution + thermometer + maybe estimates of errors introduced from reading the thermometer and anything. Since these can be considered random you may sum them in quadrature, i.e. the total thermometer uncertainty is $$
\Delta T = \sqrt{\left ( 0.5 \right )^{2}+ (accuracy)^{2} +\sum (\Delta ?)^{2
}}
$$

 

[Lloyd_Towler]

okay sweet, ill get onto that now. Thanks for all the help!

 

[Steve4Physics]

Relevant Equations:: c = 1448.96 + 4.591T - 5.304 x 10-2T2 + 2.374 x 10-4T3 + 1.340 (S-35) + 1.630 x 10-2D + 1.675 x 10-7D2 - 1.025 x 10-2T(S - 35) - 7.139 x 10-13TD3

Temperature= 7.5°C ±0.5
Salinity= 0 (no error)
Depth= 17.5cm ±0.5

I need to calculate the overall uncertainty of the value I have obtained for the speed of sound in water, using Mackenzies equation... I am not sure in what way to combine the above uncertainties. Any help would be greatly appreciated!

Are you required to provide a formal/rigorous calculation of the uncertainty? If not, here’s what I'd do.

Set-up a spreadsheet for calculating speed using the Mackenzie Equation with the values of T, D (and S if you want) in separate cells.

Use the spreadsheet to calculate the 4 speeds corresponding to the 4 different combinations of the ‘extreme’ values of T and D:
T = 7.0°C, D = 0.170m
T = 8.0°C, D = 0.170m
T = 7.0°C, D = 0.180m
T = 8.0°C, D = 0.180m

Take the uncertainty in speed as half the difference between the largest and smallest speeds (from the 4 calculated speeds).

(I sit back an await the insults!)

As an aside, the format "Depth= 17.5cm ±0.5" isn’t correct. Better would be "Depth= (17.5±0.5)cm". The "0.5" is the uncertainty in cm.

Edit: Are you sure the value for depth is really 17.5cm? Check in case it is 17.5m (which seems more likely).