Jeulin 223 008 Mode D'emploi page 39

Ultrasounds
Moduson table
Ref :
223 008
Note: The graphical study can also be performed by using a graphic calculator.
In worst-case conditions, the values of a and D are a=8 cm and D=15 cm.
Our working domain is comprised between these limits and in this case.
The representative curve of Φ(θ) versus θ yields the following graph:
31,2
30,8
30,6
30,4
30,2
29,8
In worst-case conditions, the values of a and D are, a=2 cm and D=30 cm.
Then the representative curve of Φ(θ) versus θ yields the following graph:
60,035
60,025
60,015
60,005
59,995
ENGLISH
•
Approximations
o The computerised study of the denominator Φ(θ) of δ by means of ACTILAB or of
a spreadsheet, e.g. EXCEL, yields the denominator variations in the experimental
conditions.
o varies at most between –60 deg. and +6 deg. (-π/3 and +π/3)
φ(θ)
31
30
−π/3 −π/6
o For the value 2D=30 cm, the relative deviation (Φ(θ)-2D)/2D is 1.05/30 or 3.5%
maximum at the angular abscissa θ=0. The relative deviation drops to approx.
0.8/30 or 2.7% for θ=+π/6 or -π/6 and to 0.3/30 or 1% for θ=+π/3 or -π/3.
φ(θ)
60,03
60,02
60,01
60
−π/3 −π/6
o For the value 2D=60 cm, the relative deviation is 0.033/60 or 0.055% maximum
at the angular abscissa θ=0.
o The conclusion is that 2D can be reasonably substituted for Φ(θ) with an error
that remains acceptably low.
0 +π/6
D=15 cm ; a=8 cm
0 +π/6
D=30 cm ; a=2 cm
δ θ
≈
a sin( )
36
θ
+π/3
θ
+π/3