Doppler Log general formula with only one transducer-
A transducer is fitted on the ship’s keel which transmits a beam of acoustic wave at an
angle c usually 60° to the keel in the forward direction, this gives the component “ v cos
𝜶” of the ship’s velocity towards the sea bed thus causing the Doppler shift and the
received frequency
fr = ft ( c + v Cos 𝜶)/(c - v Cos 𝜶.)
Acoustic beam transmitted at an angle a towards seabed.
If the waves are transmitted directly towards the seabed perpendicular to the keel, there will be no Doppler shift and the transmitted and received frequency will be the same. This is because the component of ship’s speed towards the seabed is zero.
Dividing numerator and denominator of equation (i) by C, we get
fr = ft {1 + (v cos α)/c} x 1/( (1 - v cos α)/c)
By Binomial expansion Theorem, we have
Hence,
since v cos α << C, neglecting higher powers of v cos α/c we get,
fr = ft + 2 v ft cos α / c
fr-ft = 2 v ft cos α / c ................(ii)
v=C (fr - ft) / 2 ft Cos a ............... (iii)
With the help of this formula we can calculate the speed of the ship, considering that there is no vertical motion.
Janus Configuration-
In practice the ship has some vertical motion and the Doppler shift measurement will have a component of this vertical motion. In this case Doppler shift measurement will be
fr-ft = 2 v ft cos α / c + 2 Vv ft sin α / c ................
fr-ft = (2 v ft cos α + 2 Vv ft sin α) / c ................ (iv)
where Vv represents the vertical motion of the ship.
This problem is overcome by installing two transducers, one transmitting in the forward direction and another in the aft direction at the same angle. This arrangement is known as Janus configuration.
In this case the forward transducer will give Doppler shift
i.e. frf - ft = 2 v ft cos α / c + 2 Vv ft sin α / c
frf - ft = (2 v ft cos α + 2 Vv ft sin α) / c ................(v)
where frf is the frequency received by the forward transducer while the aft transducer will have the component “v cos α” with negative sign since the transducer is moving away from the reflecting surface i.e. the seabed and hence the Doppler shift measured will be,
fra- ft = - 2 v ft cos α / c + 2 Vv ft sin α / c
fra- ft = (- 2 v ft cos α + 2 Vv ft sin α) / c ................ (vi)
where fra represents frequency received by the aft transducer.
In formula (v) and (vi) Vv will have the same sign since both the forward and aft
transducers will move upwards or downwards together. By measuring the difference
between the two Doppler shift frequencies, the vertical component will cancel out while
the horizontal will add hence,
(frf - ft) - (fra - ft) = (2v ft cos α + 2Vv ft sin a )/c - (-2 v ft cos α + 2 Vv ft sin α)/c
(frf - ft - fra + ft) = 2v ft cos α / c + 2Vv ft sin α/c + 2 v ft cos α/c - 2Vv ft sin α/c
frf - fra = 4 v ft cos α/c
v = c (frf - fra) / 4 ft cos a ...........(vii)
Above formula is used for calculating vessel's speed whe using Janus Configuration.