Geology Reference
In-Depth Information
Figure 5.3 Generation of sinusoidal waves by an anticlockwise-rotating
arm. See text for explanation of symbols.
by π/ 2 radians, and the sine wave is said to lag the cosine wave by π/ 2
radians (it looks, at first sight, as if it should be the other way round, but
what the graphs actually show is that the cosine wave reaches its maximum
amplitude a quarter of a cycle before the sine wave).
A completely general sinusoidal wave is described by the equation:
z = a · sin( ω t + ϕ )
where ϕ is the phase angle . Any curve of this type can be resolved into
separate sine and cosine waves with amplitudes s and c related to the
original amplitude and phase by the equations:
= ( s 2
c 2 )
a
+
and
tan
ϕ =
s
/
c
These equations can be derived from the picture in Figure 5.3 of a rotating
arm, but in this case OP would begin its rotation (at time t = 0) with an
initial phase angle ϕ , which translates into a difference in time equal to ϕ/ω .
A minor complication is that in some geophysical applications (notably in
seismic processing), phase angles are referenced to the symmetrical cosine
wave rather than the anti-symmetric sine wave.
In EM surveys, the induced currents and their associated secondary mag-
netic fields differ in phase from the primary field and can therefore be
resolved into components that are in-phase and 90 (or π/ 2 radians) out of
phase with the primary. The out-of-phase component can also (more accu-
rately and less confusingly) be described as being in phase-quadrature with
the primary signal. The fact that the two components can be represented by
vectors drawn at right angles (orthogonal) to each other also allows them to
be described in terms of the mathematics of complex numbers, which uses
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