082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

140 Chapter 3 The trigonometric functions
from which
phase angle = φ =
2π ×horizontal shift
λ
This result is important in the engineering application that follows because, more generally,
when any two waves arrive at a receiver it enables the difference in their phases,
φ, tobe calculated from knowledge of the horizontal shift between them.
Thepresenceof φ iny =Asin(kx + φ)causesahorizontal(left)shiftof φ k = φλ
2π .
Note that adding any multiple of 2π onto the phase angle φ will result in the same
graphbecauseoftheperiodicityofthesinefunction.Consequently,aphaseanglecould
(
be quoted as φ + 2nπ. For example, the wave sin 2x + π )
is the same wave as
3
sin
(2x + π )
3 +2π , sin
(2x + π )
3 +4π and so on. Normally, we would quote a value
of the phase thatwas lessthan 2πby subtracting multiples of 2πasnecessary.
Engineeringapplication3.5
Two-raypropagationmodel
Itisveryusefultobeabletomodelhowanelectromagneticwaveemittedbyatransmitter
propagates through space, in order to predict what signal is collected by the
receiver. This can be quite a complicated modelling exercise. One of the simplest
models is the two-ray propagation model. This model assumes that the signal received
consists of two main components. There is the signal that is sent direct from
the transmitter to the receiver and there is the signal that is received after being reflectedoff
the ground.
Figure 3.19 shows a transmitter with height above the groundh t
together with a
receiver with height above the groundh r
. The distance between the transmitter and
the receiver alongthe ground isd.
T
S
h t
Transmitter
O
Ground
d d
d r
d
A
R
Receiver
P
h r
Q
Figure3.19
Atransmitterandreceiver at
different heightsabove the
ground.
Note thatthe quantitiesh t
,h r
andd areall known.