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

142 Chapter 3 The trigonometric functions
Now the binomial expansion for √ (1 +x)is (see Section 6.4)
√
(1+x)=(1+x) 1/2 =1+ x 2 − x2
8 + x3
16 −···≈1+x 2 , if|x|<1
Usingthisexpansionintheexpressionfor φ andnotingthatthemoduliofboth(h t
+
h r
)/d and (h t
−h r
)/d arelessthan 1,we have
φ ≈ 2πd (1 + (h t +h r )2
−1− (h t −h )
r )2
λ 2d 2 2d 2
Expanding the bracketed termsgives
φ ≈ 2πd
(
h
2
t
+2h t
h r
+h 2 r −h2 t
+2h t
h r
−hr)
2
λ 2d 2
So
φ ≈ 4h t h r π
λd
This is a simplified approximation for the phase difference between the direct wave
andthereflectedwave.Notethatitdependsontheheightofthetransmitter,theheight
of the receiver and the distance between the transmitter and the receiver.
Thiscalculationisimportantbecauseundersomeconditionsthephasedifference
between the two paths means that the directed and reflected waves destructively
interfere.Inseverecasesthiscausesthesignaltodecreaseatthereceiverenoughso
that the communications link is lost. The effect is often termed multipath-induced
fading.
EXERCISES3.7
1 Statethe amplitude,angularfrequency, frequency,
phase angleandtime displacement ofthe following
waves:
(a) 3sin2t (b) 1 sin4t
2
(c) sin(t +1)
(d) 4cos3t (e) 2sin(t −3) (f) 5cos(0.4t)
(g) sin(100πt) (h) 6cos(5t +2) (i) 2 3 sin(0.5t)
(j) 4cos(πt −20)
2 Statethe period of
(a) 2sin7t (b) 7sin(2t +3)
(c) tan t 2
(e) cosec(2t −1)
(d) sec3t
( )
2t
(f) cot
3 +2
3 Avoltage sourceproduces atime-varying voltage,
v(t),given by
v(t) =15sin(20πt +4)
(a) Statethe amplitudeof v(t).
t 0
(b) Statethe angularfrequency of v(t).
(c) Statethe periodof v(t).
(d) Statethe phaseof v(t).
(e) Statethe time displacement of v(t).
(f) Statethe minimumvalue of v(t).
4 Asinusoidalfunctionhas an amplitude of 2 3 anda
periodof2. Stateapossibleform ofthe function.
5 Statethe phase angleandtime displacementof
(a) 2sin(t +3) relative to 2sint
(b) sin(2t −3) relative to sin2t
( )
t
(c) cos
2 +0.2 relative to cos t 2
(d) cos(2 −t)relative to cost
( )
3t+4
(e) sin relative to sin 3t
5 5
(f) sin(4 −3t) relative to sin3t