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

3.7 Modelling waves using sint and cost 139
y
1
–2p
–p
0
p
2p
x
–1
Figure3.17
A graphofthe wavey = sin2x.
(
Example3.14 Figure 3.18 shows a graph ofsin 2x + π )
.
3
(a) State the phase angle.
(b) BycomparingFigures3.17and3.18weseethattheintroductionofthephaseangle
has caused a horizontal shift of the graph (tothe left).Calculate thisshift.
y
1
–2p
–p 0
p 2p x
–1
p –6
Figure3.18
Agraph(
ofthe wave )
y=sin 2x + π .
3
(
Solution (a) By comparing sin 2x + π )
with sin(kx + φ) wesee thatthe phase angle is π 3
3 .
(
(b) By writingy = sin 2x + π ) (
as sin2 x + π )
we note that this isy = sin2x
3 6
shifted tothe left by a horizontal distance π 6 units.
The results of this example can be generalized. The wave y = Asin(kx + φ) can be
writteny = Asink(x + φ/k) so that a phase angle of φ introduces a horizontal shift of
length φ/k. (Compare thiswith the expression fortime displacement inSection3.7.)
Noting that λ = 2π k ,thenk=2π and we may write Asin(kx + φ) equivalently
( )
λ 2πx
as Asin
λ + φ . Again, using k = 2π λ , the horizontal shift, φ , may similarly be
k
written as
horizontal shift = φ k =
φ
2π/λ = φλ
2π