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

138 Chapter 3 The trigonometric functions
Equating coefficients ofcosωt and then sinωt gives
a=Rcosθ (3.7)
b=Rsinθ (3.8)
Squaring these equations and adding gives
that is
a 2 +b 2 =R 2
R = √ a 2 +b 2
Division of (3.8)by (3.7) gives
b
a =tanθ
as required.
Note that this example demonstrates that adding two waves of angular frequency ω
forms another wave having the same angular frequency but with a modified amplitude
and phase.
3.7.2 Wavelength,wavenumberandhorizontalshift
The sine and cosine waves described earlier in this section hadt as their independent
variable because the waves commonly met in engineering vary with time. There are
occasionswheretheindependentvariableisdistance,xsay,andinthiscasesomeofthe
terminology changes.Consider the wave
y =Asin(kx+φ)
Asbefore,Aistheamplitudeofthewave.Thequantitykiscalledthewavenumber.It
playsthesameroleasdidtheangularfrequency, ω,whent wastheindependentvariable.
The length of one cycle of the wave, that is the wavelength, commonly denoted λ, is
related tokby the formula λ = 2π . The phase angle is φ and its introduction has the
k
effect of shiftingthe graph horizontally.
Example3.13 Figure 3.17 shows a graph ofy = sin2x.
(a) State the wave numberforthiswave.
(b) Findthe wavelength ofthe wave.
(c) State the phase angle.
Solution (a) Comparingy = sin2x withy = sinkx wesee thatthe wave number,k, is2.
(b) The wavelength, λ = 2π = π. Note by observing the graph that this result is
k
consistent inthat the distance required for one cycle of the wave is π units.
(c) Comparingy = sin2x with sin(kx + φ)we see thatthe phase angle, φ, is 0.