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

3.7 Modelling waves using sint and cost 131
3.7 MODELLINGWAVESUSINGSINt ANDCOSt
Examiningthegraphsofsinxandcosxrevealsthattheyhaveasimilarshapetowaves.
In fact, sine and cosine functions are often used to model waves and we will see in
Chapter 23 that almost any wave can be broken down into a combination of sine and
cosinefunctions.Themainwavesfoundinengineeringareonesthatvarywithtimeand
sot isoften the independent variable.
The amplitude of a wave is the maximum displacement of the wave from its mean
position.So,forexample,sint andcost haveanamplitudeof1,theamplitudeof2sint
is2,and the amplitude ofAsint isA(see Figure 3.15).
The amplitude ofAsint isA. The amplitude ofAcost isA.
AmoregeneralwaveisdefinedbyAcos ωt orAsin ωt.Thesymbol ω representsthe
angularfrequencyofthewave.Itismeasuredinradianspersecond.Forexample,sin3t
hasanangularfrequencyof3rads −1 .Ast increasesby1secondtheangle,3t,increases
by 3 radians. Note thatsint has anangular frequency of1rad s −1 .
The angular frequency ofy =Asin ωt andy =Acos ωt is ω radians per second.
The sine and cosine functions repeat themselves at regular intervals and so are periodic
functions. Looking at Figure 3.7 we see that one complete cycle of sint is completedevery
2πseconds.Thetimetaken tocomplete one fullcycle iscalled theperiod
andisdenotedbyT.Hencetheperiodofy = sint is2πseconds.Similarlytheperiodof
y = cost is2πseconds.Mathematicallythismeansthataddingorsubtractingmultiples
of 2πtot does notchange the sineor cosine of thatangle.
sint=sin(t±2nπ)
cost=cos(t±2nπ)
n=0,1,2,3,...
n=0,1,2,3,...
Inparticular we note that
sint = sin(t +2π)
cost = cos(t +2π)
We now considery =Asin ωt andy =Acos ωt. Whent = 0 seconds, ωt = 0 radians.
Whent = 2π ω seconds, ωt = ω2π = 2π radians. We can see that ast increases from 0
ω
f (t)
A
2
A sin t
2 sin t
–2
2p
t
–A
Figure3.15
The amplitude of f (t) =Asint isA.