Chapter 4. Vibration, Wave motion and SoundChapter 4Vibration,
wave motion and soundVibration, wave motion and sound4.1 Simple
harmonic motion (SHM)4.1.1 Equation of SHM1. Definition of
SHMSimple harmonic force: he !orce on a bod" i# proportional to it#
di#placement !rom theori$in and al%a"# directed to%ard# the ori$in.
&! %e choo#e the direction o! di#placement a#the x'a(i#, the
e)uation i# $iven b"F * ' k x, (4.1)the minu# #i$n denote# that the
!orce i# a re#torin$!orce and al%a"# point# to the ori$in (x =
+).SHM &! a bod" move# in a #trai$ht line under the #imple
harmonic !orce, the motion o! the bod" i# called simple harmonic
motion.,i$. 4.1 vibrational motion. Equation of SHM-enerall" a
Hoo.e/# la% #prin$ #ati#!ie# the e)uation (4.1), but k i# called
spring constant. &! a bod"/# ma## i# m and it i# e(erted b" a
#imple harmonic !orce, it# e)uation o! motion can be obtained b"
u#in$ 0e%ton/# #econd la% o! motion11dtx dm ma F .2n the other
hand, con#iderin$ e). (4.1), %e have kxdtx dm 11orxmkdtx d
113e!ine1 * k/m and %e have +111 + xdtx d (4.1)hi# i# the
di!!erential e)uation o! the #imple harmonic motion. &t#
#olution can be e(pre##ed a# ) co#( + t A x(4.4)he motion de#cribed
b" a co#ine or #ine !unction o! time i# called Simple Harmonic
Motion.&t i# nece##ar" to point out that the t%o de!inition#
!or SHM are the e)uivalent. 2ne i# !rom the !orce t"pe and the
other i# !rom the e)uation o! motion.3i!!erentiatin$ the e)uation
(4.4) %ith re#pect to t, the velocit" and acceleration o! the SHM
can be obtained.45xChapter 4. Vibration, Wave motion and Sound + )
#in ( ) (co# ) (co# )6 co#( 7AdtdddAdtdAt Adtddtdxv(#et + t)
) #in( + t A(4.4)) co#( ) (co# ) (#in ) (#in111 + t
AAdtdddAdtdAdtdvdtx da x1 (4.8)&t canbeprovedthat
thee)uation(4.8) i# e)uivalent tothe e)uation# (4.1)
and(4.4).here!ore the e)uation (4.4) i# indeed the #olution o!
(4.1).4.1. !he characteristic quantities of SHM&n the e)uation
o! SHM, 9, and are con#tant# and an" individual SHM can be
determined b" them.1. 9 i# called "mplitude #$. &t i# the
ma(imum di#placement o! a vibratin$ bod" !rom e)uilibrium
po#ition.. %eriod and frequenc&!heperiod, denotedb"T, i#
thetimeta.en!or acompletevibration%hichi#independent o! the
po#ition cho#en !or the #tartin$ point o! the complete
vibration.!he frequenc&, denoted b" f, i# the number o!
complete vibration# per #econd, it i#the reciprocal ()o! the
periodTf1(4.5)he an$ular !re)uenc" or an$ular velocit" i# de!ined
a#Tf 11 (4.:)'. %hase and initial phase #$&n the e)uation o!
SHM, t + i# called the phase o! SHM, %here i# the pha#e at t * +,
called initial phase (unit radian). 9t t * +, e)uation# (4,4) and
(4.4) become# re#pectivel" #inco#++A vA x (4.;)S)uarin$ both #ide#
o! the above e)uation#, the amplitude o! the SHM can be !ound1 1
1+co# A x 1 1 1 1+#in v A 1 1 1 111+ 1+) #in (co# A Avx + + kv mxvx
A1+ 1+11+ 1++ + . (4. 1+'1 .$ i# in SHM at the end o! a #prin$ %ith
#prin$ con#tant k * 8+.+ 0?m.he initial di#placement and velocit"
o! the particle i# 4.++ > 1+'1 m and @1.41 m?#re#pectivel".
Calculate (1) the an$ular !re)uenc"A (1) the initial pha#eA (4) the
amplitude o! the vibrationA (4) the periodA (8) the
!re)uenc".Solution: &n order to #olve the problem, %e have to
be clear %hat thin$# have been $iven in the problemB hat i# the
.no%n condition#.he )uantitie# %e .no% are: m = 1.++ > 1+'1 .$ .
* 8+.+ 0?m(+ * 4.++ > 1+'1 m v+ * @1.41 m?#0o% u#in$ the
!ormulae %e have learned, the problem can be #olved ea#il".(1).
&n order to !ind the an$ular !re)uenc", the !ormula
repre#entin$ the relation amon$ thean$ular !re)uenc", ma## and
#prin$ con#tant ha# to be u#ed. We have s radmk? + . 8+1+ ++ . 1+ .
8+1 (1). he initial pha#e o! the vibration can be !ound u#in$ the
initial di#placement and initialvelocit". 9t t * +, %e .no%(+ * 9
co# * 4.++ > 1+'1 m v+ * ' 9 % #in * '1.41 m?#he can be obtained
b" #olvin$ above e)uation#. 2n the other hand, it can be
calculateddirectl" b" e). (4.1+)4 . 41+ . 8+ 1+ ++ . 441 . 1arctan
arctan1++
,_
,_
xv(4). he amplitude can be calculated b" the !ormulamvx A111+
1+1+ ++ . 4 + (4). he period can be !ound throu$h the relation
bet%een the an$ular !re)uenc" and the periodAs T 115 . ++ . 8+1 1
(8). ,re)uenc" can be !ound a# f * 1?T * 1?+.115 * :.1+'1m and
@1.41 m?# re#pectivel". Calculate (1) the an$ular !re)uenc"A (1)
the initial pha#eA(4) the amplitude o! the vibrationA (4) the
periodA (8) the !re)uenc". (e(ample in lecture)nm Suppo#e that an
electron move# in the addition o! t%o vibration# %hich are alon$
('a(i# and "'a(i# re#pectivel" and then the vibrational e)uation#
are $iven a# 8;) co#() co#(1 11 1 + + t A #t A xChapter 4.
Vibration, Wave motion and Soundheir compo#itive orbit in ('" plane
i# Suppo#e that A1 *1, A1 *4 and 1 ' 1 * ?4. =lea#e tr" to(1) dra%
the electron path o! motion on ('" planeA (1) determine it#
direction o! motion (cloc.%i#e or anticloc.%i#e). H. Wave#m 9
#ource o! %ave move# in SHM. &t# e)uation o! motion i# s * +.+4
co#(1.8 t) (m). hi# %ave propa$ate# in a medium alon$ po#itive
('direction at the #peed o! 1++ m #'1. r" to !ind: (1) %ave
e)uation o! motionA (1) the di#placement and velocit" o! the point
ma## %hich i# 1+ meter# a%a" !rom the %ave #ource at the time o!
1.+ #econd a!ter the %ave #ource #tart# it# motion. lm lm here are
t%o coherent %ave #ource# propa$atin$ in the #ame medium. heir
!re)uenc" i# 55+HC, their amplitude i# A * +.8m and the propa$atin$
velocit" i# 44+ m?#. he t%o %ave# inter!ere at point =. (1). ,or
t%o #ource# %hich are in pha#e, calculate the amplitudeat = %hen 9=
* 11 m and H= * 18 mA (1) !or the t%o #ource# %hich are out o!
pha#e, calculate the amplitude at the #ame point =.8. 9n ob#erver
#tandin$ on the rail%a" #ide hear# a train movin$ a%a" at the #peed
o! 4+.; m?# %ith a horn !re)uenc" o! 4:8HC. &t i# .no%n that
the velocit" o! #ound i# 44+ m?# in air. ,ind the ori$inal
!re)uenc" o! the horn on the train. 8
