WAVE MOTION Mechanical waves: for example sound wave, water wave, string wave -­‐  Caused by an ini=al disturbance at some loca=on. -­‐  Energy travels through media without the displacement of the ma@er. Types of waves Types of waves can be categorized by different means. (1)  Direc=on of par=cle mo=on Longitudinal waves (eg. Sound wave):-­‐ Wave propaga=on direc=on same as the par=cle mo=on direc=on. Transverse waves (eg, string):-­‐ Wave propaga=on direc=on is perpendicular to the par=cle mo=on direc=on. (2) Dimension (3)  Periodicity Periodical wave:-­‐ All par=cles in media are in periodic mo=on Single wave train (or wave pulse) (4)  Shape (eg. Sine wave, triangular wave, square wave) Travelling waves Consider a wave travels in +ve direc=on such that at =me zero: y(x, 0) = f (x)
At a later =me t, y(x,t) = f (x − vt)
Consider the mo=on of a par=cular point P (i.e. phase) of the wave at =me zero. At =me t, this par=cular phase P’ is at a distance vt away from P. v is called the phase velocity. For wave travelling to the leT side (i.e. –ve x direc=on) y(x,t) = f (x + vt)
Sinusoidal waves In par=cular if the wave form is sinusoidal func=on, y(x, 0) = f (x) = y sin 2π x
and thus
m
λ
2π
y(x,
t)
=
y
sin
( x − vt )
m
λ
-­‐ v is the phase velocity of the wave. -­‐  For a fixed =me t, y(x,t) repeat itself for every distance of λ, the wavelength. Define period T as the =me required at any par=cular point such that a complete cycle of wave mo=on is observed, i.e. λ = vT
2π
More defini=ons: k =
= wave number
λ
ω =2π f =
2π
= angular frequency
T
Travelling to the left: y(x, t) = ym sin ( kx − ω t ) or =ym sin (ω t − kx )
Travelling to the right: y(x, t) = ym sin ( kx + ω t ) or = ym sin "#− ( kx + ω t )$%
Transverse velocity of media par=cle Consider at some posi=on x, we monitor the ver=cal mo=on a the media par=cle, y(x, t) = ym sin ( kx − ω t )
∂y
vy (x, t) = = −ymω cos ( kx − ω t )
∂t
∂2 y
2
2
ay (x, t) =
=
y
ω
sin
kx
−
ω
t
=
ω
y
(
)
m
2
∂t
The par=cle is in ver=cal SHM with angular frequency of ω. Phase y(x, t) = y sin kx − ω t − φ
(
)
m
kx − ω t − φ = phase of the wave
φ = phase constant
Two waves with the same phase (or differed by 2nπ) is called in phase. ( " φ%
+
y(x, t) = ym sin *k $ x − ' − ω t k&
) #
,
(
" φ %+
y(x, t) = ym sin *kx − ω $ t + '# ω &,
)
(1)
(2)
Two interpreta=ons of the phase constant ϕ (1)  Consider at a fixed =me t (i.e. taking photo of the two waves) ( " φ%
+
y(x,
t)
=
y
sin
k
x
−
−
ω
t
m * $ ' - is a wave going ϕ/k ahead of y(x,
t)
= y m sin
[ kx
− ω t ] . k&
) #
,
(2) Consider at a fixed posi=on x (i.e. we observe the =me varia=on of y at this posi=on) (
" φ %+
y(x,
t)
=
y
sin
kx
−
ω
y(x,
t)
= y m sin
[ kx
−
ω
t ] leads m * $ t +
' - by a =me of ϕ/ω. # ω &,
)
The Wave Equa=on ∂2 y 1 ∂2 y
= 2 2
2
∂x
v ∂t
No=ce that y(x,t)=f(x±vt) is a solu=on to the wave equa=on. Let z = x ± vt = z(x, t) and y = f (z)
Pf: ∂y df ∂z df
=
=
∂x dz ∂x dz
∂2 y d " df % ∂z d 2 f
= $ ' =
∂x 2 dz # dz & ∂x dz 2
∂y df ∂z
df
=
= ±v
∂t dz ∂t
dz
2
∂2 y d " df % ∂z
2 d f
= $ ±v ' = v
∂t 2 dz # dz & ∂t
dz 2
∂2 y 1 ∂2 y
Thus implying 2 = 2 2
∂x
v ∂t
y(x,
t)
= y m sin
[ kx
−
ω
t ] is one of the func=on sa=sfying the wave equa=on. Transverse wave on a stretched string Considering the y-­‐component of the force ac=ng on the segment δx: ∑F
y
= F sin θ1 − F sin θ 2 ≈ F ( tan θ1 − tan θ 2 )
% ∂y (
∵tan θ i = ' *
& ∂x )x1
% ∂y (
∴ ∑ Fy = Fδ ' *
& ∂x )
By Newton's 2nd law:
∂2 y
∑ Fy = δ may = µδ x ∂t 2
Combining with (1):
(1)
(µ =linear density of string)
% ∂y (
∂2 y
Fδ ' * = µδ x 2
& ∂x )
∂t
% ∂y (
δ' *
2
& ∂x ) µ ∂ y
=
δx
F ∂t 2
∂2 y µ ∂2 y
=
∂x 2 F ∂t 2
The wave equation with : v =
F
µ
Energy of transverse wave on string Consider a transverse sine wave transmiang through a string. For the par=cular segment dx at posi=on x, it is undergoing SHM in y-­‐direc=on: y(x, t) = y sin [ kx − ω t ]
m
Mass of the segment: dm = µ dx
KE of the segment: 2
1
1
2
"
$
dK = dmvy = µ dx #−ω ym cos ( kx − ω t )%
2
2
dK 1
dx
∴
= µω 2 ym2
cos2 ( kx − ω t )
dt 2
dt
1
= µω 2 ym2 v p cos2 ( kx − ω t )
2
vP = phase velocity
(1)
PE of the segment is the work done by the force F of stretching the string from its equilibrium length to its present status " dx 2 + dy 2 − dx $
dU
=
F
dl
−
dx
=
F
(
)
#
%
2
"
$
'
*
∂y
= Fdx - 1+ ) , −1.
( ∂x +
-#
.%
n
Binomial expansion: (1+ z ) ≈ 1+ nz
2
* 1 # ∂y &2 - 1
# ∂y &
dU ≈ Fdx ,1+ % ( −1/ = Fdx % (
$ ∂x '
,+ 2 $ ∂x '
/. 2
2
dU 1 dx *
1
∴
= F + ym k cos ( kx − ω t )-. = FvP ym k 2 cos2 ( kx − ω t )
dt 2 dt
2
1
= µω 2 ym2 vP cos2 ( kx − ω t )
2
dK
=
dt
E =U + K
dE dU dK
⇒
=
+
= µω 2 ym2 vP cos2 ( kx − ω t )
dt
dt
dt
In general, dE/dt ≠ 0, energy not constant with respect to =me. No=ce it is not an isolated system. There is work done by the neghboring string segment. ! dE $
dE
As P =
and Pav = # &
" dt %av
dt
* T cos2 kx − ω t dt .
(
) , 1 2 2
⇒ P = µω 2 y 2 v ,+ ∫ 0
/ = µω ym vP
av
m P
T
,,0 2
Intensity of wave is defined as: Pav
A
A = area of wavefront for 3D-wave.
I=