Physics 214Physics 214
2: Waves in General
•Travelling Waves •Waves in a string•Basic definitions •Mathematical representation
•Transport of energy in waves•Wave Equation•Principle of Superposition•Interference
•Standing waves
Propagating vibrations
Forcing vibration
•Material of string is vibrating perpendicularly to direction of propagation•TRANSVERSE WAVE
•If the vibrations were in same direction•LONGITUDINAL
•Each part of vibration produces an oscillating force on string atoms & molecules, which cause neighboring atoms to vibrate
wavelength
amplitude A
frequency number of waves passing a fixed point in one second
cpsHz
period T = time taken for one wave to pass a fixed point
T=1
T s; m; A m
speed of wave v =
y
x
y = f(x,t) = ymsin(kx-t)
the position, x, of points on the wave are functions of time i.e. x = x(t)
phase
consider points of a fixed amplitude
y fixed y x , t ym sin kx t for these points
kx tconstant
as t increases x must increase
kdx
dt- = 0 kv = v =
k phase velocity
If the wave is propagating to left
y x, t ym
sin kx t
v k
Energy Transport
If the waves are of small amplitude Hookes Law holds
F = -ky k is the force constant of string medium and the waves are made up of propagating
simple harmonic vibrations Linear Waves
each string element of mass dm has K. E.
K 12dm
yt
2
=12
dx 2ym2cos 2 kx t
where yt
= ymcos kx t & is mass per unit length
dKdt
12
v 2 ym2cos 2 kx t
dK
dt one cycle
1
2v2y
m2 cos2 kx t
1
4v2y
m2
dU
dt one cycle
where U is potential energy
dEtotaldt one cycle
1
2v2ym
2 1
2v2A2 power transmitted
power transmitted A2 &2
From the theory of SHM of a mass connected to a spring
=kdm
kdx
dEtotaldt one cycle
dx 1
2vkA2
i.e. average power transmitted per unit length=1
2vkA2
For transverse waves (e.g. strings) that are
LINEAR, propagating vibrations perform SHM one gets
by differentiating
2y
t22Asin kx t and
2y
x 2 k 2A sin kx t
2y
t22
k 2
2y
x2v2
2y
x2
Linear Wave Equation
SUPER POSITION PRINCIPLE
Ftotal Fi ky
i k yi kytotal
for 2 waves
ytotal x, t y1 x, t y2 x, t
ytotal x, t A1 sin k1x 1t 1 A2 sin k2x 2t 2 e.g. let A1 A2 A; k1 k2 k; 1 2 ; 1 0; 2
ytotal x, t A sin kx t sin kx t
2Acos2
sin kx t
if Atotal 2A cos2
0
DESTRUCTIVE INTERFERENCE
if 0 Atotal2A cos0
2
2A
CONSTRUCTIVE INTERFERENCE
BEATS
ytotal x,t y1 x, t y2 x,t
ytotal x, t A1 sin k1x 1t 1 A2 sin k2x 2t 2 let A1 A2 A; 1 2 0
ytotal 2Acosk
1 k
2 2
x
1
2
2
t
sin
k1 k
2 2
x
1
2
2
t
2Acos kbx
bt
modulated amplitude beat
sin k
sx
st
interference wave
b
1
2
2
2; k
bk
2
STANDING WAVES
produced by interference of waves travelling
in opposite directions
ytotal x,t y1 x, t y2 x, t ytotal x, t Asin kx t Asin kx t
2A sinkx cost This is not a travelling wave-- -different form
Amplitude of standing wave
2Asin kx note that it varies with x
The amplitude is zero = positions of NODES
sinkx 0 kx 0, ,2 , ,,n,
xn 2
The amplitude is a max. = positions of ANTINODES
x n 1 2
Standing waves are Standing waves are formed by formed by
incident wave + incident wave + reflected wavereflected wave
•Length of the string must be half Length of the string must be half integer multiples of the wavelength integer multiples of the wavelength
L 1
21 2
3
2 3 2 4
Ln
2 n
•The wave with wave length 1 is called the •FUNDAMENTAL wave•The the other waves are called• OVERTONES or HIGHER HARMONICS•2 is called the •First Overtone•Second Harmonic
•3 is called the •Second Overtone•Third Harmonic
speed s
; =tension in string
1 s
1
s2L
2 s
2
s
L21
3 s
3
s2L3
3s
2L3
1
n s
2Ln
ns2L
n s
2L
n1
•Frequency of a HARMONIC Frequency of a HARMONIC FAMILY of standing waves is FAMILY of standing waves is • 33nn......................•HARMONIC SEQUENCEHARMONIC SEQUENCE•The overtone level is The overtone level is characterized by the number of characterized by the number of NodesNodes
•standing wave frequencies in string depend on
•geometry of string •length: L
•inertial property•density: •elastic property •tension:
Every object can vibrate in the form of standing waves,
whose frequencies form harmonic families and are characteristic of the object
and depend on the geometry, inertial and elastic properties
of the object i.e. on the geometry and forces (external and internal) experienced by
the object.
•A forcing vibration can make an object vibrate and produce waves in the object
•These waves have the frequency of the forcing vibration
•These waves will die out unless they can form standing waves
•i.e are vibrating at the natural frequencies of the object
•When this is the case most energy is transferred from the forcing vibration to the object
•Then the amplitude of the standing waves increases
•RESONANCE
