Transcript Document

WAVE PROPAGATION
Via an Elastic Medium
:
WAVE EQUATION
A. K. THAKUR
Objectives & Motivation
After elementary discussion on the basic concepts
related to wave motion our next discussion
1. ENERGY TRANSPORT IN WAVE MOTIoN
2. POWER TRANSPORTD BY WAVES
3. PROPAGATION OF LONGITUDINAL WAVES IN RODS
4. SETTING UP OF WAVE EQUATION IN ONE DIMENSION
5. ACOUSTIC WAVES & ITS EQUATION
6. EQUATION FORTRANSVERSE WAVES IN A STRING
Mechanical waves
Mechanical waves can appear when an initial disturbance is made to the mediums.
On a microscopic level, the forces between atoms in the mediums are responsible
for the propagation of the waves.
Elastic mediums are needed for the travel of mechanical waves.
The particles of the medium do not experience any net displacement in the direction
of the wave-as the wave passes, the particles simply move back and forth through
small distance about their equilibrium position.
The propagating oscillation in space transmit Energy, momentum, phase of the
created disturbance but not the particles. .
Waves transfer energy, NOT matter.
Typical Examples are : Sound waves, water waves, and the waves transmitting in a
guitar’s strings.
Wave Motion : Classification
WAVE MOTION
Mechanical Waves
E. M. WAVES
Traveling/Progressive Wave
Longitudinal Wave
Standing/Stationary Wave
Transverse Wave
Progressive waves – Disturbance
moves along the direction of wave propagation Waves can be characterized as
Standing waves - Disturbance
oscillates about a fixed point.
Transverse or Longitudinal.
Snapshot of Longitudinal Wave
l
x


y  A cos  2   
 l

y could refer to pressure or density
Longitudinal Traveling Waves
Longitudinal Traveling waves
Longitudinal waves:
The particles of the
disturbed medium
move parallel to the
wave motion
x  vt


y  A cos  2
 


l
Replace x with x-vt if wave moves to the right.
Replace with x+vt if wave should move to left.
Fixing x=0,
v


y  A cos  2 t   


l
Mechanism of Longitudinal Wave Propagation
Example of
Longitudinal
Sound Wave
Propagation:
Transverse Traveling Waves
Transverse Traveling Waves
particle
Transverse waves:
wave
The particles of the disturbed
medium move perpendicular to
the wave motion
Snapshot of a Transverse Wave
x


y  A cos  2   
 l

wavelength
x
Energy in wave motion
Wave transmits energy.
Energy in wave motion
Fig. a shows a wave traveling
along the string at times t1 and
t 2 ( a timeT later ).
4
A
B
y
(a)
uy
x
uy
time t1 time t 2
What do we want to calculate?
• dK/dt – the rate at which kinetic
energy is transported by wave.
• dU/dt – the rate at which
potential energy is transported.
dl
dx
(b)
dy
Kinetic Energy in Wave Motion
For dK/dt :
1
1
2
2
dK  dm  u y  ( dx ) [ ym  cos (k x   t )]
2
2
1 2 2
2
 ym  dx cos (kx  t )
2
dK 1
2 dx
2
2
  ym
cos (kx  t )
dt 2
dt v
dK 1
2
2
2
  ym v cos (kx  t )
dt 2
Potential Energy in Wave Motion
For dU/dt :
dU  F (dl  dx)  dl  (dx ) 2  (dy ) 2
dy 2
du  F[ (dx)  (dy)  dx]  Fdx[ 1  ( )  1]
dx
dy
2
2
The quantity dx is the slope of the string, and if the amplitude
of the wave is not too large this slope will be small.
1
using (1  z)  1  z       
2
1 y 2
1
y 2
du  F dx [1  ( )  1] 
F dx ( )
2 x
2
x
y
2
2
dx
 F  v   ( / k ) ;  v &   y m k cos( kx  t )
1/2
dt
x
dU 1
dK
2
2
2
  ym v cos (kx  t ) 
dt
2
dt
Total Mechanical Energy Transported by Wave
1
2
2
2
dK  dU   ym dx cos (kx  t )
2
2
2
2
dE  dK  dU   ym dxcos (kx  t )
Note that:
(a) dK and dU are both zero when the element has its maximum
displacement ( the element at relaxed length ).
(b) The total mechanical energy dE  dU  dK is not constant,
because the mass element is not an isolated system—
neighboring mass elements are doing work on it to change
its energy.
Power and Intensity of Wave Motion
Power: The rate at which mechanical energy is transmitted
 dE  dU  dK  2dK
dE
P
dt
Average power
Intensity I:
  ym v cos (kx  t )
2
2
Pav :
2
1
2
1 dE
2
Pav  
dt   ym v
T 0 dt
2
Pav
1
2 2
I 
OR I  v y m
A
2
For spherical wave:
T
1
Pav
1
I
 2 ; ym 
2
r
4r
r
Inverse Square Law in Wave Motion
Variation of Intensity & Power for Spherical Waves Obey Inverse Square
Law
P
I  Power / Area 
A
P

2
4r
Power from sound source decreases with distance squared
Energy Transport in Standing Waves
For standing waves, the energy can not be transported along it, because the
energy cannot flow past the nodes, which are permanently at rest.
U k
U k
U k
U k
Propagation of Elastic Waves
Elastic Rod
Medium Elasticity on Wave Propagation
Medium Elasticity Analogy with Spring Constant
Physical Changes in Rod during Wave Propagation
As the disturbance travels along the length of the rod, vibration of the atoms
occurs along their mean position and facilitates the propagation of the
disturbance across the rod.
Identical Situation for Longitudinal Traveling Waves
Wave
propagation via
any elastic
medium such as
air, fluid, metallic
elements bears
resemblance
Traveling Disturbance in a Rod : Physical Situation of Force
As the waves travel, forces act on the medium in terms of compression and
rarefaction and it can be estimated at any instant using Newton’s Equation:
Forces Acting on the Rod During Wave Propagation
NET FORCE :
Total Force on Rod During Wave Propagation : Origin of Wave Equation
Total Force :
Equation of Motion:
Wave Equation:
Wave Velocity:
Wave Equation for Longitudinal Wave Propagation: Alternative Approach
2
2
d y ( x, t ) 1 d y ( x, t )
 2
2
2
d x
v
d t
General Wave Equation in 1-Dimension
Wave Equation in 3 Dimensions
GENERAL
WAVE
EQUATION
Laplacian
Operator
SUMMARY
The Salient Features of Propagation of Waves
through an Elastic Medium are: