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Wave Equation II and Solution. Length of the curved element.
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Transcript of Wave Equation II and Solution. Length of the curved element.
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Wave Equation II and Solution
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22 1/ 2
1/ 2 1/ 22 22 2
[( ) ]
1
ds d dx
dx dx dxx x
Length of the curved element
ds dx
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Perpendicular force on the element dx
sin sinT d T
x dx x
Tx x
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Equation of motion of small element x
2 2
2 2T dx dxx t
Wave equation
2 2
2 2or, x T t
2 2
2 2 2
1or, x c t
= mass density
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2
22
2
2
xc
t
Gen. Wave equation
Tc
Yc
(String wave) (Elastic wave)
(Acoustic wave through fluid)
0
1
c: compressibility0:equilibrium density
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2
1 22
22
1 22
''( ) ''( )
''( ) ''( )
f ct x f ct xx
c f ct x f ct xt
Therefore, (nothing but wave equation)2
22
2
2
xc
t
General Solution)xct(f)xct(f 21
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Physical significance of =f(ct+x)
t
wave moving to the left
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For a sinusoidal plane wave
2cos ( )a ct x
Therefore,
22
c
: frequency of oscillation in time
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for pattern repeats
: Wavelength
2 :wavenumber
x n
k
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2 2c
c v
C: wave velocity
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1c
: period of oscillation
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Different forms of solution of wave equations
( )
2cos ( ) cos 2
cos cos( )
i t kx
xa ct x a t
xa t a t kxc
ae
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t
Particle velocity : cx
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Dispersion Relation
A link between spatial and temporal oscillations
For wave equation
2
22
2
2
xc
t
Plane wave solution2 ( )f ct x
Frequency of oscillation ckc
2
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For monochromatic wave in a non-dispersive medium
ckPlot dispersion relation
Slope (c) phase velocity of the wave
= ck
k
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Wave equation in spherical polar coordinates (r, , f)
For spherically symmetric wave
2
2
22
2
11tcr
rrr s
2
2
2
2
2
2222
2
1
sin1sin
sin11
tc
rrrr
rr
s
f
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Solution of spherically symmetric wave
rctrf
rctrftr )()(),( 21
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•Wave equation
2 2
2 2 2
1 x c t
•General solution of wave equation
)xct(f)xct(f 21
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•Solution of Plane wave of form
2cos ( )a ct x
: wavelength
c : wave velocity
a : amplitude
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1. LECTURE NOTES FOR PHYSICS ISASTRY AND SARASWAT
2. THE PHYSICS OF VIBRATIONS AND WAVESAUTHOR: H.J. PAINIIT KGP Central LibraryClass no. 530.124 PAI/P