wave equation ii and solution. length of the curved element
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Wave Equation II and Solution
22 1/ 2
1/ 2 1/ 22 22 2
[( ) ]
1
ds d dx
dx dx dxx x
Length of the curved element
ds dx
Perpendicular force on the element dx
sin sinT d T
x dx x
Tx x
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
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
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
Physical significance of =f(ct+x)
t
wave moving to the left
For a sinusoidal plane wave
2cos ( )a ct x
Therefore,
22
c
: frequency of oscillation in time
for pattern repeats
: Wavelength
2 :wavenumber
x n
k
2 2c
c v
C: wave velocity
1c
: period of oscillation
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
t
Particle velocity : cx
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
For monochromatic wave in a non-dispersive medium
ckPlot dispersion relation
Slope (c) phase velocity of the wave
= ck
k
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
Solution of spherically symmetric wave
rctrf
rctrftr )()(),( 21
•Wave equation
2 2
2 2 2
1 x c t
•General solution of wave equation
)xct(f)xct(f 21
•Solution of Plane wave of form
2cos ( )a ct x
: wavelength
c : wave velocity
a : amplitude
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
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