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