derivation of wave equation

derivation of wave equation

Let a string of matter be tightened between the points

x

=

0

and

x

=

p

of the

x

-axis and let the string be made vibrate in the

x

⁢

y

-plane.  Let the of the string be the constant

σ

.  We suppose that the amplitude of the vibration is so small that the tension

T

→

of the string can be regarded to be constant.

The position of the string may be represented as a function

y

=

y

⁢

(

x

,

t

)

where

t

is the time.  We consider an element

d

⁢

m

of the string situated on a tiny interval

[

x

,

x

+

d

⁢

x

]

;  thus its mass is

σ

⁢

d

⁢

x

.  If the angles the vector

T

→

at the ends

x

and

x

+

d

⁢

x

of the element forms with the direction of the

x

-axis are

α

and

β

, then the scalar force

F

→

of all on

d

⁢

m

(the gravitation omitted) are

F

x

=

-

T

⁢

cos

⁡

α

+

T

⁢

cos

⁡

β

,

F

y

=

-

T

⁢

sin

⁡

α

+

T

⁢

sin

⁡

β

.

Since the angles

α

and

β

are very small, the ratio

F

x

F

y

=

cos

⁡

β

-

cos

⁡

α

sin

⁡

β

-

sin

⁡

α

=

-

2

⁢

sin

⁡

β

-

α

2

⁢

sin

⁡

β

+

α

2

2

⁢

sin

⁡

β

-

α

2

⁢

cos

⁡

β

+

α

2

,

having the

expression

-

tan

⁡

β

+

α

2

, also is very small.  Therefore we can omit the horizontal

component

F

x

and think that the vibration of all elements is

strictly

vertical.  Because of the smallness of the angles

α

and

β

, their sines in the expression of

F

y

may be replaced with their tangents, and accordingly

F

y

=

T

⋅

(

tan

⁡

β

-

tan

⁡

α

)

=

T

⁢

[

y

x

′

⁢

(

x

+

d

⁢

x

,

t

)

-

y

x

′

⁢

(

x

,

t

)

]

=

T

⁢

y

x

⁢

x

′′

⁢

(

x

,

t

)

⁢

d

⁢

x

,

the last form due to the mean-value theorem.

On the other hand, by Newton the force equals the mass times the acceleration:

F

y

=

σ

⁢

d

⁢

x

⁢

y

t

⁢

t

′′

⁢

(

x

,

t

)

Equating both expressions, dividing by

T

⁢

d

⁢

x

and denoting

T

σ

=

c

,  we obtain the

partial differential equation

y

x

⁢

x

′′

=

1

c

2

⁢

y

t

⁢

t

′′

(1)

for the equation of the

vibrating string

.

But the equation (1) don’t suffice to entirely determine the vibration.  Since the end of the string are immovable,the function

y

⁢

(

x

,

t

)

has in to satisfy the

boundary conditions

y

⁢

(

0

,

t

)

=

y

⁢

(

p

,

t

)

=

0

(2)

The vibration becomes completely determined when we know still e.g. at the beginning

t

=

0

the position

f

⁢

(

x

)

of the string and the initial velocity

g

⁢

(

x

)

of the points of the string; so there should be the

initial conditions

y

⁢

(

x

,

0

)

=

f

⁢

(

x

)

,

y

t

′

⁢

(

x

,

0

)

=

g

⁢

(

x

)

.

(3)

The equation (1) is a special case of the general

wave equation

∇

2

⁡

u

=

1

c

2

⁢

u

t

⁢

t

′′

(4)

where

u

=

u

⁢

(

x

,

y

,

z

,

t

)

.  The equation (4) rules the spatial waves in

ℝ

.  The number

c

can be shown to be the velocity of propagation of the wave motion.

References

1

K. Väisälä:

Matematiikka IV

.  Handout Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).

Title

derivation of wave equation

Canonical

name

DerivationOfWaveEquation

Date of creation

2013-03-22 18:46:36

Last modified on

2013-03-22 18:46:36

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Derivation

Classification

msc 35L05

Related topic

Slope

Related topic

MeanValueTheorem

Related topic

DerivationOfHeatEquation

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