Derivation Of The Wave Equation F=qe 3z3160

Derivation of the wave equation. The wave equation is derived for a very thin uniform string. the string has mass density

and is in

equilibrium under tension T. Now we displace the string slightly from its equilibrium position. And exaggerated view of a small (infinitesimal) segment is shown below:

The length of the segment is

x , hence its mass is:

m

(0.1)

x

The motion of the string is in the y direction. This means that the horizontal components of the tension TR and TL must be equal (and in opposite directions) to some constant

TL cos

TR cos

L

:

(0.2)

R

In the vertical direction we can write Newton's second law equation, namely F the acceleration is the second derivative of y with respect to time:

ma noting that

2

Fy

TR sin

Fy

TR sin

R

R

TL sin TL sin

L

L

y t2

m qE

The mass is given in equation (1.1) and we also divide equation (1.3) by

TR sin

TR sin

TL sin

R

R

TL sin

However, according to (1.2) we can write it as:

L

L

(0.3)

x

Velectron

:

2

y t2

(0.4) 2

y t2

TR sin

R

TL sin

L

x

y t2

TR sin

R

TL sin

L

x

2

TR sin

R

TL sin

L

TR cos

R

TL cos

L

tan

tan

2

L

y t2

2

y t2

L

Velectron

tan

R

y t2

x

x

tan

R

2

(0.5)

2

y t2

But the tangents of the angles are, by definition, the derivatives of y with respect to x, hence:

y x

Dividing both sides by

x

y x

x

x x

2

y t2

(0.6)

x we get:

1 x 1 Velectron

y x

x

y x

x

y x

x

2

y t2

x

y x

x

If we take the limit of an infinitesimal segment, x partial derivative with respect to x (by definition).

(0.7)

2

x

y t2

0 , the left hand side becomes the second

Can’t get double derivative here because of Velectron Furthermore, when the segment is very short and the string is not far from equilibrium, the tension is for all practical reasons equals the original tension T. This turns equation (1.7) into the well known one dimensional wave equation: 2

y x2 Or:

2

T

y t2

(0.8)

2

T

The units of the constant coefficient T

y x2

2

y t2

T

(0.9)

are:

T

kg m s -2 kg m -1

m2 s2

(0.10)

This are units of the square of a speed. Hence we can define the wave's propagation speed:

c2

T

(0.11)

And the one dimensional wave equation is:

c2

2

y x2

2

y t2

(0.12)