8/11/2019 Derivation of the wave equation F=qE

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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 x (0.1)

The motion of the string is in the y direction. This means that the horizontal components of the

tensionR

T andL

T must be equal (and in opposite directions) to some constant :

cos cosL L R R

T T (0.2)

In the vertical direction we can write Newton's second law equation, namelyF manoting that

the acceleration is the second derivative of ywith respect to time:

2

2sin sin

y R R L L

yF T T m

t (0.3)

sin siny R R L L

F T T qE

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

2

2

sin sinR R L L

T T x y

t (0.4)

2

2

sin sinR R L L electron

T T V y

t

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

8/11/2019 Derivation of the wave equation F=qE

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2

2

2

2

2

2

sin sin

sin sin

sin sin

cos cos

R R L L

R R L L

R R L L

R R L L

T T x y

t

T T x y

t

T T x y

tT T

2

2tan tan

R L

x y

t (0.5)

2

2tan tan electron

R L

V y

t

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

2

2

x x x

y y x y

x x t (0.6)

Dividing both sides by x we get:

2

2

1

x x x

y y y

x x x t (0.7)

2

2

1

electron x x x

y y y

V x x t

If we take the limit of an infinitesimal segment, 0x , the left hand side becomes the second

partial derivative with respect to x(by definition).

Cant get double derivative here because ofelectron

V

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 2

2 2

y y

Tx t (0.8)

Or:

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2 2

2 2

T y y

Tx t (0.9)

The units of the constant coefficient T are:

-2 2

-1 2

kg m s m

kg m s

T (0.10)

This are units of the square of a speed.

Hence we can define the wave's propagation speed:

2 T

c (0.11)

And the one dimensional wave equation is:

2 22

2 2

y yc

x t (0.12)