Wave Equation

Topics: Wave, Interference, Light Pages: 10 (2366 words) Published: April 11, 2013
Physical Optics

Chapter-1One Dimensional Wave Equation
Wave equation in one dimension

Chapter-2Three Dimensional Wave Equation
Total energy of a vibrating particle
Superposition of two waves acting along the same line
Graphical methods of adding disturbances of the same frequency

Chapter – 1

The branch of Physics based on the wave concept of light is called ‘Wave Optics’ or ‘Physical Optics’.

Mathematical representation of a wave:

One-Dimensional Wave Equation:
The most general form of a travelling wave and the differential equation it satisfies can be determined in the following way: Let O(x,y) be a fixed coordinate system.
Consider a One-Dimensional wave of an arbitrary shape in a coordinate system O’(x’, y’) The wave is described by
y’ = f(x’)
Let the O’(x’, y’)system move to the right along the x-axis with a uniform speed ‘ v ’ relative to the coordinate system O(x, y). Let the shape of the wave be constant.
Consider a point P on the wave shape.
It can be described by both the coordinates ,
Along the x-coordinate
along the y-coordinate
y = y’ = f(x’) = f(x-vt) Wave moving to the right of O Similarly
y = y’ = f(x’) = f(x+vt) Wave moving to the left of O
or the general form of a traveling wave is written as
y = y’ = f(x’) = f(x±vt)
Examples of Traveling Waves:
The function f is any function, for example:
y = A sin(x-vt) or y =A (x+vt)2 or y = e(x-vt)
Any such wave or shape that repeats itself in space with time is called a “PERIODIC WAVE” We will derive the Equation that represents the motion of periodic wave such as ‘f’ of any arbitrary shape One Dimensional Wave Equation Derivation:

Consider the motion of a periodic wave fixed with respect to a coordinate system O’(x’,y’) and travelling to the right with respect to a fixed coordinate system O(x,y). There is no motion along y-coordinate.

The O’-system moves to the right with respect to the O-system. Y is a function of TWO variables x and t.
Y = f(x’) and y = y’ = f(x’) = f(x±vt) (1)
As there are TWO variables we shall take the partial derivatives with respect to each of the variables. ∂x'∂x=1 and ∂x'∂t=±v(2)
Employing the chain rule from eqn.(1) the space derivative is ∂y∂x=∂f∂x'∂x'∂x=∂f∂x(3)
Repeating the procedure to find the second derivative,
∂2y∂x2= ∂∂x∂y∂x=∂∂y∂x∂x'∂x'∂x=∂∂x'∂f∂x=∂2f∂x'2 (4) Similarly the Time-Derivatives is given by
∂y∂t=∂f∂x'∂x'∂t=±v∂f∂x (5) First derivative ∂2y∂t2= ∂∂t∂y∂t=∂∂y∂t∂x'∂x'∂t=∂∂x'±v∂f∂x'=v2∂2f∂x'2 (6)Second Derivative Combining the results in the equations (4) and (6)
We can write the One-Dimensional differential wave equation as ∂2y∂x2=1v2∂2f∂t2 (7) One-D Wave Equation
This is the general form of the wave equation in One-Dimension. Solution of the wave equation in one-dimension:

The solution to this second order partial differential equation can be expressed in terms of a sine or cosine function of position The displacement of successive particles of the medium is given by sine or cosine function of position. The displacement y at t = 0 is given by

Where A and k are constants.
Disturbance is propagating along positive x-direction then
The waveform in (i) is a sine function, it repeats itself at regular distances. The first repetition would take place when
This distance after which repetition takes place is called the wavelength Hence or
Constant k is called propagation constant or wave vector.
Equation (ii) turns into
The equation which leads the wave represented by (iii) by half the wavelength is given by

y = …….(iv)

Relation between wavelength and velocity of propagation :

Time taken for one complete of wave to pass any point is the time period (T). If in the equation (iv) we put the term undergoes one complete cycle i.e, increased by ....
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