Wave Equation
Physical Optics
UNIT -I
Chapter-1 One Dimensional Wave Equation Introduction Wave equation in one dimension Chapter-2 Three 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
Introduction:
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 x’=x-vt 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
UNIT -I
Chapter-1 One Dimensional Wave Equation Introduction Wave equation in one dimension Chapter-2 Three 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
Introduction:
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 x’=x-vt 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