The Wave Equation
What is the Wave Equation?
The wave equation for a plane wave traveling
in the x direction is:
where v is the phase velocity of the wave and
y represents the variable which is changing as
the wave passes.
Wave Equation in 2-Dimensions
The mathematical description of a wave
makes use of partial derivatives.
In two dimensions the wave Equation takes
The Wave in an Ideal String
We start with a small portion of that rope.
We have Tension ‘T’ and mass per unit length
‘µ’. If our displacement is not absurdly high
then the Tension is the same on the both side.
Derivation of the Wave Equation
We will only consider the motion in the Y
That’s for small angles other wise all our
assumptions will be wrong. When
So now the Equation becomes:
Appling Newton's Second Law
The amount of mass that is in
We know that since the length of
is the amount of mass per unit
Since in a limiting case we are going to make
, We will find the tangent for θ,
We know that tangent of “theta” is always equal
to the derivative in space (position). We used
the partial derivative since we assume that its
all happening on a instant in time.
Taking Derivative on both sides of (i),
For Small Angle Approximation,
The Solution to the differential
Where c is the constant. We know that the
dimensions of ct are the same as that of x.
Above function will satisfy the differential
When we take the 2nd derivative in time we get
the out and we get the 2nd derivative of the
Take the 2nd derivative in x and we only get the
2nd derivative of the function.
The only thing required is…
The value of ‘c’
The dimensions of c is meter per seconds i.e. c is
Thus we change our equation into a
more uniform way
This is the Equation that is generally called the
We know that x is the displacement of wave
along x axis.
is a point back in time at t=0 and
The Other Solution…
Since The Wave Equation Involves a square of v,
So we can generate another class of solutions by
simply changing the sign of velocity i.e.
Most surprising Result…
The most general solution to the wave Equation
is the sum of a wave to the left and a wave to
Thus wave Equation is Linear since the sum of
two solutions is itself a solution.
How the wave travels in a string
Suppose that two persons are holding a rope of
constant thickness. The person the right side jerks
the rope and a wave forms in the rope. While
travelling towards the guy on left It travels, let’s
say, as a Mountain. While On its way back it’s a
Why is it so?
We know that the point B is Fixed so when the
wave passes through point B then to have zero
displacement point B moves exactly the same
distance downwards. Thus producing a valley
as a result of a mountain.