14.3 Classical Waves
631
wave nature of light was established experimentally when interference and diffraction
of light were observed.
The Flexible String
In order to illustrate the mathematics of classical waves, we now analyze the vibrations
of a flexible string, which is a model system designed to resemble a real vibrating
string. It is defined to have the following properties:
1. The string is uniform. All parts have the same mass per unit length, denoted by ρ.
2. There is a tension force of magnitude T pulling at each end of the string.
3. The string is perfectly flexible.
4. There is no friction.
5. The string undergoes only small displacements, so that the total length of the
string remains nearly constant and the magnitude of the tension force T is nearly
constant.
The state of the string at time t is specified by giving the displacement and velocity at
each point of the string.
z z(x, t)
(14.3-1)
ν z ν z (x, t)
∂z
∂t
(14.3-2)
The displacement and velocity are functions of x as well as functions of t. The derivative
∂z/∂t is a partial derivative, taken with a fixed value of x. The classical equation of
motion of the flexible string is derived in Appendix E from Newton’s second law. From
Eq. (E-10) the wave equation is
∂ 2 z
∂x 2
ρ
T
∂ 2 z
∂t 2
1
c 2
∂ 2 z
∂t 2
(14.3-3)
where we let
c
2
T /ρ
(14.3-4)
Since Eq. (14.3-3) contains partial derivatives, it is called a partial differential
equation. Its solution gives z as a function of x and t.
Let the string be fixed at x 0 and x L so that nodes occur at these locations.
We assume that at time t 0 the string is displaced into some shape in the x−z plane
and released to vibrate freely in this plane. We obtain a solution of Eq. (14.3-3) by
separation of variables. We assume a trial solution that is a product of a function of x
and a function of t:
z(x, t) ϕ(x)η(t)
(14.3-5)
We substitute the trial solution into the differential equation and perform algebraic operations that produce an equation with terms that are functions of only one independent
variable. We substitute the trial function of Eq. (14.3-5) into Eq. (14.3-3):
η
d 2 ϕ
dx 2
1
c 2 ϕ
d 2 η
dt 2
(14.3-6)
631
wave nature of light was established experimentally when interference and diffraction
of light were observed.
The Flexible String
In order to illustrate the mathematics of classical waves, we now analyze the vibrations
of a flexible string, which is a model system designed to resemble a real vibrating
string. It is defined to have the following properties:
1. The string is uniform. All parts have the same mass per unit length, denoted by ρ.
2. There is a tension force of magnitude T pulling at each end of the string.
3. The string is perfectly flexible.
4. There is no friction.
5. The string undergoes only small displacements, so that the total length of the
string remains nearly constant and the magnitude of the tension force T is nearly
constant.
The state of the string at time t is specified by giving the displacement and velocity at
each point of the string.
z z(x, t)
(14.3-1)
ν z ν z (x, t)
∂z
∂t
(14.3-2)
The displacement and velocity are functions of x as well as functions of t. The derivative
∂z/∂t is a partial derivative, taken with a fixed value of x. The classical equation of
motion of the flexible string is derived in Appendix E from Newton’s second law. From
Eq. (E-10) the wave equation is
∂ 2 z
∂x 2
ρ
T
∂ 2 z
∂t 2
1
c 2
∂ 2 z
∂t 2
(14.3-3)
where we let
c
2
T /ρ
(14.3-4)
Since Eq. (14.3-3) contains partial derivatives, it is called a partial differential
equation. Its solution gives z as a function of x and t.
Let the string be fixed at x 0 and x L so that nodes occur at these locations.
We assume that at time t 0 the string is displaced into some shape in the x−z plane
and released to vibrate freely in this plane. We obtain a solution of Eq. (14.3-3) by
separation of variables. We assume a trial solution that is a product of a function of x
and a function of t:
z(x, t) ϕ(x)η(t)
(14.3-5)
We substitute the trial solution into the differential equation and perform algebraic operations that produce an equation with terms that are functions of only one independent
variable. We substitute the trial function of Eq. (14.3-5) into Eq. (14.3-3):
η
d 2 ϕ
dx 2
1
c 2 ϕ
d 2 η
dt 2
(14.3-6)