14.3 Classical Waves
633
where B, D, F , and G are constants. The product of the coordinate factor ϕ and the
time factor η is a function that represents the displacement of the string as a function
of position and time. We call z(x, t) a wave function. Our present wave function is not
necessarily a general solution to the original wave equation, since solutions can exist
that are not the product of two factors.
Exercise 14.5
Show that the product of the factors in Eq. (14.3-12) and (14.3-13) satisfies Eq. (14.3-3).
The wave function z(x, t) does not yet apply to our case that z vanishes at x 0 and
at x L. We call this condition a boundary condition since it pertains to the ends of the
string. If z(x, t) vanishes at these points, the coordinate factor ϕ(x) must also vanish at
these points, since it contains all of the x dependence of z. The condition that ϕ(0) 0
requires that B 0, since sin(0) 0 and cos(0) 1. The sine function vanishes if its
argument is an integral multiple of π(90 ◦ ) so that the condition ϕ(L) 0 requires that
κL nπ
(14.3-14)
where n is an integer. We can write a different coordinate factor for each value of n:
ϕ(x) ϕ n (x) Dsin
nπx
L
(14.3-15)
The coordinate factor ϕ has been determined partly by the differential equation and
partly by the boundary conditions.
The values of the constants D, F , and G are now chosen to match our initial conditions. A second-order differential equation requires two initial conditions. We choose as
the first initial condition that the string is passing through its equilibrium position (z 0
for all x) at time t 0. This requires that F 0, since sin(0) 0 and cos(0) 1. The
time factor η is now determined by Eq. (14.3-14) and our initial conditions. The wave
function can now be written as
z(x, t) z n (x, t) ϕ n (x)η n (t) DGsin
nπx
L
sin
nπct
L
Asin
nπx
L
sin
nπct
L
(14.3-16)
Where we replace the constant product DG by a single symbol, A.
We now choose the second initial condition to be that the maximum value of z is
equal to z 0 :
z(x, t) z n (x, t) z 0 sin
nπx
L
sin
nπct
L
(14.3-17)
This is a set of solutions, one for each value of n. Each solution corresponds to a
different standing wave. The differential equation, the boundary conditions, and the
initial conditions have completely determined the set of wave functions.
The velocity is given by
ν z
∂z
∂t
z 0
nπc
L
sin
nπx
L
cos
nπct
L
(14.3-18)
633
where B, D, F , and G are constants. The product of the coordinate factor ϕ and the
time factor η is a function that represents the displacement of the string as a function
of position and time. We call z(x, t) a wave function. Our present wave function is not
necessarily a general solution to the original wave equation, since solutions can exist
that are not the product of two factors.
Exercise 14.5
Show that the product of the factors in Eq. (14.3-12) and (14.3-13) satisfies Eq. (14.3-3).
The wave function z(x, t) does not yet apply to our case that z vanishes at x 0 and
at x L. We call this condition a boundary condition since it pertains to the ends of the
string. If z(x, t) vanishes at these points, the coordinate factor ϕ(x) must also vanish at
these points, since it contains all of the x dependence of z. The condition that ϕ(0) 0
requires that B 0, since sin(0) 0 and cos(0) 1. The sine function vanishes if its
argument is an integral multiple of π(90 ◦ ) so that the condition ϕ(L) 0 requires that
κL nπ
(14.3-14)
where n is an integer. We can write a different coordinate factor for each value of n:
ϕ(x) ϕ n (x) Dsin
nπx
L
(14.3-15)
The coordinate factor ϕ has been determined partly by the differential equation and
partly by the boundary conditions.
The values of the constants D, F , and G are now chosen to match our initial conditions. A second-order differential equation requires two initial conditions. We choose as
the first initial condition that the string is passing through its equilibrium position (z 0
for all x) at time t 0. This requires that F 0, since sin(0) 0 and cos(0) 1. The
time factor η is now determined by Eq. (14.3-14) and our initial conditions. The wave
function can now be written as
z(x, t) z n (x, t) ϕ n (x)η n (t) DGsin
nπx
L
sin
nπct
L
Asin
nπx
L
sin
nπct
L
(14.3-16)
Where we replace the constant product DG by a single symbol, A.
We now choose the second initial condition to be that the maximum value of z is
equal to z 0 :
z(x, t) z n (x, t) z 0 sin
nπx
L
sin
nπct
L
(14.3-17)
This is a set of solutions, one for each value of n. Each solution corresponds to a
different standing wave. The differential equation, the boundary conditions, and the
initial conditions have completely determined the set of wave functions.
The velocity is given by
ν z
∂z
∂t
z 0
nπc
L
sin
nπx
L
cos
nπct
L
(14.3-18)