14.3 Classical Waves
637
We can find the speed of a traveling wave by following the motion of one of the nodes.
At time t 0 there is a node in the wave function of Eq. (14.3-26) at x 0. At time t
this node will be located at the point where x − ct 0. Thus
x(node) ct
(14.3-27)
The node is moving toward the positive end of the x axis with a speed equal to c.
From Eq. (14.3-4) the speed is determined by the tension force T and the mass per unit
length ρ:
c
T
ρ
(14.3-28)
Exercise 14.9
What change would have to be made in the mass per unit length to quadruple the speed of
propagation? What change would have to made in the tension force to double the speed of
propagation?
If the function of Eq. (14.3-26) is replaced by
z(x, t) Asin(κx + κct)
(14.3-29)
the wave travels toward the negative end of the x axis with speed c. This function
satisfies the same wave equation as the function shown in Eq. (14.3-26) and the speed
is the same.
Exercise 14.10
a. Show that the function of Eq. (14.3-29) satisfies Eq. (14.3-3).
b. Show that the speed of the wave is equal to c and that the wave moves to the left.
In one wavelength, the argument of the ϕ function changes by 2π if t is fixed, so that
the same relationship occurs as in Eq. (14.3-21) for a standing wave:
κ
2π
λ
(14.3-30)
The relationship between the frequency and the wavelength can be obtained by observing that in time t, the length of the wave “train” that passes a fixed point is
Length ct
(14.3-31)
where c is the speed. The number of wavelengths in this wave train is equal to
Number
ct
λ
(14.3-32)
In time t, the number of oscillations is equal to
Number νt
(14.3-33)
637
We can find the speed of a traveling wave by following the motion of one of the nodes.
At time t 0 there is a node in the wave function of Eq. (14.3-26) at x 0. At time t
this node will be located at the point where x − ct 0. Thus
x(node) ct
(14.3-27)
The node is moving toward the positive end of the x axis with a speed equal to c.
From Eq. (14.3-4) the speed is determined by the tension force T and the mass per unit
length ρ:
c
T
ρ
(14.3-28)
Exercise 14.9
What change would have to be made in the mass per unit length to quadruple the speed of
propagation? What change would have to made in the tension force to double the speed of
propagation?
If the function of Eq. (14.3-26) is replaced by
z(x, t) Asin(κx + κct)
(14.3-29)
the wave travels toward the negative end of the x axis with speed c. This function
satisfies the same wave equation as the function shown in Eq. (14.3-26) and the speed
is the same.
Exercise 14.10
a. Show that the function of Eq. (14.3-29) satisfies Eq. (14.3-3).
b. Show that the speed of the wave is equal to c and that the wave moves to the left.
In one wavelength, the argument of the ϕ function changes by 2π if t is fixed, so that
the same relationship occurs as in Eq. (14.3-21) for a standing wave:
κ
2π
λ
(14.3-30)
The relationship between the frequency and the wavelength can be obtained by observing that in time t, the length of the wave “train” that passes a fixed point is
Length ct
(14.3-31)
where c is the speed. The number of wavelengths in this wave train is equal to
Number
ct
λ
(14.3-32)
In time t, the number of oscillations is equal to
Number νt
(14.3-33)