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14 Classical Mechanics and the Old Quantum Theory
The velocity at t 0 is
ν z (0) z 0
nπc
L
sin
nπx
L
(14.3-19)
Our second initial condition has specified the initial velocity. Instead of specifying the
maximum amplitude we could have specified that the initial velocity is given by this
function.
Exercise 14.6
a. Show by substitution that Eq. (14.3-16) satisfies Eq. (14.3-3).
b. What is the effect on the wave function of replacing n by its negative?
c. What is the relationship between the value of n and the number of nodes?
The wavelength λ for our standing wave is the distance such that the argument of
the sine function in Eq. (14.3-15) changes by 2π. This gives
λ
2L
n
(14.3-20)
The wavelength is quantized. That is, it can take on a certain value for each value of
n, but no values between these allowed values. There is a relation between κ and λ:
λ
2L
n
2π
κ
(14.3-21)
The relation between λ and κ is independent of the value of n.
The period τ of our standing wave is the time required for the argument of the time
factor sin(nπct/L) to change by 2π, so that
2π
nπcτ
L
or τ
2L
nc
(14.3-22)
The frequency ν is the number of oscillations per unit time and equals the reciprocal
of the period:
ν
nc
2L
n
2L
T
ρ
(14.3-23)
The frequency is quantized. It is directly proportional to the integer n, inversely proportional to the length of the string, directly proportional to the square root of the tension
force, and inversely proportional to the square root of the mass per unit length.
The strings of real musical instruments obey our formulas to a good approximation.
In musical acoustics, the standing wave with n 1 is called the fundamental or the
first harmonic, the standing wave with n 2 is called the first overtone or the second
harmonic, and so on. Figure 14.7 represents the coordinate factor ϕ for several values
of n. Each graph in the figure represents the position of the string at a time when the
time factor η equals unity. For other times, the string oscillates between the position
given by this graph and its negative.
14 Classical Mechanics and the Old Quantum Theory
The velocity at t 0 is
ν z (0) z 0
nπc
L
sin
nπx
L
(14.3-19)
Our second initial condition has specified the initial velocity. Instead of specifying the
maximum amplitude we could have specified that the initial velocity is given by this
function.
Exercise 14.6
a. Show by substitution that Eq. (14.3-16) satisfies Eq. (14.3-3).
b. What is the effect on the wave function of replacing n by its negative?
c. What is the relationship between the value of n and the number of nodes?
The wavelength λ for our standing wave is the distance such that the argument of
the sine function in Eq. (14.3-15) changes by 2π. This gives
λ
2L
n
(14.3-20)
The wavelength is quantized. That is, it can take on a certain value for each value of
n, but no values between these allowed values. There is a relation between κ and λ:
λ
2L
n
2π
κ
(14.3-21)
The relation between λ and κ is independent of the value of n.
The period τ of our standing wave is the time required for the argument of the time
factor sin(nπct/L) to change by 2π, so that
2π
nπcτ
L
or τ
2L
nc
(14.3-22)
The frequency ν is the number of oscillations per unit time and equals the reciprocal
of the period:
ν
nc
2L
n
2L
T
ρ
(14.3-23)
The frequency is quantized. It is directly proportional to the integer n, inversely proportional to the length of the string, directly proportional to the square root of the tension
force, and inversely proportional to the square root of the mass per unit length.
The strings of real musical instruments obey our formulas to a good approximation.
In musical acoustics, the standing wave with n 1 is called the fundamental or the
first harmonic, the standing wave with n 2 is called the first overtone or the second
harmonic, and so on. Figure 14.7 represents the coordinate factor ϕ for several values
of n. Each graph in the figure represents the position of the string at a time when the
time factor η equals unity. For other times, the string oscillates between the position
given by this graph and its negative.