Download Φ21 Fall 2006 HW19 Solutions

Φ21 Fall 2006
1
HW19 Solutions
Problem K20.7
The wave speed on a string under tension is 160 m/s. What is the speed if the tension is doubled?
√
T /µ, where T√is the tension
√ and µ is the mass density
2, so v = 2 · 160 m/s = 226.3 m/s
speed will go up by
Solution: The speed of the waves on a string is
of the string. So, if the tension is doubled, the
2
v=
Problem K20.11
A wave travels with speed 208 m/s.
Its wave number is 2.00 rad/m.
Parts A and B. What are its
wavelength and frequency?
Solution: The wave number is inversely proportional to the wavelength.
frequency and wavelength are related to the velocity by
3
v = f λ,
so
f=
v
λ
2π
2π
k = 2 rad/m =
208 m/s
3.14 m = 66.2 Hz.
λ=
=
3.14 m.
The
Problem K20.13
The displacement of a wave traveling in the negative y-direction is
where
y
is in m and
t
Solution: From the formula, we can identify the angular frequency
f=
ω
2π
=
62.0 s−1
2π
D(y, t) = (4.90 cm) sin(6.50y + 62.0t),
is in s. Part A. What is the frequency of this wave?
ω = 62.0 s−1 = 2πf .
The frequency is
= 9.87 Hz.
Part B. What is the wavelength?
Solution: From the formula, identify the wavenumber
k = 6.50 m−1 =
2π
λ . Then
λ =
2π
k
=
2π
6.50 m−1
=
0.97 m.
Part C. What is the speed of the wave?
Solution: The speed is
ω
4
and
v = fλ =
ω
k
=
62.0 s−1
6.50 m−1
= 9.54 m/s.
(It's more accurate to use the original numbers
k .)
Problem K20.42
Part A. What is the wave speed?
Solution:
From the gure, the wavelength is 2.0 m.
So the speed is
v = f λ = (5.0 Hz) (2.0 m) = 10 m/s.
Part B. What is the phase constant of the wave?
Solution: The phase constant in the travelling wave solution D(x, t) =
D0 sin (kx ± ωt + φ0 ) serves to shift the wave to the right. In this case,
◦
at t = 0 and x = 0, D(x, t) = D0 sin (φ0 ) = D0 /2, so φ0 = 30 .
Figure
1:
graph at
This
t = 0s
is
traveling to the left.
1
a
snapshot
of a 5.0 Hz wave
5
Problem K20.52
Earthquakes are essentially sound waves traveling through the earth. They are called seismic waves. Because
the earth is solid, it can support both longitudinal and transverse seismic waves. These travel at dierent
speeds. The speed of longitudinal waves, called P waves, is 8000 m/s. Transverse waves, called S waves,
travel at a slower 4500 m/s. A seismograph records the two waves from a distant earthquake.
If the S wave arrives 2.0 min after the P wave, how far away was the earthquake? You can assume that the
waves travel in straight lines, although actual seismic waves follow more complex routes.
Solution: Since we know the dierence in times, start from there.
tS − tP
=
d
=
(
)
d
d
1
1
−
=d
−
vS
vP
vS
vP
120 s
t − tP
)=(
) = 1.234 × 106 m
( S
1
1
1
1
vS − vP
4500 m/s − 8000 m/s
2