Physics For Scientists And Engineers 6E - part 128
Problems
509
what factor? This phenomenon led to the collapse of part
of the Nimitz Freeway in Oakland, California, during the
Loma Prieta earthquake of 1989.
Section 16.6 The Linear Wave Equation
43. (a) Evaluate A in the scalar equality (7 # 3)4 ! A.
(b) Evaluate A, B, and C in the vector equality 7.00iˆ #
3.00kˆ ! Aiˆ # B jˆ # C kˆ. Explain how you arrive at the an-
swers to convince a student who thinks that you cannot
solve a single equation for three different unknowns.
(c) What If? The functional equality or identity
A # B cos(Cx # Dt # E ) ! (7.00 mm) cos(3x # 4t # 2)
is true for all values of the variables x and t, which are mea-
sured in meters and in seconds, respectively. Evaluate the
constants A, B, C, D, and E. Explain how you arrive at the
answers.
44. Show that the wave function y ! e
b(x"vt)
is a solution of the
linear wave equation (Eq. 16.27), where b is a constant.
Show that the wave function y ! ln[b(x " vt)] is a solution
to Equation 16.27, where b is a constant.
46.
(a) Show that the function y(x, t) ! x
2
#
v
2
t
2
is a solution
to the wave equation. (b) Show that the function in part
(a) can be written as f (x # vt) # g(x " vt), and determine
the functional forms for f and g. (c) What If? Repeat parts
(a) and (b) for the function y(x, t) ! sin(x)cos(vt).
Additional Problems
47. “The wave” is a particular type of pulse that can propagate
through a large crowd gathered at a sports arena to watch
a soccer or American football match (Figure P16.47). The
elements of the medium are the spectators, with zero posi-
45.
tion corresponding to their being seated and maximum
position corresponding to their standing and raising their
arms. When a large fraction of the spectators participate in
the wave motion, a somewhat stable pulse shape can de-
velop. The wave speed depends on people’s reaction time,
which is typically on the order of 0.1 s. Estimate the order of
magnitude, in minutes, of the time required for such a
pulse to make one circuit around a large sports stadium.
State the quantities you measure or estimate and their
values.
48. A traveling wave propagates according to the expression
y ! (4.0 cm) sin(2.0x " 3.0t), where x is in centimeters
and t is in seconds. Determine (a) the amplitude, (b) the
wavelength, (c) the frequency, (d) the period, and (e) the
direction of travel of the wave.
The wave function for a traveling wave on a taut string
is (in SI units)
y(x, t) ! (0.350 m) sin(10&t "3&x # &/4)
(a) What are the speed and direction of travel of the wave?
(b) What is the vertical position of an element of the string
at t ! 0, x ! 0.100 m? (c) What are the wavelength and
frequency of the wave? (d) What is the maximum magni-
tude of the transverse speed of the string?
50.
A transverse wave on a string is described by the equation
y(x, t) ! (0.350 m) sin[(1.25 rad/m)x # (99.6 rad/s)t]
Consider the element of the string at x ! 0. (a) What is
the time interval between the first two instants when this
element has a position of y ! 0.175 m? (b) What distance
does the wave travel during this time interval?
51. Motion picture film is projected at 24.0 frames per second.
Each frame is a photograph 19.0 mm high. At what con-
stant speed does the film pass into the projector?
52.
Review problem. A block of mass M, supported by a string,
rests on an incline making an angle $ with the horizontal
(Fig. P16.52). The length of the string is L, and its mass is
m // M. Derive an expression for the time interval re-
quired for a transverse wave to travel from one end of the
string to the other.
49.
Figure P16.47
Gregg Adams/Getty Images
M
m, L
θ
Figure P16.52
53.
Review problem. A 2.00-kg block hangs from a rubber
cord, being supported so that the cord is not stretched.
The unstretched length of the cord is 0.500 m, and its
mass is 5.00 g. The “spring constant” for the cord is
100 N/m. The block is released and stops at the lowest