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Physics 2214 Assignment 3
Concepts:
traveling waves
harmonic traveling waves waves on a string
sound waves Reading: Lecture notes 4 and 5; Y&F, Vol. 1, Chapters 15 and 16.
Assignment: Due in homework boxes opposite 125 Clark Hall before 430 pm on Tuesday,
February 15.
Please turn in this sheet stapled to the top of your work.
Physics Problems:
1. Mechanical Impedance
In physics 2213, you learned about electrical impedance Z=V/I, which is the ratio of the
(complex) potential difference V driving charge motion through a circuit element to the
(complex) current I or charge flow that occurs. The impedance is defined only in the case
of steady state sinuosoidal driving, when all currents and voltages are sinusoidal at the
same frequency.
Similarly, when we a drive a mechanical system with a sinusoidal force, all other
quantities characterizing the mechanical response will be sinusoidal, so we again use a
complex representation. We can define a mechanical impedance Z = F/v, the ratio of the
(complex) force driving an object’s motion to the (complex) velocity v of that motion.
(a) What is the mechanical impedance of the driven damped oscillator that we analyzed in
lecture 3?
(b) For a lightly damped oscillator, at what frequency is this impedance minimized?
(c) In analogy with electrical circuits, the average power dissipated by the system is
where θ is the phase difference between F and v. Determine the average power
dissipated at resonance in an underdamped system.
2. Wave motion. A transverse harmonic wave travels down a long steel wire at 2500 m/s.
The frequency of the wave is 2 kHz, and the amplitude of the displacement is 10 mm.
Let x=0 be at one end of the wire, such that the wave propagates along the wire in the +x
direction. Assume that at t=0 the wire has a maximum of its displacement at x=0.
(a) Write an equation describing this wave in form
Physics 2214, Spring 2011 1 Cornell University y(x,t) = A cos[k(xvt)+φ]
Give numerical values for A, φ, k, ω, f and λ.
(b) Sketch a snapshot of the wire at t=0.00075 s, i.e., y(x,t=0.00075 s). Identify one point on
your graph for each of the following conditions: (A) maximum particle velocity upward, (B)
zero particle velocity, (C) maximum particle acceleration downward, and (D) zero particle
acceleration.
(c) Sketch the motion y(t) of a piece of the wire at x=2.5 m.
(d) What is the maximum y velocity of the wire at x=2.5 m?
(e) Given an expression for the slope of the wire at x=2.5 m. When the wire has its maximum y velocity as calculated in (d), what is the value of this slope?
(f) Are your answers to (d) and (e) consistent with the pulse equation?
(g) Determine expressions for (i) the curvature of the wire
acceleration of the wire and (ii) the transverse . At any x and t, what is their ratio? 3. Why do waves propagate? For an ideal transverse wave on a string, each piece of
string oscillates up and down, in the y direction. How then does the wave propagate in
the x direction? What sets the string into motion far from where you first pluck it?
4. Tsumanis. On December 26, 2004, a magnitude Mw=9.3 earthquake struck off the coast
of Sumatra, 30 km below the surface of the Indian Ocean. The motion of the sea floor
created a tsunami. The water waves propagated across the ocean and then inundated
coastal areas as far away as Sri Lanka and India, killing roughly 200,000 people. In the
open ocean, the tsunami wavelength measured by the TOPEX/Poseidon and Jason1
satellites was roughly 800 km, and the wave period was roughly 1 hour.
(a) What was the speed of these waves?
(b) A tsunami is an example of a shallow water wave, i.e., a wave whose wavelength is
large compared with the water depth. The wave speed is given by
, where
g=9.8 m/s2 and H is the depth of the water in m. From your answer to (a), what was the
water depth in open ocean?
(c) The tsunami amplitude in open ocean was roughly 1 m. If you were in a boat on that
ocean, what would be your maximum vertical speed as a tsunami wave passed?
Would you notice this? (Near the shore, the wave speed dropped but its amplitude grew
to as much as 16 m. See http://www.bom.gov.au/info/tsunami/tsunami_info.shtml for a
nice discussion of tsunamis. Scroll down to the 2004 tsunami, and look at the animation
showing how the waves propagated.)
5. Waves on wires. One end of a steel wire is attached to a fixed support. The other end
passes over a pulley and supports an object of mass 5 kg. The length of the wire
between the pulley and the fixed support is 4m and the mass of the wire is 40 g.
(a) If the wire is struck with a traverse blow at one end, how long does it take for the resulting
disturbance to reach the other end?
Physics 2214, Spring 2011 2 Cornell University (b) Let's suppose that the blow creates a triangular pulse of height 10 cm and width 20 cm.
What will be the maximum transverse speed
of the wire as the pulse moves down
its length?
(c) In (a) and (b) we made our usual assumption that the weight of the wire could be
neglected, because the tension was much greater than the weight. But now suppose that
you have a heavy steel cable that hangs vertically from a support at its upper end. The
lower end is free. How would you expect the wave speed to vary from the bottom to the
top of the cable?
(d) In case (c), if you launched a single squareish pulse from the bottom of the cable toward
the top, how would you expect its shape to change as it propagated?
6. Forces due to waves on strings. An ideal transverse wave on a string (no forces other
than tension) is described by
y(x,t) = Acos(kx + ωt)
The string is under a tension τ.
(a) Draw a free body diagram of a segment of the string. What is the maximum vertical force
on a segment of string due to the neighboring segment pulling on one side only?
Express your answer in terms of A, k, ω, τ as needed.
(b) What is the maximum net vertical force on a segment of string of length Δx due to the
neighboring segments pulling on both sides? Assume that the segment is short (Δx << λ),
so use an approximation rather than trying to find the exact answer.
7. Characteristic impedance. For a transverse sinusoidal mechanical wave propagating
through a medium, we can define a characeristic impedance Z=Fy/vy, where
is the (complex) transverse velocity of a piece of the medium and
is the (complex) applied oscillating force in the y direction on one side
that drives the motion. Show that the characteristic impedance for a transverse wave on
a string is
, where v is the wave speed (not the particle speed.) Hint: In our
derivation of the wave equation for a string, we saw that the y component of the tension,
τy, drives the transverse motion of the string. Imagine cutting the string, and applying an
oscillating force Fy that is exactly the same as would have been applied by the piece of
string you've cut away. What is the resulting y velocity? We've already calculated y(x,t),
so just take a derivative with respect to time. (We'll see that the characteristic impedance
governs wave reflection and transmission at interfaces between different media.)
8. Dispersion on Piano strings. Most piano “strings” are steel wires. To accurately
describe transverse waves on a piano string (and on steel cables used in construction),
the stiffness of the steel—that is, its resistance to bending—has to be taken into account.
The propagation of transverse waves along such strings is described by the modified
wave equation
Physics 2214, Spring 2011 3 Cornell University where µ and τ are the the linear mass density and tension, respectively, and Γ is an
elastic parameter that depends on the diameter of the wire and the stiffness of the
material, but is independent of the tension.
(a) Show, by direct substitution into the modified wave equation, that a traveling harmonic
wave of the form
y(x,t) = Re[Aei(kx ± ωt)]
can exist on such a string, where A in general is complex. Find the relationship between
ω and k required for this y(x,t) to be a solution.
(b) The phase speed of a harmonic traveling wave is defined as Find as a function of k. (Since is not constant, harmonic waves with different k’s have different phase speeds. This is called dispersion. For any kind of wave, the relation
between ω and k is called the dispersion relation.
If ω∝k, the waves are
dispersionless.)
9. Sound waves in air and water. The longitudinal displacement produced by a sinusoidal
sound wave in air (v=340 m/s, ρ=1.2 kg/m3) is given by
s(x,t) = A cos (kx  ωt)
where A = 2 µm, and the wavelength is 0.5 m.
(a) Graph the displacement s(x,t) and the pressure modulation p(x,t)p0 versus x at time t=0.
(b) What is the amplitude of the pressure modulation?
(c) Suppose that a sound wave with the same frequency and the same pressure modulation
amplitude is propagating through water (v=1480 m/s, ρ=1000 kg/m3). What will be the
amplitude of the displacements s(x,t) in this case?
Questions, Exercises and Problems from Y&F 12th Ed. for study and review:
(Do not turn these in!)
Volume 1, Chapter 15: Questions 1, 7, 10, 11, 12, 15
Exercises 3, 4, 6, 7, 8, 9, 11, 12, 16, 18
Problems 53, 57, 80
Volume 1, Chapter 16:
Questions 1, 2. Exercises 1, 2, 3, 6, 13
Physics 2214, Spring 2011 4 Cornell University ...
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This note was uploaded on 10/25/2011 for the course PHYS 2214 at Cornell University (Engineering School).
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