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the transmitted and reflected waves is straightforward but beyond the scope of this class (it starts to involve real math and studies of boundary conditions). However, the following qualitative properties of the transmitted and reflected waves should be learned:

•Light string (medium) to heavy string (medium): Transmitted pulse right side up, reflected pulse inverted. (A fixed string boundary is the limit of attaching to an “infinitely heavy string”).

•Heavy string to light string: Transmitted pulse right side up, reflected pulse right side up. (a free string boundary is the limit of attaching to a “massless string”).

10.5: Energy

Clearly a wave can carry energy from one place to another. A cable we are coiling is hung up on a piece of wood. We flip a pulse onto the wire, it runs down to the piece of wood and knocks the wire free. Our lungs and larnyx create sound waves, and those waves trigger neurons in ears far away. The sun releases nuclear energy, and a few minutes later that energy, propagated to earth as a light wave, creates sugar energy stored inside a plant that is still later released while we play basketball202. Since moving energy around seems to be important, perhaps we should figure out how a wave manages it.

Let us restrict our attention to a harmonic wave of known angular frequency ω, although many of our results will be more general, because arbitrary wave pulses can be fourier decomposed as noted above.

Consider a small piece of the string of length dx and mass dm = µdx. This piece of string, displaced to its position y(x, t), will have kinetic energy:

	1		µ	dy		¶	2
dK =		dm							(944)
	2			dt
=	1	2		2			2
		A µω			cos			(kx − ωt)dx	(945)
	2

We can easily integrate this over any specific interval. Let us pick a particular time t = 0 and integrate it over a single wavelength:

K =	Z0	KdK =		Z0		2 A2µω2 cos2		(kx)dx
					λ 1
			Z0
=	2k A2µω2			cos2(kx)kdx
	1			λ
=	2k A2µω2		Z0	cos2(θ)dθ
	1			2π

=14 A2µω2λ

(where we have used the easily proven relation:

Z 2π

sin2(θ)dθ = π

0

(946)

(947)

(948)

(949)

(950)

to do the final form of the integral.) If we assume that the string has length L λ, we can average this over many wavelengths and get the average kinetic energy per unit length

κ = K/L ≈ K/λ =	1	2	2
		A µω		(951)
	4

202Painfully and badly, in my case. As I’m typing this, my ribs and ankles hurt from participating in the Great Beaufort 3 on 3 Physics Basketball Tournament, Summer Session 1, 2011! But what the heck, we won and are in the finals. And it was energy propagated by a wave, stored (in my case) as fat, that helped get us there...

444	Week 10: The Wave Equation

In the limit of an infinitely long string, or a length of string that contains an integer number of wavelengths, this expression is exact. Note also that this answer does not depend on time, because ωt only corresponds to a di erent phase and the integral of sin2(θ) + δ) does not change.

The average potential energy of the string is more di cult. This is because there are two possible interpretations for the ”potential energy of a piece of string”.

a)Treating it as a mass, we talk of the potential energy of a piece of mass as the work done moving it from some reference position to the position we find it in. For example, we might speak of the potential energy of a rock sitting on the surface of the earth as an integral over the force exerted to move it from a reference position to its position on the surface, given the gravitational field of the Earth.

b)Treating it as the work done stretching the piece of string! We often speak of the potential energy stored in a spring, that is, the work required to stretch the spring from a reference position to the final position. Note that in this approach, we don’t really care whether or not a mass is present at all.

These two interpretations are quite di erent. We don’t describe the gravitational potential energy of the rock as the work required to assemble the rock out of small chunks of mass, although we certainly could and in certain problems (especially in electromagnetism) we do talk about the “binding energy”, or the work done assembling a mass or collection of charge. But this internal potential energy is irrelevant to the force exerted on the rock itself by all the rest of the mass that we use to compute its potential energy.

A second di erence is that the first definition doesn’t really tell you where the energy is. The total work done moving the mass around involves contributions between every bit of mass in both the rock and in the Earth. Precisely where the energy is stored in space is di cult (but possible, with enough work and insight) to answer. The total energy in the second case tells you exactly where the energy is, and the argument extends all the way to the microscopic description given in the section on elastic materials and Young’s Modulus, where we discover that the energy is in the chemical (electromagnetic) bonds between the molecules that make up the material.

The two interpretations become one at the level of field theory. If you continue on and take introductory electricity and magnetism, you will learn that we can indeed describe the potential energy of a configuration of mass (for example) as the integral over a field energy per unit volume over all space! But this is far beyond the scope of this course.

This leaves us with a choice. We can compute the amount of work required to move a (chunk of) string from its unstretched equilibrium position to its final position in a waveform, or we can compute the amount of work required to stretch the (chunk of) string itself.

which has exactly the same integral:

K =	1	2	2
		A µω		λ	(952)
	4

so that the total energy in a wavelength of the wave is:

	Etot =	1	µω2A2λ	(953)
		2

Study the dependences in this relation. Energy depends on the amplitude squared! (Emphasis to convince you to remember this! It is important!) It depends on the mass per unit length times the length (the mass of the segment). It depends on the frequency squared.

This energy moves as the wave propagates down the string. If you are sitting at some point on the string, all the energy in one wavelength passes you in one period of oscillation. This lets us

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compute the power carried by the string – the energy per unit time that passes us going from left to right:

P =	E	=	1		µω2A2λf =	1		µω2A2v	(954)
	T		2			2

We can think of this as being the energy per unit length (the total energy per wavelength divided by the wavelength) times the velocity of the wave. This is a very good way to think of it as we prepare to study light waves, where a very similar relation will hold.

446	Week 10: The Wave Equation

Homework for Week 10

Problem 1.

Physics Concepts: Make this week’s physics concepts summary as you work all of the problems in this week’s assignment. Be sure to cross-reference each concept in the summary to the problem(s) they were key to, and include concepts from previous weeks as necessary. Do the work carefully enough that you can (after it has been handed in and graded) punch it and add it to a three ring binder for review and study come finals!

Problem 2.

This (and several of the following) problems are from resonance and damped and/or driven oscillation, not from waves! Be sure to use the correct concepts on your concept summary!

Roman soldiers (like soldiers the world over even today) marched in step at a constant frequency – except when crossing wooden bridges, when they broke their march and walked over with random pacing. Why? What might have happened (and originally did sometimes happen) if they marched across with a collective periodic step?

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Problem 3.

										(A)
	10
	9
	8
	7
	6
P(ω)	5
	4
	3									(B)
	2
	1
										(C)
	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
										ω

In the figure above, three resonance curves are drawn showing the power delivered to a steadystate driven oscillator, P (ω). In all three cases the resonance frequency ω0 is the same. Put down an estimate of the Q-value of each oscillator by looking at the graph. It may help for you to put down the definition of Q most relevant to the process of estimation on the page.

a)

b)

c)

448	Week 10: The Wave Equation

Problem 4.

k

b

m

A mass m is attached to a spring with spring constant k and immersed in a medium with damping coe cient b. (Gravity, if present at all, is irrelevant as shown in class). The net force on the mass when displaced by x from equilibrium and moving with velocity vx is thus:

Fx = max = −kx − bvx

(in one dimension).

a)Convert this equation (Newton’s second law for the mass/spring/damping fluid arrangement) into the equation of motion for the system, a “second order linear homogeneous di erential equation” as done in class.

b)Optionally solve this equation, finding in particular the exponential damping rate of the solution (the real part of the exponential time constant) and the shifted frequency ω′, assuming that the motion is underdamped. You can put down any form you like for the answer; the easiest is probably a sum of exponential forms. However, you may also simply put down the solution derived in class if you plan to just memorize this solution instead of learn to derive and understand it.

c)Using your answer for ω′ from part b), write down the criteria for damped, underdamped, and critically damped oscillation.

d)Draw three qualitatively correct graphs of x(t) if the oscillator is pulled to a position x0 and released at rest at time t = 0, one for each damping. Note that you should be able to do this part even if you cannot derive the curves that you draw or ω′. Clearly label each curve.

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Problem 5.

a

b

c

m	4m	9m

Three strings of length L (not shown) with the same mass per unit length µ are suspended vertically and blocks of mass m, 4m and 9m are hung from them. The total mass of each string µL m (the strings are much lighter than the masses hanging from them). If the speed of a wave pulse on the first string is v0, fill in the following table with entries for b) and c):

a) v0

b)

c)

450	Week 10: The Wave Equation

Problem 6.

				a
		2
				b
		3
4T				c	4T
		4
				d

In the figure above, the neck of a stringed instrument is schematized. Four strings of di erent thickness (and hence di erent µ as shown) and the same length are stretched in such a way that the tension in each is about the same (T ) in each string. This produces a total of 4T between the end bridges – if this were not so, the neck of the guitar or ukelele or violin would tend to bow towards the side with the greater tension.

If the speed of a wave pulse on the first (lightest) string is v0, fill in the following table for the speed of a wave pulse for the other three:

a) v0

b)

c)

d)

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Problem 7.

ava

bvb

Two combinations of two strings with di erent mass densities are drawn above that are connected in the middle. In both cases the string with the greatest mass density is drawn darker and thicker than the lighter one, and the strings have the same tension T in both a and b. A wave pulse is generated on the string pairs that is travelling from left to right as shown. The wave pulse will arrive at the junction between the strings at time ta (for a) and tb (for b). Sketch reasonable estimates for the transmitted and reflected wave pulses onto (copies of) the a and b figures at time 2ta and 2tb respectively. Your sketch should correctly represent things like the relative speed of the reflected and transmitted wave and any changes you might reasonably expect for the amplitude and appearance of the pulses.

452	Week 10: The Wave Equation

Problem 8.

y	T,
	x
0	L

A string of mass density µ is stretched to a tension T and is fixed at both x = 0 and x = L. The transverse string displacement is measured in the y direction. All answers should be given in terms of these quantities or new quantities you define in terms of these quantities.

a)Following the text, derive the wave equation (the equation of motion) for waves on a string and identify the wave velocity squared in terms of T and µ. This one derivation su ces for this and the next problem.

b)Write down the equation yn(x, t) for a generic standing wave on this string with mode index n, assuming that the string is maximally displaced at t = 0. Verify that it is a solution to the ODE in a). Remember that the string is fixed at both ends!

c)Find kn, ωn, fn, λn for the first three modes supported by the string. Sketch them in on the axes below, labelling nodes and antinodes. Note that you should be able to draw the modes and find at least the wavelengths from the pictures alone.

y

x

0

L

y

x

0

L

y

x

0

L

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Problem 9.

y	T,
	x
	0
	L

A string of mass density µ is stretched to a tension T and is fixed at x = 0 and free (frictionless loop) at x = L. The transverse string displacement is measured in the y direction. All answers should be given in terms of these quantities or new quantities you define in terms of these quantities.

a)Write down the equation yn(x, t) for a generic standing wave on this string with mode index n, assuming that the string is maximally displaced at t = 0. Verify that it is a solution to the ODE in a). Remember that the string is free at one end!

b)Find kn, ωn, fn, λn for the first three modes supported by the string. Sketch them in on the axes below, labelling nodes and antinodes. Note that you should be able to draw the modes and find at least the wavelengths from the pictures alone.

y

x

0

L

y

x

0

L

y

x

0

L

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Problem 10.

L

This problem will help you learn required concepts such as:

•Speed of Wave on String

•Static Equilibrium

•Relationship betweeen Distance, Velocity, and Time

so please review them before you begin.

A string of total length L with a mass density µ is shown hanging from the ceiling above.

a)Find the tension T (y) in the string as a function of y, the distance up from its bottom end. Note that the string is not massless, so each small bit of string must be in static equilibrium.

b)Find the velocity v(y) of a small wave pulse cast into the string at the bottom that is travelling upward.

c)Find the amount of time it will take this pulse to reach the top of the string, reflect, and return to the bottom. Neglect the size (width in y) of the pulse relative to the length of the string.

Hint for last part. Set v = dy/dt, rearrange to get all the y-dependent parts on one side and dt and some constants on the other side, then integrate both sides, the y part from 0 to L, the t part from 0 to t0. Solve for t0, double it. This is what calculus is for!

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Problem 11.

0	L

A string of total mass M and total length L is fixed at both ends, stretched so that the speed of waves on the string is v. It is plucked so that it harmonically vibrates in its n = 4 mode:

y(x, t) = Asin(k4x)cos(ω4t).

Find the instantaneous total energy in the string in terms of M , L, n = 4, v and A (although it will simplify matters to use λ4 and ω4 once you define them in terms of the givens).

If you are a physics or math major, the word ”find” should be interpreted as “derive”, following the derivation presented in the book. All others can get by from remembering the general way the total energy scales with the givens in understandable ways to lead to the formula derived in the book above.

For those who attempt the derivation, remember (or FYI):

Z0	sin2(u)du = Z0	cos2	(u)du =	2
nπ		nπ		nπ

456	Week 11: Sound

Optional Problems

Continue studying for the final exam! Only one more week of class (and one chapter) to go in this textbook! Don’t wait until the last moment to start!