AP Physics 2 · Topic 14.1

Topic 14.1: Properties of Wave Pulses and Waves

Unit 14: Waves, Sound, and Physical Optics12-15% of the multiple-choice section

A wave transfers energy between two locations without transferring matter. A pulse is a single disturbance; a wave is a continuous, periodic one. Mechanical waves need a medium and electromagnetic waves do not. Speed is set by the medium, and amplitude is the maximum displacement from equilibrium.

AP Physics: Unit 14 (topics 14.1 Properties of Wave Pulses and Waves). AP Physics 2 Unit 14, Topic 14.1. One learning objective, 14.1.A, describe the physical properties of waves and wave pulses. Six essential knowledge statements: 14.1.A.1 (waves transfer energy between two locations without transferring matter between those locations), with 14.1.A.1.i (a wave pulse is a single disturbance that transfers energy without transferring matter between two locations) and 14.1.A.1.ii (a wave is modeled as a continuous, periodic disturbance with well-defined wavelength and frequency); 14.1.A.2 (mechanical waves or wave pulses require a medium in which to propagate, electromagnetic waves or wave pulses do not); 14.1.A.3 (the speed at which a wave or wave pulse propagates through a medium depends on the type of wave and the properties of the medium), with 14.1.A.3.i (the speed of all electromagnetic waves in a vacuum is a universal physical constant, c = 3.00 x 10^8 m/s), 14.1.A.3.ii (the speed along a string depends on the tension F_T and the mass per length, relevant equation v_string = sqrt(F_T / (m/l))) and 14.1.A.3.iii (in a given medium, the speed of sound waves increases with the temperature of the medium); 14.1.A.4 (in a transverse wave the direction of the disturbance is perpendicular to the direction of propagation); 14.1.A.5 (in a longitudinal wave the direction of the disturbance is parallel to the direction of propagation), with 14.1.A.5.i (sound waves are modeled as mechanical longitudinal waves) and 14.1.A.5.ii (the regions of high and low pressure in a sound wave are called compressions and rarefactions, respectively); and 14.1.A.6 (amplitude is the maximum displacement of a wave from its equilibrium position), with 14.1.A.6.i (the amplitude of a longitudinal pressure wave may be determined by the maximum increase or decrease in pressure from equilibrium pressure), 14.1.A.6.ii (the loudness of a sound increases with increasing amplitude) and 14.1.A.6.iii (the energy carried by a wave increases with increasing amplitude). The topic prints no boundary statement; Unit 14's only three sit under Topics 14.4, 14.5 and 14.9. Suggested skills are 1.C, 2.C, 3.B and 3.C, listed identically on the topic page and in the Unit at a Glance table, and 2.B is not among them. Unit 14 is weighted at 12 to 15 percent of the multiple-choice section across a suggested 14 to 23 class periods. Of the 15 equations in the Waves, Sound, and Optics group of the equation sheet, the string speed equation is the one belonging to this topic; c is printed separately in the Constants and Conversion Factors group. No speed of sound and no wave-energy equation is printed anywhere on the AP Physics 2 sheet.

What Topic 14.1 requires

One learning objective, 14.1.A: describe the physical properties of waves and wave pulses. Six essential knowledge statements sit under it, and between them they define almost every word the rest of Unit 14 uses.

14.1.A.1. Waves transfer energy between two locations without transferring matter between those locations. Sub-statements: a wave pulse is a single disturbance that transfers energy without transferring matter between two locations (14.1.A.1.i), and a wave is modeled as a continuous, periodic disturbance with well-defined wavelength and frequency (14.1.A.1.ii).
14.1.A.2. Mechanical waves or wave pulses require a medium in which to propagate. Electromagnetic waves or wave pulses do not require a medium in which to propagate.
14.1.A.3. The speed at which a wave or wave pulse propagates through a medium depends on the type of wave and the properties of the medium. Sub-statements cover the speed of light in a vacuum (14.1.A.3.i), the speed of a wave on a string (14.1.A.3.ii), and the temperature dependence of the speed of sound (14.1.A.3.iii).
14.1.A.4. In a transverse wave, the direction of the disturbance is perpendicular to the direction of propagation of the wave.
14.1.A.5. In a longitudinal wave, the direction of the disturbance is parallel to the direction of propagation of the wave. Sound waves are modeled as mechanical longitudinal waves (14.1.A.5.i), and the regions of high and low pressure in a sound wave are called compressions and rarefactions, respectively (14.1.A.5.ii).
14.1.A.6. Amplitude is the maximum displacement of a wave from its equilibrium position. Three sub-statements attach pressure amplitude, loudness and energy to it.

Suggested skills for this topic: 1.C create qualitative sketches of graphs that represent features of a model or the behavior of a physical system, 2.C compare physical quantities between two or more scenarios or at different times and locations in a single scenario, 3.B apply an appropriate law, definition, theoretical relationship, or model to make a claim, and 3.C justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws. Those four are printed identically on the topic page and in the Unit at a Glance table.

Topic 14.1 prints no boundary statement. Unit 14 has exactly three, and they sit under Topics 14.4, 14.5 and 14.9.

Notice what 14.1.A is: it is a describe objective. Every verb in the essential knowledge is definitional. The one place a number can be demanded is the string speed equation, and 14.1's skill list does not even include 2.B, calculate an unknown quantity.

Energy moves, matter does not

Everything in this topic hangs off one sentence, 14.1.A.1: waves transfer energy between two locations without transferring matter between those locations.

Watch a rope. Snap one end and a hump travels to the far end. Nothing about the rope has moved from your hand to the wall. Each bit of rope went up, came back down, and finished where it started. What arrived at the wall was energy.

The same is true of a duck on a lake. A boat's wake reaches the duck and the duck bobs; it does not get carried across the water. If waves moved matter, standing in the ocean would move you steadily out to sea.

This distinction is worth stating carefully because it is the source of a common wrong answer. When a question asks what a wave transports, the answer is energy. Matter is not transported, and neither is the medium. The medium is what oscillates, in place, so that the disturbance can move through it.

The CED then splits that disturbance in two:

A wave pulse is a single disturbance (14.1.A.1.i). One snap of the rope, one clap, one flash.
A wave is modeled as a continuous, periodic disturbance with well-defined wavelength and frequency (14.1.A.1.ii). Keep shaking the rope at a steady rate and you have one.

Read the wording of 14.1.A.1.ii closely, because it is doing more work than it looks. It says modeled as. A real disturbance is neither perfectly continuous nor perfectly periodic; the wave is the model you lay over it. And it says with well-defined wavelength and frequency, which is why those two quantities exist for a wave and not for a single pulse. A pulse has an amplitude, a speed and a shape, and asking for its wavelength is a category error. Everything about periodic quantities is deferred to Topic 14.2.

Mechanical or electromagnetic: which waves need something to travel in

14.1.A.2 gives the classification in one sentence each way. Mechanical waves or wave pulses require a medium in which to propagate. Electromagnetic waves or wave pulses do not require a medium in which to propagate.

That is the whole test. Ask whether the wave needs stuff.

	Needs a medium	Examples in Unit 14
Mechanical	Yes	Sound, waves on a string, water waves
Electromagnetic	No	Light, radio, X-rays

Two consequences follow that show up constantly.

Sound cannot cross a vacuum. Sound is a mechanical longitudinal wave (14.1.A.5.i), so with nothing to compress there is no wave. This is the physics behind the ringing bell in a bell jar losing its sound as the air is pumped out, while the bell stays visible the whole time: the light still gets through.

Light crosses a vacuum at a fixed speed. 14.1.A.3.i states that the speed of all electromagnetic waves in a vacuum is a universal physical constant, $c = 3.00 \times 10^8$ m/s. Note the two strong words. All: radio and gamma rays travel at the same $c$ in a vacuum despite a wavelength ratio of many orders of magnitude. Universal physical constant: it is not a property of a medium the way the speed of sound in air is.

The value $3.00 \times 10^8$ m/s is printed in the Constants and Conversion Factors group of the AP Physics 2 equation sheet, so you do not have to remember it. Topic 14.4 picks up what an electromagnetic wave actually consists of.

Wave speed is a property of the medium

14.1.A.3 is the statement students most often contradict without noticing: the speed at which a wave or wave pulse propagates through a medium depends on the type of wave and the properties of the medium.

Read the list of what speed depends on. Type of wave, properties of the medium. That is all. Amplitude is not on it. Frequency is not on it. How hard you shook the rope is not on it.

The CED supplies one worked case, 14.1.A.3.ii: the speed at which a wave pulse or wave propagates along a string is dependent upon the tension in the string, $F_T$ , and the mass per length of the string. The relevant equation is printed on the equation sheet:

v_{\text{string}} = \sqrt{\frac{F_T}{m/\ell}}

Both quantities in it belong to the string, not to you. Tighten the string and waves move faster. Use a heavier string at the same tension and they move slower. Because the relationship is a square root, a factor of four in either quantity is a factor of two in speed, which is the shape practice 2.C questions like.

Then 14.1.A.3.iii, for sound: in a given medium, the speed of sound waves increases with the temperature of the medium. The CED states the direction of the effect only. It gives no equation for it and no table of sound speeds, so a question that needs a numerical speed of sound has to hand it to you. The CED's own sample instructional activity for Topic 14.6 supplies one that way, describing a room where the speed of sound in the air is $343$ m/s.

Three habits follow from all of this:

Change the medium, change the speed. Nothing else in the problem changes it.
Do not use amplitude to argue about speed. Amplitude belongs to 14.1.A.6 and controls energy, not speed.
Do not use frequency to argue about speed either. Topic 14.2 relates speed, frequency and wavelength, and the causal direction there runs from a fixed speed and a chosen frequency to a resulting wavelength, not the other way round.

Transverse and longitudinal: one question tells them apart

14.1.A.4: in a transverse wave, the direction of the disturbance is perpendicular to the direction of propagation of the wave. 14.1.A.5: in a longitudinal wave, the direction of the disturbance is parallel to the direction of propagation of the wave.

Both definitions compare the same two directions, so there is exactly one question to ask: which way does a bit of the medium move, compared with which way the wave goes?

Perpendicular means transverse. A rope shaken up and down while the pulse runs sideways.
Parallel means longitudinal. A spring pushed and pulled along its own length while the compression runs along it.

Two things follow that the CED states explicitly.

Sound is longitudinal (14.1.A.5.i): sound waves are modeled as mechanical longitudinal waves. Air molecules move back and forth along the direction the sound travels, not across it.

Compressions and rarefactions are the names of the two extremes (14.1.A.5.ii): the regions of high and low pressure in a sound wave are called compressions and rarefactions, respectively. High pressure is a compression. Low pressure is a rarefaction. The word order in the CED's sentence tells you which goes with which, and mixing them up is an easy mark to lose.

The transverse and longitudinal split is not just vocabulary. It decides whether polarization is possible: Topic 14.3 states that transverse waves can be polarized and oscillate in a single plane, and that longitudinal waves cannot be polarized. So "is this wave transverse?" is a question you will be answering again, two topics later, for a different reason.

Where does light fall? Topic 14.4 settles it: electromagnetic waves are transverse waves because the oscillations of the electric and magnetic fields are perpendicular to the direction of propagation.

Amplitude, and exactly how far the CED goes with it

14.1.A.6: amplitude is the maximum displacement of a wave from its equilibrium position. The word maximum matters, and so does from equilibrium. Amplitude is measured from the rest position out to a crest, not from a trough up to a crest. The trough-to-crest distance is twice the amplitude.

For a longitudinal wave there is nothing visibly displaced sideways, so 14.1.A.6.i gives the equivalent: the amplitude of a longitudinal pressure wave may be determined by the maximum increase or decrease in pressure from equilibrium pressure. Same idea, measured in pascals instead of metres, and again measured from the equilibrium value.

Then two consequences:

14.1.A.6.ii. The loudness of a sound increases with increasing amplitude.
14.1.A.6.iii. The energy carried by a wave increases with increasing amplitude.

Now read what those two statements do not say. Both are worded as increases with. Neither gives a proportionality, and there is no equation on the AP Physics 2 equation sheet relating a wave's energy to its amplitude. So on an AP Physics 2 question the defensible claim is that doubling the amplitude increases the energy. The claim that it multiplies the energy by four is not supported by anything in this course's framework or on its sheet, and asserting it is a way to lose a justification mark on a 3.C question.

There is one more sentence about energy, and it lives in the next topic rather than this one: 14.2.A.1.iv says the energy of a wave increases with increasing frequency. Taken together the CED gives you two levers on energy, amplitude in 14.1 and frequency in 14.2, both stated as directions rather than as formulas. And 14.2.A.1.iii keeps them separate: the amplitude of a wave is independent of the period and the frequency of that wave.

Sketching the wave (skill 1.C)

Topic 14.1 is the only place in Unit 14 outside Topic 14.6 where skill 1.C, create qualitative sketches of graphs that represent features of a model or the behavior of a physical system, is listed. It is worth knowing what a good sketch has to show.

For a transverse wave, the picture almost draws itself: displacement on the vertical axis, and the medium visibly humped. Label the equilibrium line, mark the amplitude as the distance from that line to a crest, and put an arrow on the wave showing the direction of propagation and a second arrow on one particle of the medium showing that it moves perpendicular to the first.

For a longitudinal wave, a picture of the medium is a row of dots, bunched at compressions and spread at rarefactions. That is hard to read amplitudes off, so the standard move is to graph pressure instead: pressure on the vertical axis against position along the direction of travel, with the equilibrium pressure as the horizontal line. Compressions sit above it, rarefactions below, and the amplitude is the maximum departure from the line in either direction, exactly as 14.1.A.6.i says.

Two cautions about that second graph, because it looks like the transverse picture and means something different:

The sine-like curve is a plot of pressure, not a picture of the air moving up and down. The air is moving left and right.
Nothing about the graph being smooth and wave-shaped makes the wave transverse. The graph is a representation, the wave is a longitudinal disturbance.

A sketch that carries a labelled axis, an equilibrium line, an amplitude arrow and a propagation arrow answers a 1.C prompt. A sketch of a squiggle does not. Which axis a wave graph carries, position or time, is the subject of Topic 14.2, where it decides whether you can read a wavelength or a period off it.

How Topic 14.1 shows up on the exam, and where it goes wrong

Topic 14.1 is almost entirely conceptual on the multiple-choice section. Its four suggested skills are 1.C, 2.C, 3.B and 3.C, and of those only 2.C, 3.B and 3.C are assessed on Section I at all: skill 1 is not assessed on the multiple-choice section. Practice 2.B, calculate an unknown quantity, is not on this topic's list, so a Topic 14.1 question is more likely to ask you to compare two strings than to hand you a tension and a mass.

The errors that cost marks:

Saying a wave carries matter. The exact wording in 14.1.A.1 is that waves transfer energy between two locations without transferring matter between those locations. Energy moves; the medium oscillates in place.
Giving a pulse a wavelength. 14.1.A.1.ii reserves well-defined wavelength and frequency for a wave, which the CED models as continuous and periodic. A single pulse has amplitude, speed and shape.
Letting amplitude change the speed. 14.1.A.3 lists the type of wave and the properties of the medium. A bigger wave on the same string is not a faster wave.
Quantifying "increases with". 14.1.A.6.iii says the energy carried by a wave increases with increasing amplitude, and stops. No exponent is supplied, and none is printed on the sheet.
Swapping compressions and rarefactions. 14.1.A.5.ii pairs them in order with high and low pressure.
Assuming denser means slower. For a string at fixed tension a larger mass per length does mean a lower speed, and the printed equation shows why. That is a result of the equation for that specific case, not a general law about density, and Topic 14.3 states its own rules in terms of speed rather than density for exactly this reason.

Where to go next. Topic 14.2 adds period, frequency and wavelength to the disturbance defined here. Topic 14.3 asks what happens when it reaches the edge of the medium. The unit hub lays out how all nine topics fit together, and if you want the calculation routine for wave speed rather than the CED framing, the wave speed, frequency and wavelength guide covers it.

Speed on a string, and what happens when you change it (skill 2.C)

A guitar-like string of length $1.50 \text{ m}$ and mass $24.0 \text{ g}$ is stretched to a tension of $60.0 \text{ N}$ . (a) Find the speed of a wave pulse on the string. (b) A second string of the same length and mass is tightened to $240. \text{ N}$ . By what factor does the pulse speed change? (c) A third string, at the original $60.0 \text{ N}$ tension, has four times the mass per length. What is the pulse speed on it?

(a) The printed equation needs mass per length, not mass, so convert first: $\dfrac{m}{\ell} = \dfrac{24.0 \times 10^{-3} \text{ kg}}{1.50 \text{ m}} = 1.60 \times 10^{-2} \text{ kg/m}$ .
Substitute into the relevant equation from 14.1.A.3.ii: $v = \sqrt{\dfrac{F_T}{m/\ell}} = \sqrt{\dfrac{60.0 \text{ N}}{1.60 \times 10^{-2} \text{ kg/m}}} = \sqrt{3.75 \times 10^{3} \text{ m}^2/\text{s}^2}$ .
$v = 61.2 \text{ m/s}$ , to three significant figures.
(b) Do not recompute. The tension goes from $60.0 \text{ N}$ to $240. \text{ N}$ , a factor of $4$ , and $v$ depends on the square root of $F_T$ , so the speed changes by a factor of $\sqrt{4} = 2$ . The new speed is $2 \times 61.2 = 122 \text{ m/s}$ .
(c) Mass per length is under the square root in the denominator, so multiplying it by $4$ divides the speed by $\sqrt{4} = 2$ : $v = 61.2 / 2 = 30.6 \text{ m/s}$ .
Check the reasoning against 14.1.A.3 before finishing. Both changes were changes to the string, which is the medium. Nothing here required knowing the amplitude, the frequency, or how the pulse was made, and that is the point of the essential knowledge statement.

(a) $v = 61.2 \text{ m/s}$ . (b) The speed doubles, to $122 \text{ m/s}$ , because $v$ depends on $\sqrt{F_T}$ . (c) $v = 30.6 \text{ m/s}$ , half the original, because $v$ depends on $1/\sqrt{m/\ell}$ .

Reading the amplitude of a sound wave off its pressure trace

In a room where the equilibrium air pressure is $1.01000 \times 10^{5} \text{ Pa}$ , a microphone records a pure tone whose pressure swings between $1.01032 \times 10^{5} \text{ Pa}$ and $1.00968 \times 10^{5} \text{ Pa}$ . (a) What is the amplitude of this pressure wave? (b) A second tone in the same room swings between $1.01048 \times 10^{5} \text{ Pa}$ and $1.00952 \times 10^{5} \text{ Pa}$ . Which tone is louder, and what can you say about the energy each carries?

(a) Use 14.1.A.6.i: the amplitude of a longitudinal pressure wave may be determined by the maximum increase or decrease in pressure from equilibrium pressure. Measure from the equilibrium value, not from peak to peak.
Maximum increase: $1.01032 \times 10^{5} - 1.01000 \times 10^{5} = 32 \text{ Pa}$ . Maximum decrease: $1.01000 \times 10^{5} - 1.00968 \times 10^{5} = 32 \text{ Pa}$ . The two agree, as they should for a wave oscillating about equilibrium, so the amplitude is $32 \text{ Pa}$ .
If you had taken the full swing, $1.01032 \times 10^{5} - 1.00968 \times 10^{5} = 64 \text{ Pa}$ , you would have twice the amplitude. That is the standard slip on this kind of question, and it comes from measuring trough to crest instead of from equilibrium.
(b) Same arithmetic for the second tone: $1.01048 \times 10^{5} - 1.01000 \times 10^{5} = 48 \text{ Pa}$ , so its amplitude is $48 \text{ Pa}$ , which is $1.5$ times the first.
By 14.1.A.6.ii, the loudness of a sound increases with increasing amplitude, so the second tone is louder. By 14.1.A.6.iii, the energy carried by a wave increases with increasing amplitude, so the second tone also carries more energy.
Stop there on the energy. The CED states the direction of the relationship and no more, and no equation linking amplitude to energy is printed on the AP Physics 2 equation sheet. Saying the second tone carries $1.5^2 = 2.25$ times the energy goes beyond what this course gives you.

(a) The amplitude is $32 \text{ Pa}$ , measured as the maximum departure from the equilibrium pressure. (b) The second tone has an amplitude of $48 \text{ Pa}$ , so it is louder (14.1.A.6.ii) and carries more energy (14.1.A.6.iii). The CED supports "more", not a numerical factor.

Thunder and lightning: one needs a medium, one does not

A lightning strike happens $1.50 \text{ km}$ away on a day when the speed of sound in the air is $343 \text{ m/s}$ . (a) How long does the flash take to reach you? (b) How long does the thunder take? (c) Explain, using the CED's own classification, why the two answers differ by so much, and say which of the two figures would change if the air were warmer.

(a) Light is an electromagnetic wave, and 14.1.A.3.i gives its speed in a vacuum as the universal constant $c = 3.00 \times 10^8 \text{ m/s}$ , which is also printed in the Constants group of the equation sheet. Air is close enough to a vacuum for this estimate.
$t_{\text{light}} = \dfrac{1.50 \times 10^{3} \text{ m}}{3.00 \times 10^{8} \text{ m/s}} = 5.00 \times 10^{-6} \text{ s}$ , five microseconds.
(b) Sound is a mechanical longitudinal wave (14.1.A.5.i), travelling here at $343 \text{ m/s}$ : $t_{\text{sound}} = \dfrac{1.50 \times 10^{3} \text{ m}}{343 \text{ m/s}} = 4.37 \text{ s}$ .
The ratio is $\dfrac{4.37 \text{ s}}{5.00 \times 10^{-6} \text{ s}} \approx 8.7 \times 10^{5}$ . The sound takes close to a million times longer.
(c) The classification is 14.1.A.2. Sound is mechanical, so it requires a medium, and its speed is a property of that medium. Light is electromagnetic, so it requires no medium, and its vacuum speed is a universal physical constant rather than a property of anything the wave passes through.
Which figure moves with temperature: the sound one. 14.1.A.3.iii says that in a given medium the speed of sound waves increases with the temperature of the medium, so warmer air means a faster sound and a shorter delay. The CED gives the direction only, so predict "shorter" and do not attempt a new number. The light figure does not move, because $c$ is not a property of the air.

(a) $5.00 \times 10^{-6} \text{ s}$ . (b) $4.37 \text{ s}$ . (c) Sound is mechanical and needs a medium, so its speed is set by the air; light is electromagnetic and needs none, so it travels at the universal constant $c$ . Warmer air raises the speed of sound (14.1.A.3.iii) and shortens the thunder delay, while the flash time is unchanged.

Frequently asked questions

Do waves transfer matter?

No. Essential knowledge 14.1.A.1 in the AP Physics 2 CED states that waves transfer energy between two locations without transferring matter between those locations. The medium oscillates in place while the disturbance moves through it: each bit of a rope goes up and comes back down as a pulse passes, and finishes where it started. What arrives at the far end is energy, not rope. The same holds for sound in air and for a duck bobbing on a wake.

What is the difference between a wave pulse and a wave?

A wave pulse is a single disturbance that transfers energy without transferring matter between two locations (14.1.A.1.i). A wave is modeled as a continuous, periodic disturbance with well-defined wavelength and frequency (14.1.A.1.ii). The practical difference is that wavelength, period and frequency only exist for the repeating case. A pulse has an amplitude, a speed and a shape, so a question asking for the wavelength of a single pulse is asking for something that is not defined.

What is the difference between a transverse and a longitudinal wave?

It is the angle between two directions: the direction the medium is disturbed and the direction the wave travels. In a transverse wave the disturbance is perpendicular to the direction of propagation (14.1.A.4), as on a rope shaken up and down. In a longitudinal wave the disturbance is parallel to the direction of propagation (14.1.A.5), as in a spring pushed along its length. AP Physics 2 models sound as a mechanical longitudinal wave and electromagnetic waves as transverse.

Which waves need a medium to travel through?

Mechanical ones. Essential knowledge 14.1.A.2 states that mechanical waves or wave pulses require a medium in which to propagate, and that electromagnetic waves or wave pulses do not require a medium in which to propagate. So sound, waves on a string and water waves all need matter to travel in, and light, radio waves and X-rays do not. That is why a bell ringing inside a jar goes silent as the air is pumped out while remaining perfectly visible.

Does amplitude affect the speed of a wave?

No. Essential knowledge 14.1.A.3 says the speed at which a wave or wave pulse propagates through a medium depends on the type of wave and the properties of the medium. Amplitude is not on that list, and neither is frequency. For a wave on a string the printed equation makes the point exactly: the speed is the square root of the tension divided by the mass per length, and both of those belong to the string. What amplitude does control, per 14.1.A.6, is the energy the wave carries and, for sound, its loudness.

Does doubling a wave's amplitude quadruple its energy in AP Physics 2?

The AP Physics 2 CED does not say so. Essential knowledge 14.1.A.6.iii states only that the energy carried by a wave increases with increasing amplitude, with no proportionality given, and no equation relating wave energy to amplitude is printed on the AP Physics 2 equation sheet. So the supportable answer on this exam is that the energy increases. Claiming a factor of four goes beyond the course framework, and a justification question is graded against the framework.

Is the wave speed equation for a string on the AP Physics 2 equation sheet?

Yes. The Waves, Sound, and Optics group of the AP Physics 2 equation sheet prints 15 equations, and one of them is the string speed, the square root of the tension divided by the mass per length. The CED cites it as the relevant equation for essential knowledge 14.1.A.3.ii. The speed of light in a vacuum, 3.00 times 10 to the 8 metres per second, is printed separately in the Constants and Conversion Factors group and is cited by 14.1.A.3.i. No speed of sound is printed anywhere on the sheet.