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AP® Physics 2: Algebra-Based · Unit 14 · Topic 14.1

Properties of Wave Pulses and Waves: Study Guide (Physics 2 14.1)

What a wave is and what it carries: pulses and continuous waves, mechanical and electromagnetic waves, transverse and longitudinal motion, wave speed on a string, sound and amplitude.

Energy moves, matter stays

Tie one end of a long rope to a post and flick the other end up and down once. A hump runs along the rope to the post. That hump is a wave pulse: a single disturbance that travels through a medium.

Watch one small piece of the rope, such as a knot tied in it. As the hump passes, the knot moves up and then back down. It ends where it started. The knot never travels to the post. What reaches the post is energy: the post feels a tug when the pulse arrives.

This is the central idea of the whole unit:

A wave transfers energy between two places without transferring matter between them.

If you keep moving your hand up and down in a steady rhythm, you send a continuous train of humps and dips. This is a wave. In this course a wave is modelled as a continuous, repeating (periodic) disturbance with a well-defined wavelength and frequency. Topic 14.2 makes those two quantities precise. A pulse is the simplest case: one disturbance, not a repeating pattern.

A floating ball on a pond shows the same idea. Ripples pass under it and the ball bobs up and down, but it does not ride the ripples to the shore. The ripples carry energy outwards; the water mostly stays where it is.

Mechanical and electromagnetic waves

Waves fall into two families.

  • Mechanical waves are disturbances of a material: a rope, a spring, water, air, a steel rail. They need a medium. Each part of the medium pushes or pulls on the next part, and that is how the disturbance is passed on. Sound, ripples on water and pulses on strings are all mechanical.
  • Electromagnetic (EM) waves are disturbances of electric and magnetic fields. They do not need a medium. Light, radio waves and X-rays are EM waves. Light reaches Earth from the Sun across almost empty space.

Put a ringing alarm clock inside a sealed glass jar and pump the air out. You can still see the clock, because light does not need the air. You hear it less and less, because sound has no medium to travel through. The clock still vibrates; the sound simply cannot leave the jar.

Transverse and longitudinal waves

Compare two directions: the direction in which the medium is disturbed, and the direction in which the wave travels (propagates).

  • In a transverse wave the disturbance is perpendicular to the direction of travel. The rope pulse is transverse: the rope moves up and down while the pulse moves along the rope.
  • In a longitudinal wave the disturbance is parallel to the direction of travel. Push and pull the end of a stretched spring along its length. Coils bunch together and spread apart, and these regions travel along the spring.

Snapshot of a transverse pulse

Figure 1 shows a pulse on a string at two instants. The pulse travels to the right at 4.0 m/s, so in 0.50 s it moves 4.0 m/s × 0.50 s = 2.0 m. Its shape stays the same.

A transverse pulse on a string at two times Horizontal position along the string from 0 to 8 metres. A solid line shows a single upward hump centred at 2.0 metres at time zero. A dashed line shows the same hump centred at 4.0 metres at time 0.50 seconds. An arrow above the pulses labelled v equals 4.0 metres per second points to the right. On the solid pulse, point P on the right-hand (leading) side has an arrow pointing up, and point Q on the left-hand (trailing) side has an arrow pointing down. 0 2.0 4.0 6.0 8.0 Position along the string (m) v = 4.0 m/s P Q t = 0 (solid) t = 0.50 s (dashed)
Figure 1. A transverse pulse moving right at 4.0 m/s. Solid line: the pulse at t = 0. Dashed line: the same pulse 0.50 s later, 2.0 m further on. At t = 0, point P (on the front edge) is moving up and point Q (on the back edge) is moving down. Each point of the string moves only up and down.

To find how a point of the string is moving, imagine the pulse shifted a little in its direction of travel. P is on the front of the hump: as the hump moves right, the string at P is about to be higher, so P is moving up. Q is on the back of the hump: the hump is leaving Q, so Q is moving down. Neither point moves sideways.

Sound is a longitudinal wave

A loudspeaker cone pushes forward and squeezes the air in front of it. Then it pulls back and leaves a region of thinner air. In this course, sound is modelled as a mechanical, longitudinal wave in the medium.

  • A region where the particles are pushed together, and the pressure is higher than normal, is a compression.
  • A region where the particles are spread apart, and the pressure is lower than normal, is a rarefaction.
A sound wave as layers of air and as a pressure graph Top: a row of thin vertical lines representing layers of air at one instant. The lines are crowded together near the labels C, at about one quarter and three quarters of the way across, and spread out near the labels R, at the left end, the middle and the right end. An arrow shows the wave travelling to the right. Bottom: a graph of air pressure against position for the same instant. The horizontal dashed line is the normal air pressure P zero. The curve rises to a maximum directly below each C and falls to a minimum directly below each R. A dotted guide line runs right from the maximum below the second C, and a vertical arrow from P zero up to this guide line is marked as the amplitude. CCRRR direction of travel P₀ pressure amplitude Position x (same instant)
Figure 2. Top: layers of air in a sound wave at one instant. C marks compressions (layers crowded, pressure above normal); R marks rarefactions (layers spread out, pressure below normal). Bottom: pressure against position at the same instant. Peaks line up with compressions and troughs with rarefactions. The amplitude is the largest change in pressure from the normal value P₀.

Each layer of air moves back and forth along the direction of travel, around its own equilibrium position. The compressions and rarefactions travel; the air does not flow away from the speaker.

What sets the speed of a wave

The speed of a wave or pulse is set by what kind of wave it is and by the properties of the medium it travels through. It does not depend on how hard or how far you shake the source.

Pulses on a string

For a pulse or wave on a stretched string:

v = √(F_T / μ), where μ = m/ℓ

  • F_T is the tension in the string (N).
  • μ is the mass per length of the string (kg/m): the mass m of a piece of string divided by its length ℓ.

Units check: √(N ÷ (kg/m)) = √((kg·m/s²)(m/kg)) = √(m²/s²) = m/s.

The two quantities pull in opposite directions:

  • More tension, faster wave. A tighter string pulls each piece back more strongly.
  • More mass per length, slower wave. Heavier string has more inertia, so each piece responds more slowly.

Because of the square root, you need four times the tension to double the speed.

Sound and temperature

Sound travels through solids, liquids and gases, at different speeds in each. In a given medium, the speed of sound increases with the temperature of the medium. In warmer air the molecules move faster, so a compression is passed on more quickly. In dry air at 20 °C sound travels at about 343 m/s.

Light

All EM waves travel at the same speed in a vacuum, a universal constant: c = 3.00 × 10⁸ m/s. That is about 875 000 times the speed of sound in air at 20 °C.

Amplitude, energy and loudness

The amplitude A of a wave is the maximum displacement of the medium from its equilibrium position. For the rope, it is the height of the hump above the undisturbed rope. It is not the full height from the bottom of a dip to the top of a hump.

For sound, it is often easier to measure pressure than displacement. The amplitude of a pressure wave is the largest increase or decrease in pressure from the equilibrium pressure. If the pressure at a microphone swings between 101 000.6 Pa and 100 999.4 Pa around a normal 101 000 Pa, the pressure amplitude is 0.6 Pa. Note how small this is: a loud sound changes the air pressure by a tiny fraction.

Two links to remember:

  • The energy carried by a wave increases with its amplitude. To make a bigger pulse on a rope, you must do more work on the rope.
  • For sound, a larger amplitude is heard as a louder sound.

(Background: for a given wave the energy grows with the square of the amplitude, but this course asks only that you know energy increases with amplitude.)

Worked example 1: a pulse on a stretched string

Question. A string is 2.0 m long and has a mass of 50 g. It is held at a tension of 90 N. (a) How fast does a pulse travel along it? (b) How long does a pulse take to travel the full length? (c) What tension would make pulses travel at 120 m/s? (d) The string is replaced by a thicker one with four times the mass per length, at 90 N. What is the new pulse speed?

  1. Mass per length: μ = m/ℓ = 0.050 kg ÷ 2.0 m = 0.025 kg/m. (Convert grams to kilograms first.)
  2. (a) v = √(F_T/μ) = √(90 N ÷ 0.025 kg/m) = √3600 m²/s² = 60 m/s.
  3. (b) t = distance ÷ speed = 2.0 m ÷ 60 m/s = 0.033 s (0.0333 s).
  4. (c) Rearrange: F_T = μv² = (0.025 kg/m)(120 m/s)² = 360 N. Check with factors: the speed doubles, so the tension must rise by 2² = 4, and 4 × 90 N = 360 N.
  5. (d) v = √(90 ÷ 0.10) = √900 = 30 m/s. Four times the mass per length halves the speed.

Interpretation and check. The answer to (a) has units of m/s, as it must. Doubling the speed needed four times the tension, and four times μ halved the speed. Both follow from the square root. Nothing in the calculation used the size of the pulse, so a pulse twice as tall would still take 0.033 s.

Worked example 2: seeing and hearing the same event

Question. At an athletics track, a timer stands 170 m from the starter. The starter fires a pistol, which makes a flash and a bang at the same moment. On this day the speed of sound in the air is 343 m/s. (a) How long after the flash does the timer hear the bang? (b) How long does the light take? (c) Explain why the timer should start the clock on the flash, not the bang. (d) On a hotter day the speed of sound is 353 m/s. Does the delay get longer or shorter? By how much?

  1. (a) Sound: t = 170 m ÷ 343 m/s = 0.496 s.
  2. (b) Light: t = 170 m ÷ (3.00 × 10⁸ m/s) = 5.7 × 10⁻⁷ s, less than a millionth of a second.
  3. (c) The flash reaches the timer almost instantly. The bang arrives about half a second late. Starting on the bang would make every runner’s recorded time about 0.50 s too short. In a sprint, that is a very large error.
  4. (d) Sound travels faster in warmer air, so the delay is shorter: t = 170 m ÷ 353 m/s = 0.482 s. The delay is shorter by 0.496 s − 0.482 s = 0.014 s.

Interpretation and check. Two kinds of wave carry news of the same event. Light is an EM wave and needs no medium; its speed is fixed at c. Sound is a mechanical wave whose speed depends on the air and its temperature. The light time is so short that it is safe to treat the flash as instant.

Common misconceptions

  • “The wave carries the water (or air, or rope) with it.” The medium moves around its equilibrium position and returns. Only energy travels from source to receiver.
  • “Transverse means up and down; longitudinal means sideways.” The labels compare two directions: disturbance and travel. A wave on a horizontal string can be transverse with sideways motion, if the string moves side to side while the wave moves along it.
  • “Sound can travel through space.” Sound is mechanical and needs a medium. Light and other EM waves can cross a vacuum.
  • “A harder flick makes a faster pulse.” A harder flick gives a larger amplitude and more energy. The speed is set by the medium: on a string, by F_T and μ.
  • “Double the tension, double the speed.” v depends on the square root of F_T. Doubling the tension multiplies the speed by √2 ≈ 1.41.
  • Using the total mass instead of the mass per length. μ = m/ℓ. A long string and a short string of the same material have the same μ.
  • Taking amplitude as the distance from crest to trough. Amplitude is measured from equilibrium to the maximum: half of that distance.
  • “The air in a compression is pushed all the way to the listener.” Compressions are regions of high pressure and crowded particles. The particles in them oscillate about their own positions; they do not travel with the compression.

Where this leads

Topic 14.2 turns the continuous wave into a periodic wave with a period, frequency and wavelength, and links them to the speed through v = fλ. Continue with Periodic Waves. Before that, test yourself with the practice questions, then use the revision notes and the topic checklist to consolidate.

Quick questions

If the medium does not travel with the wave, what does move?

Each small piece of the medium moves a little around its own equilibrium position and then returns. The disturbance, and the energy it carries, is handed on from one piece to the next. So the pattern travels a long way while each particle stays close to home.

Does a bigger pulse on a string travel faster?

No. In the model used in this course the speed depends only on the medium: for a string, the tension and the mass per length. A tall pulse and a small pulse sent along the same string at the same tension arrive at the same time. The tall pulse simply carries more energy.

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