---
title: "Mechanical Waves"
book: "High School Physics"
subject: physics
language: en
chapter: 20
exercises: 15
source: https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves
---

# Chapter 20 — Mechanical Waves

Saturday evening, a full stadium: a ripple of standing spectators sweeps around the stands in under a minute, yet nobody leaves their seat. A pebble’s ring crosses the pond while the cork only bobs. This chapter studies the traveler itself — the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) — and the four numbers that tame it: speed, [delay](#def-g12-mechanical-waves-delay), [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period), [wavelength](#def-g12-mechanical-waves-wavelength) — enough to read an earthquake off a strip of paper.

## 20.1 The disturbance travels, the medium stays

**Definition 20.1 (Mechanical wave).**

A *mechanical wave* is a disturbance propagating through a material medium — rope, water, air, rock — carrying [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) *without transporting matter*: each piece of the medium swings briefly in place, hands the motion on, and settles back.

**Example 20.2 (No matter transport).**

In the stadium [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) the “medium” is the crowd: what travels is the act of standing up, while every spectator keeps their seat. On the pond, the ring expands but a floating cork only bobs in place.

**Definition 20.3 (Transverse and longitudinal waves).**

A [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) is *transverse* when the medium moves *perpendicular* to the propagation: a hump races along a flicked rope while each point moves only up and down. It is *longitudinal* when the medium moves *along* the propagation: a zone of squeezed [coils](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid) travels down a pushed spring, each [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid) swinging in place.

![Transverse (rope): medium moves perpendicular to propagation. Longitudinal (spring): compressions travel, coils swing in place.](https://one-course.com/images/onecourse/chapters/physics-2/g12-mechanical-waves/fig-79877a900902.svg)

*[Transverse](#def-g12-mechanical-waves-transverse) (rope): medium moves perpendicular to propagation. [Longitudinal](#def-g12-mechanical-waves-transverse) (spring): compressions travel, [coils](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid) swing in place.*

**Remark 20.4 (Sound is longitudinal).**

[Sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) is the [longitudinal wave](#def-g12-mechanical-waves-transverse) par excellence: compressed air — higher [pressure](https://one-course.com/books/physics/2/en/chapter/7-pressure-from-sport-to-diving#def-g10-pressure-pressure) ([Chapter 7](https://one-course.com/books/physics/2/en/chapter/7-pressure-from-sport-to-diving#ch-g10-pressure)) — travels from source to eardrum; [Chapter 21](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#ch-g12-sound-acoustics) is devoted to it. Earthquakes send both kinds: P [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) compress the rock, S [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) shake it sideways.

## 20.2 Speed and delay

**Definition 20.5 (Propagation speed).**

The *propagation speed* of a [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) is the distance covered by the disturbance divided by the time taken: $v = d/t$. It is the speed of the traveling *shape*, not of the medium.

![A rope pulse photographed twice: the crest moved 2.4\, m in 0.30\, s, so v = 8.0\, m/ s — the shape travels, the rope stays.](https://one-course.com/images/onecourse/chapters/physics-2/g12-mechanical-waves/fig-81d13f885aa8.svg)

*A rope pulse photographed twice: the crest moved $2.4\,\mathrm{m}$ in $0.30\,\mathrm{s}$, so $v = 8.0\,\mathrm{m}/\mathrm{s}$ — the shape travels, the rope stays.*

**Proposition 20.6 (The medium sets the speed).**

For small disturbances, the [propagation speed](#def-g12-mechanical-waves-speed) depends only on the *medium*, not on the shape or size of the disturbance. On a stretched rope, $v$ increases with [tension](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) and decreases with mass per [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit): taut and light is fast, slack and heavy is slow.

**Proof.** *Admitted at this level.* ∎

**Remark 20.7 (Where the formula lives).**

The rope’s exact speed formula is derived in the Year 1 volume; this year the experimental facts above suffice.

**Example 20.8 (Sound speeds).**

[Sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) travels at $340\,\mathrm{m}/\mathrm{s}$ in air, $1500\,\mathrm{m}/\mathrm{s}$ in water, $5900\,\mathrm{m}/\mathrm{s}$ in steel — the stiffer, the faster. Hence the western-movie ear on the rail: steel announces the train first.

**Definition 20.9 (Delay).**

A [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) of speed $v$ reaches a point at distance $d$ after the *delay* $\tau = d/v$: two points a distance $d$ apart repeat the same motion, the farther lagging by $\tau$ — the [echo](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#rem-g10-signals-and-waves-honest) ranging of [Chapter 8](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#ch-g10-signals-and-waves), read backwards.

**Proposition 20.10 (Superposition without interaction).**

Where two [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) overlap, the displacement of the medium is at every instant the *sum* of the displacements each would cause alone; after crossing, each [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) continues unchanged — derived in the Year 2 volume from the linearity of the equations of motion.

**Proof.** *Admitted at this level.* ∎

**Example 20.11 (Crossing pulses).**

Send an upward hump from each end of a rope: where they meet, the rope briefly rises to the sum of the two heights; then two intact humps emerge — as two crossing conversations reach their listeners.

## 20.3 Periodic waves: the double periodicity

**Definition 20.12 (Periodic wave).**

When the source repeats its motion with [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) $T$ and [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) $f = 1/T$ ([Chapter 8](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#ch-g10-signals-and-waves)), it feeds the medium a *periodic wave*: every point repeats its own motion with [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) $T$, each lagging its neighbors by the travel [delay](#def-g12-mechanical-waves-delay).

**Definition 20.13 (Wavelength).**

A snapshot of a [periodic wave](#def-g12-mechanical-waves-periodic) shows a pattern repeating in space; its spatial [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) — crest to crest — is the *wavelength* $\lambda$, in [metres](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit). A [periodic wave](#def-g12-mechanical-waves-periodic) is *doubly periodic*: [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) $T$ in time at each point, [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) $\lambda$ in space at each instant.

**Theorem 20.14 (The wave relation).**

A [periodic wave](#def-g12-mechanical-waves-periodic) of speed $v$, [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) $T$ and [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) $f$ has [wavelength](#def-g12-mechanical-waves-wavelength) $\lambda = v\,T = v/f$.

**Proof.** During one [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) the pattern slides forward by $v\,T$; but the source — hence every point — is then back in its starting state, so the slid pattern coincides with the original: the repeat distance is $v\,T$. ∎

![The same sine twice: the snapshot repeats every in space, the one-point signal every T in time; = v\,T.](https://one-course.com/images/onecourse/chapters/physics-2/g12-mechanical-waves/fig-b7523830864e.svg)

*The same sine twice: the snapshot repeats every $\lambda$ in space, the one-point [signal](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-signal) every $T$ in time; $\lambda = v\,T$.*

**Remark 20.15 (The frequency belongs to the source).**

Passing from one medium into another — [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) from air into water — a [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) keeps its [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency), which the source dictates, while its speed changes; by $\lambda = v/f$ the [wavelength](#def-g12-mechanical-waves-wavelength) changes with it.

**Example 20.16 (Concert A, in air and underwater).**

The $440\,\mathrm{Hz}$ concert A of [Chapter 8](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#ch-g10-signals-and-waves) has [wavelength](#def-g12-mechanical-waves-wavelength) $340/440 = 0.77\,\mathrm{m}$ in air and $1500/440 =
3.4\,\mathrm{m}$ in water: same note, stretched by the faster medium.

**Definition 20.17 (In phase, in phase opposition).**

Two points separated by a whole number of [wavelengths](#def-g12-mechanical-waves-wavelength), $d = k\lambda$, move identically at every instant: they are *in phase*. Points separated by $d = (k + \tfrac12)\lambda$ always move oppositely — crest against trough: they are in *phase opposition*.

**Method 20.18 (Reading the double periodicity).**

1. On a recording $y(t)$ at a fixed point, read $T$ crest to crest; $f = 1/T$ .
2. On a snapshot $y(x)$ at a fixed instant, read $\lambda$ crest to crest.
3. Deduce $v = \lambda/T = \lambda f$ ; check against a direct timing $v = d/\tau$ when available.
4. Never confuse the two graphs: one axis carries seconds, the other [metres](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) .

## 20.4 Wavefronts and attenuation

**Definition 20.19 (Wavefront).**

A *wavefront* is a line (or surface) of points that the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) reaches at the same instant — a crest line on the pond. From a point source the wavefronts are expanding circles; far from any source they are nearly straight. The [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) propagates perpendicular to its wavefronts, successive crests $\lambda$ apart.

**Definition 20.20 (Attenuation).**

The decrease of a [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave)’s [amplitude](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-amplitude) as it propagates is its *attenuation*. Two causes add up: the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) of a circular [wavefront](#def-g12-mechanical-waves-wavefront) is shared over an ever-longer crest, and the medium absorbs a little in passing. Straight [wavefronts](#def-g12-mechanical-waves-wavefront) escape the first cause — how a distant storm’s swell can cross an ocean.

![Circular wavefronts, one wavelength apart, fade as energy spreads over longer crests; straight fronts (right) fade slowly.](https://one-course.com/images/onecourse/chapters/physics-2/g12-mechanical-waves/fig-a1661a6a3074.svg)

*Circular [wavefronts](#def-g12-mechanical-waves-wavefront), one [wavelength](#def-g12-mechanical-waves-wavelength) apart, fade as [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) spreads over longer crests; straight fronts (right) fade slowly.*

## 20.5 Exercises

**Exercise 20.1 ★.**

A stadium [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) runs the $320\,\mathrm{m}$ ring of the stands in $40\,\mathrm{s}$. (a) Compute its speed. (b) Does any spectator travel around the ring? What does? (c) Each spectator stands $1.5\,\mathrm{s}$ after their neighbor $12\,\mathrm{m}$ away: check the consistency.

**Solution of Exercise 20.1.**

(a) $v = 320/40 = 8.0\,\mathrm{m}/\mathrm{s}$. (b) No spectator moves along the ring: the disturbance (the act of standing) travels, not the crowd. (c) $12/1.5 = 8.0\,\mathrm{m}/\mathrm{s}$ — consistent.

**Exercise 20.2 ★.**

[Transverse](#def-g12-mechanical-waves-transverse) or [longitudinal](#def-g12-mechanical-waves-transverse)? (a) a pulse on a rope shaken up and down; (b) a compression sent along a spring; (c) [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) in air; (d) a pond ripple (surface bobs [vertically](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight), ring travels horizontally); (e) seismic P [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) (compress along the travel) and S [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) (shake sideways).

**Solution of Exercise 20.2.**

(a) [transverse](#def-g12-mechanical-waves-transverse); (b) [longitudinal](#def-g12-mechanical-waves-transverse); (c) [longitudinal](#def-g12-mechanical-waves-transverse); (d) [transverse](#def-g12-mechanical-waves-transverse); (e) P [longitudinal](#def-g12-mechanical-waves-transverse), S [transverse](#def-g12-mechanical-waves-transverse).

**Exercise 20.3 ★.**

Thunder reaches you $4.0\,\mathrm{s}$ after the flash. (a) How far is the strike ($v = 340\,\mathrm{m}/\mathrm{s}$)? (b) A diver hears the same [delay](#def-g12-mechanical-waves-delay) between a flash and an underwater boom ($v = 1500\,\mathrm{m}/\mathrm{s}$): how far is that source? (c) What must you know before turning [delay](#def-g12-mechanical-waves-delay) into distance?

**Solution of Exercise 20.3.**

(a) $d = 340 \times 4.0 \approx 1.4\,\mathrm{km}$. (b) $1500 \times 4.0 = 6.0\,\mathrm{km}$. (c) The [propagation speed](#def-g12-mechanical-waves-speed) in the medium crossed.

**Exercise 20.4 ★.**

Compute the [wavelength](#def-g12-mechanical-waves-wavelength) of a $440\,\mathrm{Hz}$ [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) in air and in water. When the [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) crosses from air into water, which of $f$, $v$, $\lambda$ change, and which stays fixed?

**Solution of Exercise 20.4.**

$\lambda = v/f$: $340/440 = 0.77\,\mathrm{m}$ in air, $1500/440 = 3.4\,\mathrm{m}$ in water. The [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) stays fixed (the source dictates it); $v$ and $\lambda$ both change.

**Exercise 20.5 ★.**

On a pond, adjacent ripple crests are $12\,\mathrm{cm}$ apart and the rings advance at $0.30\,\mathrm{m}/\mathrm{s}$. Find the [wavelength](#def-g12-mechanical-waves-wavelength), the [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) and the [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) of the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave).

**Solution of Exercise 20.5.**

$\lambda = 12\,\mathrm{cm}$; $T = \lambda/v = 0.12/0.30 = 0.40\,\mathrm{s}$; $f = 1/T = 2.5\,\mathrm{Hz}$.

**Exercise 20.6 ★★.**

A pulse on a long rope is photographed twice, $0.30\,\mathrm{s}$ apart: its crest sits at $x = 1.0\,\mathrm{m}$, then $3.4\,\mathrm{m}$. (a) The speed? (b) When does the pulse reach the wall at $x = 7.0\,\mathrm{m}$? (c) A ribbon knotted at $x = 5.0\,\mathrm{m}$: its motion as the pulse passes?

**Solution of Exercise 20.6.**

(a) $v = 2.4/0.30 = 8.0\,\mathrm{m}/\mathrm{s}$. (b) From the first photo the crest has $6.0\,\mathrm{m}$ to go: $t = 6.0/8.0 = 0.75\,\mathrm{s}$ after it. (c) The ribbon rises then falls, purely transversely, and stays at $x = 5.0\,\mathrm{m}$: the rope does not travel.

**Exercise 20.7 ★★.**

Two upward pulses, of heights $3.0\,\mathrm{cm}$ and $2.0\,\mathrm{cm}$, travel toward each other at $2.0\,\mathrm{m}/\mathrm{s}$ each, crests $4.0\,\mathrm{m}$ apart. (a) When do the crests coincide? (b) The rope’s maximal height then? (c) Same if the small pulse is downward. (d) The rope just after?

**Solution of Exercise 20.7.**

(a) Closing speed $4.0\,\mathrm{m}/\mathrm{s}$, gap $4.0\,\mathrm{m}$: $t = 1.0\,\mathrm{s}$. (b) Superposition: $3.0 + 2.0 = 5.0\,\mathrm{cm}$. (c) $3.0 - 2.0 = 1.0\,\mathrm{cm}$. (d) Two intact pulses emerge and continue unchanged, as if they had never met.

**Exercise 20.8 ★★.**

(a) Two clotheslines carry the same [tension](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory); a pulse is sent down each. Which is faster: the [thin](https://one-course.com/books/physics/2/en/chapter/10-lenses-images-and-the-eye#def-g11-lenses-and-eye-lens) one or the thick one? Why? (b) A guitarist tightens a string: what happens to the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) speed? (c) Does flicking twice as hard (a taller pulse) make the pulse travel faster?

**Solution of Exercise 20.8.**

(a) The [thin](https://one-course.com/books/physics/2/en/chapter/10-lenses-images-and-the-eye#def-g11-lenses-and-eye-lens) one: less mass per [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) at equal [tension](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) means a faster [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave). (b) It increases (higher [tension](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory)). (c) No: the speed depends only on the medium, not on the pulse’s size.

**Exercise 20.9 ★★.**

A worker strikes a rail $1.8\,\mathrm{km}$ away; you listen with one ear on the steel. Compute the arrival times through the rail ($5900\,\mathrm{m}/\mathrm{s}$) and the air ($340\,\mathrm{m}/\mathrm{s}$), and the gap between the two bangs.

**Solution of Exercise 20.9.**

Rail: $1800/5900 \approx 0.31\,\mathrm{s}$; air: $1800/340 \approx 5.3\,\mathrm{s}$; gap $\approx 5.0\,\mathrm{s}$.

**Exercise 20.10 ★★.**

A rope carries a [periodic wave](#def-g12-mechanical-waves-periodic) of [wavelength](#def-g12-mechanical-waves-wavelength) $24\,\mathrm{cm}$. [In phase](#def-g12-mechanical-waves-phase), in [phase opposition](#def-g12-mechanical-waves-phase), or neither, for point separations (a) $48\,\mathrm{cm}$; (b) $12\,\mathrm{cm}$; (c) $36\,\mathrm{cm}$; (d) $30\,\mathrm{cm}$; (e) $6.0\,\mathrm{cm}$?

**Solution of Exercise 20.10.**

With $\lambda = 24\,\mathrm{cm}$: (a) $2\lambda$, [in phase](#def-g12-mechanical-waves-phase); (b) $\lambda/2$, opposition; (c) $\tfrac32\lambda$, opposition; (d) $1.25\lambda$, neither; (e) $\lambda/4$, neither.

**Exercise 20.11 ★★.**

An earthquake sends P [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) at $6.0\,\mathrm{km}/\mathrm{s}$ and S [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) at $3.5\,\mathrm{km}/\mathrm{s}$, both of [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) $2.0\,\mathrm{s}$. Compute both [wavelengths](#def-g12-mechanical-waves-wavelength), compare them with a building, and explain why a whole neighborhood rises and falls nearly together.

**Solution of Exercise 20.11.**

$\lambda_P = 6.0 \times 2.0 = 12\,\mathrm{km}$; $\lambda_S = 3.5 \times 2.0 = 7.0\,\mathrm{km}$ — hundreds of times a building’s size. Points much closer than $\lambda$ are nearly [in phase](#def-g12-mechanical-waves-phase), so the whole neighborhood moves together.

**Exercise 20.12 ★★★.**

A recording of the water height at a buoy shows crests every $0.50\,\mathrm{s}$; an aerial photograph shows crests $0.75\,\mathrm{m}$ apart. (a) Find $T$ and $f$. (b) Find $\lambda$. (c) Deduce the speed. (d) The buoy rides a crest at $t = 0$: when next? (e) Is a point $1.875\,\mathrm{m}$ away [in phase](#def-g12-mechanical-waves-phase) with the buoy, in [phase opposition](#def-g12-mechanical-waves-phase), or neither?

**Solution of Exercise 20.12.**

(a) $T = 0.50\,\mathrm{s}$, $f = 2.0\,\mathrm{Hz}$. (b) $\lambda = 0.75\,\mathrm{m}$. (c) $v = \lambda/T = 0.75/0.50 = 1.5\,\mathrm{m}/\mathrm{s}$. (d) At $t = T = 0.50\,\mathrm{s}$. (e) $1.875/0.75 = 2.5$ [wavelengths](#def-g12-mechanical-waves-wavelength) $= (2 + \tfrac12)\lambda$: [phase opposition](#def-g12-mechanical-waves-phase).

**Exercise 20.13 ★★★.**

An anchored buoy completes $10$ full oscillations in $80\,\mathrm{s}$ on a regular swell whose crests are $120\,\mathrm{m}$ apart. (a) Find the [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) and [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency). (b) Find the speed of the swell, in $\mathrm{m}/\mathrm{s}$ and $\mathrm{km}/\mathrm{h}$. (c) How long does a crest now at the buoy take to reach the beach $900\,\mathrm{m}$ away? (d) Does the buoy drift ashore with it?

**Solution of Exercise 20.13.**

(a) $T = 80/10 = 8.0\,\mathrm{s}$, $f = 0.125\,\mathrm{Hz}$. (b) $v = \lambda/T = 120/8.0 = 15\,\mathrm{m}/\mathrm{s} = 54\,\mathrm{km}/\mathrm{h}$. (c) $t = 900/15 = 60\,\mathrm{s}$. (d) No: [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) transport no matter — the anchored buoy only bobs.

**Exercise 20.14 ★★★.**

Two identical pulses, one upward and one downward, travel toward each other at $3.0\,\mathrm{m}/\mathrm{s}$ each, centers $6.0\,\mathrm{m}$ apart. (a) When do their centers coincide, and what does the rope look like then? (b) Is the rope then at rest? Where has the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) gone? (c) What happens next? (d) Why does “momentarily flat” not mean “nothing there”?

**Solution of Exercise 20.14.**

(a) Closing at $6.0\,\mathrm{m}/\mathrm{s}$ over $6.0\,\mathrm{m}$: $t = 1.0\,\mathrm{s}$; the opposite displacements cancel — the rope is momentarily flat. (b) No: its points are moving (the [transverse](#def-g12-mechanical-waves-transverse) velocities add, not cancel); the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) is all kinetic. (c) The two pulses reappear and continue unchanged. (d) The flat rope carries velocity, hence [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy): the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) is momentarily stored in motion, not in shape.

**Exercise 20.15 ★★★.**

A pebble’s ripple spreads on a still pond. (a) Compute the crest length at $r = 0.50\,\mathrm{m}$ and $r = 8.0\,\mathrm{m}$; growth factor? (b) The crest’s [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) is roughly conserved: what happens to the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) per [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit), and to the [amplitude](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-amplitude)? (c) Why does a straight-fronted swell attenuate far less? (d) And the stadium [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave), not at all?

**Solution of Exercise 20.15.**

(a) $2\pi r$: $3.1\,\mathrm{m}$ and $50\,\mathrm{m}$ — $\times 16$. (b) [Energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) per [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) falls by $16$, so the [amplitude](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-amplitude) drops (the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) grows with the [amplitude](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-amplitude)). (c) Straight fronts do not lengthen: only absorption remains. (d) The medium is active: each spectator stands on their own muscle [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy), resupplying the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave).

## 20.6 Problem: Reading a Seismogram

**Problem 20.1.**

Weekend problem — reading a seismogram: two waves race out of a broken fault, three circles pin the epicenter, and a slow third wave gives a coastline its warning

Night shift at a seismological observatory. At 09:14:32 the needle of station A jumps: sharp fast wiggles, then slower, wider swings. An earthquake has broken rock somewhere; two [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) race out through the crust: P [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave), compressions at $v_P = 6.0\,\mathrm{km}/\mathrm{s}$, and S [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave), a sideways shaking at $v_S = 3.5\,\mathrm{km}/\mathrm{s}$.

**Part I — Two [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave), one clock.**

1. From the descriptions above, classify P and S [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) as [transverse](#def-g12-mechanical-waves-transverse) or [longitudinal](#def-g12-mechanical-waves-transverse) .
2. Which [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) arrives first at every station? Compute the travel times of both over $d = 120\,\mathrm{km}$ .
3. Show that the P–S arrival gap is $\Delta t = d\,(1/v_S - 1/v_P)$ and compute it for $d = 120\,\mathrm{km}$ .
4. Deduce the seismologist’s ranging rule $d = \dfrac{v_P\,v_S}{v_P - v_S}\,\Delta t$ and show the coefficient is about $8.4\,\mathrm{km}$ per second of gap.
5. Why does the gap grow with distance — and why is that a gift to a station that never saw the quake start?

**Part II — Circles on the map.**

6. Station A measures $\Delta t_A = 25\,\mathrm{s}$ . How far is the epicenter from A?
7. Explain why this one reading places the epicenter on a circle, not at a point.
8. Stations B and C measure $\Delta t_B = 15\,\mathrm{s}$ and $\Delta t_C = 32\,\mathrm{s}$ : their distances to the epicenter?
9. Explain how the three circles together pin the epicenter down.
10. The P [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) reached A at 09:14:32. Compute its travel time from the epicenter, and the origin time of the quake.

**Part III — The [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) that crosses the ocean.** The three circles meet offshore: the fault broke under the sea floor and set the water column moving. Long [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) on deep water travel at a speed set by the depth — about $200\,\mathrm{m}/\mathrm{s}$ in the open ocean, about $10\,\mathrm{m}/\mathrm{s}$ near a beach (admitted here; the formula is derived in the Year 1 volume). A port lies $D = 600\,\mathrm{km}$ from the epicenter.

11. Convert $200\,\mathrm{m}/\mathrm{s}$ to $\mathrm{km}/\mathrm{h}$ and name a vehicle with a comparable cruising speed.
12. How long does the tsunami take to reach the port?
13. The port’s own seismometer feels the P [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) first. After how long? If the alarm [sounds](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) at that instant, how much warning time does the coast get before the tsunami?
14. At sea the tsunami’s [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) is about $15\,\mathrm{min}$ . Compute its [wavelength](#def-g12-mechanical-waves-wavelength) there, and explain why a ship does not notice it.
15. Nearing the beach the speed falls to $10\,\mathrm{m}/\mathrm{s}$ while the [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) stays fixed. Compute the new [wavelength](#def-g12-mechanical-waves-wavelength) . What must happen to the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) as it bunches up?

**Part IV — [Energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy), and the report.** A strong quake releases on the order of $1 \times 10^{15}\,\mathrm{J}$, carried away by the [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave).

16. Compute the length of a circular [wavefront](#def-g12-mechanical-waves-wavefront) $10\,\mathrm{km}$ from the epicenter, then $210\,\mathrm{km}$ ; by what factor has it grown?
17. Deduce why shaking weakens with distance even in a lossless crust — and name the second cause a real crust adds.
18. A liquid cannot resist shear. Far-side stations record the P [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) but no S [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) : what does that reveal about Earth’s core?
19. The high-speed train of the previous year’s [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) chapter ( [Chapter 18](https://one-course.com/books/physics/2/en/chapter/18-mechanical-energy-and-its-conservation#ch-g11-mechanical-energy) ) carries about $1.5 \times 10^{9}\,\mathrm{J}$ of [kinetic energy](https://one-course.com/books/physics/2/en/chapter/18-mechanical-energy-and-its-conservation#def-g11-mechanical-energy-kinetic) : how many such trains is $1 \times 10^{15}\,\mathrm{J}$ worth?
20. The report, in two sentences: where and when was the quake, and how much warning did the port get? Give the three numbers.

**Solution of Problem 20.1.**

**1.** P compress along the travel: [longitudinal](#def-g12-mechanical-waves-transverse); S shake sideways: [transverse](#def-g12-mechanical-waves-transverse).

**2.** P, the faster, arrives first everywhere. $t_P = 120/6.0 = 20\,\mathrm{s}$; $t_S = 120/3.5 \approx 34\,\mathrm{s}$.

**3.** $\Delta t = d/v_S - d/v_P = d\,(1/v_S - 1/v_P)$; for $120\,\mathrm{km}$: $34.3 - 20 \approx 14\,\mathrm{s}$.

**4.** Solve for $d$: $d = \dfrac{v_P v_S}{v_P - v_S}\,\Delta t
= \dfrac{6.0 \times 3.5}{2.5}\,\Delta t = 8.4\,\mathrm{km}$ per second of gap.

**5.** $\Delta t \propto d$: one station’s own recording gives its distance to the epicenter without knowing the origin time.

**6.** $d_A = 8.4 \times 25 = 2.1 \times 10^{2}\,\mathrm{km}$.

**7.** The gap gives a distance, not a direction: the epicenter lies somewhere on the circle of radius $d_A$ around A.

**8.** $d_B = 8.4 \times 15 = 1.3 \times 10^{2}\,\mathrm{km}$; $d_C = 8.4 \times 32 \approx 2.7 \times 10^{2}\,\mathrm{km}$.

**9.** Two circles meet in two points; the third selects one: the common intersection is the epicenter.

**10.** $t_P = 210/6.0 = 35\,\mathrm{s}$: the quake began at $09{:}14{:}32 - 35\,\mathrm{s} = 09{:}13{:}57$.

**11.** $200\,\mathrm{m}/\mathrm{s} = 720\,\mathrm{km}/\mathrm{h}$ — a cruising jet airliner.

**12.** $t = 6.0 \times 10^{5}/200 = 3.0 \times 10^{3}\,\mathrm{s} = 50\,\mathrm{min}$.

**13.** $t_P = 600/6.0 = 100\,\mathrm{s}$. Warning: $3000 - 100 = 2.9 \times 10^{3}\,\mathrm{s} \approx 48\,\mathrm{min}$.

**14.** $\lambda = vT = 200 \times 900 = 1.8 \times 10^{5}\,\mathrm{m} =
180\,\mathrm{km}$: a [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) or so of rise spread over $180\,\mathrm{km}$ — the ship climbs an imperceptible slope over several minutes.

**15.** $\lambda = 10 \times 900 = 9.0\,\mathrm{km}$: the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) bunches up twentyfold, its [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) crams into a shorter length, so its height must grow — the tsunami rears up at the shore.

**16.** $2\pi r$: $63\,\mathrm{km}$, then $\approx 1.3 \times 10^{3}\,\mathrm{km}$ — $\times 21$.

**17.** The same [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) is shared over a $21\times$ longer front, so the [amplitude](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-amplitude) drops even in a lossless crust; a real crust adds absorption.

**18.** S [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave), [transverse](#def-g12-mechanical-waves-transverse) shear, cannot cross a liquid: the [S-wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) shadow shows the (outer) core is liquid.

**19.** $1 \times 10^{15}/1.5 \times 10^{9} \approx 6.7 \times 10^{5}$: about seven hundred thousand trains at full speed.

**20.** The quake struck at $09{:}13{:}57$, offshore at the crossing of the three circles, $210\,\mathrm{km}$ from station A. The port’s [P-wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) alarm sounded $100\,\mathrm{s}$ after the origin, about $48\,\mathrm{min}$ before the tsunami arrived.