IMAT Physics · Oscillations and Waves

Waves, Sound and the Doppler Effect

A wave carries energy without carrying matter; its speed is frequency times wavelength, and sound is a longitudinal wave that needs a medium.

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Why this matters

No past question has been set on this page yet. It is core syllabus, and a question on it is usually one line: v = fλ, an echo time, or which way a pitch shifts.

Concept 1 of 3: Transverse and longitudinal waves, and v = fλ

Flick one end of a rope and a bump runs along it, but each bit of rope only moves up and down and comes back. A wave moves energy, not material. In one period the source sends out one wavelength, so the wave travels one wavelength per period: speed equals frequency times wavelength.

Definition

In a transverse wave the particles vibrate at right angles to the direction the wave travels (waves on a string, water ripples, all electromagnetic waves).
In a longitudinal wave they vibrate along the direction of travel, making compressions and rarefactions (sound).
Wavelength $\lambda$ : distance between neighbouring crests (or compressions). Frequency $f$ : waves per second. Period $T = 1/f$ .
The source sets the frequency. When a wave passes into a new medium its frequency stays the same, while its speed and wavelength change together.

Wave speed

v = f\lambda \qquad T = \frac{1}{f}

$v$ wave speed, in m/s
$f$ frequency, in Hz
$\lambda$ wavelength, in m
$T$ period, in s

Worked example

The end of a long rope is moved up and down 15 times in 5.0 s. The crests are 0.50 m apart. Find the frequency, period and speed of the wave.

Practice this conceptself-check · 4 quick reps

The frequency does not change in a new medium

When a wave enters a new medium its speed and wavelength change, but its frequency is fixed by the source. Options in which the frequency changes at a boundary are wrong.

Concept 2 of 3: Sound: speed, pitch, loudness and echoes

Sound is a chain of squeezes and stretches passed from molecule to molecule, so it needs something to travel through. It is fastest where the particles are tightly linked, in solids, and slowest in gases. An echo is the same sound returning from a surface, and it has to go there and back.

Definition

Sound needs a medium; it cannot cross a vacuum. Typical speeds: air about 340 m/s (faster in warmer air), water about 1500 m/s, steel about 5000 m/s.
Pitch is set by the frequency; loudness by the amplitude (the intensity).
Humans hear roughly 20 Hz to 20 kHz. Ultrasound is above 20 kHz and is used in medical imaging; infrasound is below 20 Hz.
The decibel scale is logarithmic: each extra 10 dB multiplies the intensity by 10, so +20 dB is 100 times the intensity.
In an echo the sound travels to the reflector and back, so the distance is half of speed times time.

Distance from an echo

d = \frac{v\,t}{2}

$d$ distance to the reflecting surface, in m
$v$ speed of sound in the medium, in m/s
$t$ time between sending and hearing the echo, in s

Worked example

An ultrasound pulse sent into the body returns from a boundary between two organs after

6.5 \times 10^{-5}\ \text{s}

. Sound travels at 1540 m/s in soft tissue. How deep is the boundary?

Practice this conceptself-check · 4 quick reps

An echo goes there and back

The time of an echo covers the trip to the reflector and back. The distance is

vt/2

; using

vt

gives twice the true distance, and that answer is usually among the options.

Louder is not higher

Loudness depends on amplitude; pitch depends on frequency. Turning up the volume makes a note louder without changing its pitch.

Concept 3 of 3: The Doppler effect in words

A moving source sends out each new wave a little closer to the waves ahead of it, so in front of the source the waves are bunched and behind it they are stretched. Bunched waves arrive more often, so a listener ahead hears a higher pitch. Nothing about the source itself changes.

Definition

The Doppler effect is the change in observed frequency when the source and the observer move relative to each other.
Approaching: shorter wavelength reaches you, higher frequency. Receding: longer wavelength, lower frequency.
The bigger the relative speed, the bigger the shift. The frequency the source emits does not change, and the speed of sound in the air is not changed by the source's motion.
It applies to light too: a galaxy moving away shows a red shift (longer wavelengths).

Situation	Frequency heard or seen	Example
Source moving towards you	Higher than emitted	An ambulance siren as it approaches
Source moving away from you	Lower than emitted	The same siren after it has passed
You moving towards a still source	Higher: you meet the waves more often	Driving towards a ringing alarm
No relative motion	Same as emitted	Standing next to a parked car sounding its horn
Light from a galaxy moving away	Shifted to longer wavelengths (red shift)	Evidence that the universe is expanding
Ultrasound reflected from moving blood	Shifted by an amount that depends on the blood's speed	Doppler ultrasound scans of arteries and the fetal heart

Practice this conceptself-check · 3 quick reps

A moving source does not make sound travel faster

The speed of sound depends on the medium only. A moving source changes the wavelength and so the frequency heard, not the wave's speed through the air.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Transverse and longitudinal waves, and v = fλ
Wave speed
$v = f\lambda \qquad T = \frac{1}{f}$
Sound: speed, pitch, loudness and echoes
Distance from an echo
$d = \frac{v\,t}{2}$

Reference tables (1)

The Doppler effect in words6 rows

Situation	Frequency heard or seen	Example
Source moving towards you	Higher than emitted	An ambulance siren as it approaches
Source moving away from you	Lower than emitted	The same siren after it has passed
You moving towards a still source	Higher: you meet the waves more often	Driving towards a ringing alarm
No relative motion	Same as emitted	Standing next to a parked car sounding its horn
Light from a galaxy moving away	Shifted to longer wavelengths (red shift)	Evidence that the universe is expanding
Ultrasound reflected from moving blood	Shifted by an amount that depends on the blood's speed	Doppler ultrasound scans of arteries and the fetal heart

Watch out for (4)

The frequency does not change in a new medium
→ Transverse and longitudinal waves, and v = fλ
An echo goes there and back
→ Sound: speed, pitch, loudness and echoes
Louder is not higher
→ Sound: speed, pitch, loudness and echoes
A moving source does not make sound travel faster
→ The Doppler effect in words

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