IMAT Physics · Formula sheet

Oscillations and Waves formulas

6 formulas, 3 reference tables and 13 common traps for IMAT Physics Oscillations and Waves, grouped by subtopic.

Simple Harmonic Motion and Pendulums

Simple harmonic motion: displacement, velocity and acceleration

SHM from an end position

x = A\cos(\omega t) \qquad v = -A\omega\sin(\omega t) \qquad a = -\omega^2 x \qquad \omega = \frac{2\pi}{T}

$x$ displacement from the centre, in m
$A$ amplitude, in m
$\omega$ angular frequency, in rad/s
$T$ period, in s

Energy in simple harmonic motion and damping

Energy of a mass on a spring

E = \tfrac{1}{2}kA^2 = \tfrac{1}{2}m v_{\max}^2

$k$ spring constant, in N/m
$A$ amplitude, in m
$m$ oscillating mass, in kg
$v_{\max}$ speed at the centre, in m/s

The periods of a simple pendulum and a mass on a spring

Periods of the two standard oscillators

T_{\text{pendulum}} = 2\pi\sqrt{\frac{L}{g}} \qquad T_{\text{spring}} = 2\pi\sqrt{\frac{m}{k}}

$L$ length of the pendulum, in m
$g$ gravitational field strength, in N/kg
$m$ mass on the spring, in kg
$k$ spring constant, in N/m

Common traps

Periodic is not the same as simple harmonic

SHM needs a restoring force proportional to the displacement, giving a sine or cosine motion. A bouncing ball, or a puck sliding at steady speed between two walls, repeats itself but is not SHM.

Velocity is zero at the ends, not at the centre

x = \pm A

the object stops for an instant and turns round, so

v = 0

there while the acceleration is largest. When

\cos(\omega t) = \pm 1

\sin(\omega t) = 0

, so the velocity is zero.

Without friction an oscillator never stops

Nothing removes energy from a frictionless oscillator, so it keeps swinging with the same amplitude for ever. Statements that a frictionless pendulum slows down and stops are false; stopping is what damping does.

Passing the centre twice per period

A pendulum crosses its lowest point once going each way, so successive crossings are half a period apart. If a question gives the time between crossings, double it before using

T = 2\pi\sqrt{L/g}

A heavier bob does not change a pendulum's period

Mass cancels out of a pendulum's motion. It does matter for a mass on a spring, where

T \propto \sqrt{m}

. Mixing up the two formulas is a common slip.

Waves, Sound and the Doppler Effect

Learn this subtopic in the notes

Transverse and longitudinal waves, and v = fλ

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

Sound: speed, pitch, loudness and echoes

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

The Doppler effect in words

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

Common traps

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.

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.

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.

The Electromagnetic Spectrum, Diffraction and Interference

Learn this subtopic in the notes

Superposition, coherence, interference and standing waves

Interference conditions and a string fixed at both ends

\Delta = n\lambda \ \text{(constructive)} \qquad \Delta = \left(n + \tfrac{1}{2}\right)\lambda \ \text{(destructive)} \qquad \lambda_n = \frac{2L}{n}

$\Delta$ path difference from the two sources, in m
$n$ a whole number (0, 1, 2, … for interference; 1, 2, 3, … for the string)
$\lambda$ wavelength, in m
$L$ length of the string, in m

The electromagnetic spectrum: order, wavelengths and uses

Region	Typical wavelength	Use or source
Gamma rays	Below about $10^{-11}\ \text{m}$	Radiotherapy, sterilising medical equipment, radioactive decay
X-rays	About $10^{-11}$ to $10^{-8}\ \text{m}$	Imaging bones and teeth, CT scans
Ultraviolet	About 10 nm to 400 nm	Sterilising water, vitamin D in the skin, sunburn
Visible light	About 400 nm to 700 nm	Sight, optical fibres, photosynthesis
Infrared	About 700 nm to 1 mm	Heat radiation, remote controls, thermal cameras
Microwaves	About 1 mm to 30 cm	Microwave ovens, mobile phones, radar
Radio waves	Above about 30 cm	Radio and TV broadcasting, MRI scanners

Reflection, refraction, diffraction and polarisation of waves

Behaviour	What happens	What changes	Example
Reflection	The wave bounces off a boundary	Direction only	An echo; an image in a mirror
Refraction	The wave bends as it changes speed entering a new medium	Speed, wavelength and direction; not frequency	A straw looking bent in water
Diffraction	The wave spreads out after a gap or an edge	Direction spreads; speed, frequency and wavelength stay	Hearing a voice round a corner
Polarisation	Vibrations are limited to one plane	Plane of vibration	Polarising sunglasses cutting glare

Common traps

All electromagnetic waves have the same speed in a vacuum

Gamma rays and radio waves both travel at

3.00 \times 10^8\ \text{m/s}

in a vacuum. What differs is the frequency and wavelength. Radio waves are not sound waves: they are electromagnetic and cross a vacuum.

Diffraction does not change the wavelength

After a gap the wave spreads out, but its speed, frequency and wavelength are the same as before. Only refraction, a change of medium, changes the wavelength.

Coherence is about phase, not amplitude or brightness

Two waves are coherent when their phase difference stays constant in time, which needs equal frequencies. A strong wave and a weak one can be perfectly coherent, and so can two radio or sound waves: coherence is about keeping in step.

Nodes are half a wavelength apart, not a whole one

In a standing wave each loop between neighbouring nodes is

\lambda/2

long. Counting a loop as a whole wavelength halves every wavelength you work out.

Oscillations and Waves formulas

Simple Harmonic Motion and Pendulums

Simple harmonic motion: displacement, velocity and acceleration

Energy in simple harmonic motion and damping

The periods of a simple pendulum and a mass on a spring

Common traps

Waves, Sound and the Doppler Effect

Transverse and longitudinal waves, and v = fλ

Sound: speed, pitch, loudness and echoes

The Doppler effect in words

Common traps

The Electromagnetic Spectrum, Diffraction and Interference

Superposition, coherence, interference and standing waves

The electromagnetic spectrum: order, wavelengths and uses

Reflection, refraction, diffraction and polarisation of waves

Common traps

More IMAT Physics formula sheets