4.1.1 Waves transfer energy and information

Waves carry energy without carrying matter

1. A wave begins with an oscillating source, such as a loudspeaker cone, a vibrating string or a dipper in a ripple tank.
2. The source transfers energy to nearby particles or fields, and the disturbance passes from one place to the next.
3. Particles in a material medium oscillate about fixed positions and have no net movement with the wave.
4. A wave can carry information when variations in the wave form a signal, such as speech in sound or digital data in radio waves.

Water waves leave the water behind

1. A floating cork moves up and down as crests and troughs pass, but remains in approximately the same horizontal position.
2. If the water travelled with the wave, the cork would be carried continuously across the tank.
3. The travelling crests and the stationary average position of the cork show that the wave transfers energy without a net transfer of water.

Sound waves leave the air behind

1. A loudspeaker cone moves backwards and forwards, producing compressions and rarefactions in the surrounding air.
2. Each air particle oscillates about a fixed position and transfers energy to neighbouring particles.
3. Sound reaches a listener without producing a continuous wind or moving all the air towards one wall.
4. This shows that the sound disturbance travels while the air has no net movement from speaker to listener.

4.1.2 Frequency, wavelength, amplitude, period and wavefronts

Wave quantities describe size, spacing and timing

1. Frequency has symbol $f$ and unit hertz, $Hz$, where $1Hz$ means one complete oscillation each second.
2. Wavelength has symbol $\lambda$ and SI unit metre, $m$.
3. Measure wavelength from crest to crest, trough to trough or compression to compression.
4. Measure amplitude from the equilibrium line to a crest or trough.
5. The crest-to-trough height is twice the amplitude.

Period is the time for one oscillation

1. Period has symbol $T$ and SI unit second, $s$.
2. Frequency and period are reciprocals: $f=\frac{1}{T}$ and $T=\frac{1}{f}$.

Wavefronts join points moving in step

1. Neighbouring wavefronts are one wavelength apart, and the direction of travel is perpendicular to the wavefronts.
2. A straight source produces plane wavefronts, while a point source produces circular wavefronts.

4.1.3 Longitudinal and transverse waves

Wave type depends on oscillation direction

1. Transverse waves have crests and troughs when shown as displacement diagrams.
2. Electromagnetic waves, seismic S waves and water surface waves are transverse examples.
3. Longitudinal waves contain compressions, where particles are closer together, and rarefactions, where particles are farther apart.
4. Sound waves in air and seismic P waves are longitudinal examples.

Compare oscillation with energy transfer

1. For a transverse wave, the vibration arrow is at $90^{\circ }$ to the direction of travel.
2. For a longitudinal wave, the vibration arrow lies along the direction of travel.
3. Both types transfer energy without a net transfer of matter.

4.1.4 The wave speed equations

Two equations calculate wave speed

1. The frequency-wavelength equation is $v=f\lambda$.
2. The distance-time equation is $v=\frac{x}{t}$.
3. Use $v$ in $m/s$, $f$ in $Hz$, $\lambda$ and $x$ in $m$, and $t$ in $s$.

Rearrange before substituting

1. From $v=f\lambda$, use $f=\frac{v}{\lambda }$ or $\lambda =\frac{v}{f}$.
2. From $v=\frac{x}{t}$, use $x=vt$ or $t=\frac{x}{v}$.
3. Convert units first: $1kHz=10^{3}Hz$, $1MHz=10^{6}Hz$ and $1cm=10^{-2}m$.

Frequency and wavelength vary inversely

1. Wave speed is approximately constant for a wave travelling in a fixed medium.
2. Because $v=f\lambda$, doubling frequency halves wavelength when speed is constant.

4.1.5 Measuring wave velocity

Choose a method that matches the wave

1. A direct method measures distance and time, then uses $v=\frac{x}{t}$.
2. An indirect method measures frequency and wavelength, then uses $v=f\lambda$.
3. Equipment suitability depends on wave speed, available distance and whether wavefronts can be observed or detected.

Measure sound speed electronically

1. Place two microphones a measured distance apart and connect them to an oscilloscope or data logger.
2. Create a short sound near the first microphone and measure the delay between the detected signals.
3. Calculate $v=\frac{x}{t}$ from the separation and delay.
4. A larger separation reduces percentage uncertainty, while electronic timing removes human reaction time.

A boundary changes speed and wavelength

1. When sound enters another medium, frequency stays constant because the source still oscillates at the same rate.
2. Speed changes because wave speed depends on the medium.
3. Since $v=f\lambda$ and $f$ is unchanged, wavelength changes in the same ratio as speed.

4.1.6 Calculating depth or distance from waves

Echo time gives a two-way distance

1. A pulse travels to a boundary, reflects and returns to the detector.
2. The measured echo time is normally the total time for both journeys.
3. The total distance is $x=vt$, so the one-way depth is $d=\frac{vt}{2}$.
4. If the given time is explicitly one-way, do not divide by two.

Use the speed in the stated medium

1. Use the wave speed for the material through which the pulse travels.
2. Typical values are about $340m/s$ in air, $1500m/s$ in water and $3.0\times 10^{8}m/s$ for radio waves in air.
3. Convert times using $1ms=10^{-3}s$ and $1\mu s=10^{-6}s$.

Rearrange for time or speed

1. From $d=\frac{vt}{2}$, use $t=\frac{2d}{v}$ or $v=\frac{2d}{t}$.
2. Keep the factor of two attached to the two-way journey.

4.2.1 Behaviour of waves at material interfaces

Waves at boundaries

1. When a wave reaches a material interface, its energy may be reflected, transmitted or absorbed, and a transmitted wave may also be refracted.
2. A material interface is the boundary between two materials or between two regions with different wave properties.
3. The proportions of energy in each outcome depend on the two materials, the wavelength and the angle at which the wave meets the boundary.

Reflection

1. A reflected wave remains in the original material and travels away from the boundary.
2. For a straight boundary, the angle of incidence equals the angle of reflection: $i=r$.
3. Both angles are measured from the normal, an imaginary line at $90^{\circ }$ to the surface.
4. Reflection changes the direction of travel, but the wave remains in the same material, so its speed, frequency and wavelength stay unchanged.

Transmission and refraction

1. A transmitted wave carries energy beyond the boundary.
2. If its speed changes and it arrives at an angle to the normal, its direction changes by refraction.
3. The source continues to set the frequency, so frequency does not change at the boundary.
4. Because $v=f\lambda$, a change in speed at constant frequency causes the wavelength to change.
5. A wave that slows bends towards the normal, whereas a wave that speeds up bends away from the normal.

Absorption

1. Absorption reduces the amplitude and energy of the wave that continues through or returns from the material.
2. The absorbed energy commonly increases the material's temperature, although it may produce another response in a detector.
3. A surface can partly absorb and partly reflect or transmit the same incident wave, so these outcomes are not mutually exclusive.

Energy at an interface

1. Energy is conserved at the boundary, so the incident wave energy is shared between reflected, transmitted and absorbed energy.
2. In symbols, $E_{incident}=E_{reflected}+E_{transmitted}+E_{absorbed}$.
3. A wave with a smaller transmitted amplitude has not necessarily lost energy from the system; some energy may have been reflected or absorbed.

4.2.2 Refraction at a boundary

Refraction and wave speed

1. A wave changes speed when the properties of the medium that determine its propagation change.
2. The source does not change at the boundary, so the wave frequency remains constant.
3. Using $v=f\lambda$, constant $f$ means that a lower speed gives a shorter wavelength and a higher speed gives a longer wavelength.

Direction of bending

1. Draw a normal at $90^{\circ }$ to the boundary where the wave arrives.
2. If the wave slows down, it bends towards the normal, so the angle of refraction is smaller than the angle of incidence.
3. If the wave speeds up, it bends away from the normal, so the angle of refraction is larger than the angle of incidence.
4. The rule depends on the change in wave speed, not simply on which material is denser.
5. If the wave travels along the normal, every part of the wavefront changes speed together, so the direction does not change.

Why the wave turns

1. A wavefront arriving at an angle reaches the new medium one side first.
2. That side changes speed before the rest of the wavefront.
3. The unequal distances travelled by the two sides during the change rotate the wavefront.
4. The direction of travel, which is perpendicular to the wavefront, therefore changes.

Wave quantities at the boundary

1. Frequency remains unchanged because each incoming oscillation produces one transmitted oscillation.
2. Speed changes because the wave now travels through a different medium.
3. Wavelength changes in the same ratio as speed because $\lambda =\frac{v}{f}$ and $f$ is constant.

4.2.3 Wavelength-dependent behaviour of waves

Wavelength changes interactions

1. The same substance can interact differently with waves of different wavelengths.
2. For each wavelength, a material may absorb, transmit, reflect or refract a different proportion of the incident wave energy.
3. A statement that a material is transparent or reflective is incomplete unless the relevant part of the electromagnetic spectrum is clear.

Selective absorption and transmission

1. A material transmits a wavelength well when only a small fraction of that wave energy is absorbed or reflected.
2. Visible glass transmits much visible light, but different glass compositions can absorb more ultraviolet or infrared radiation.
3. A coloured filter transmits a limited range of visible wavelengths and absorbs many of the others.
4. An object appears coloured because the wavelengths reaching the eye depend on which wavelengths the object reflects or transmits.

Reflection depends on wavelength

1. A surface can be smooth compared with one wavelength but rough compared with a shorter wavelength.
2. This difference changes whether reflection is concentrated in one direction or scattered over many directions.
3. Metallic surfaces reflect many radio wavelengths effectively, which allows metal dishes to direct radio and microwave signals.

Refraction depends on wavelength

1. The speed of electromagnetic waves in a material can depend on wavelength.
2. Different wavelengths therefore change speed by different amounts at the same boundary and are refracted through different angles.
3. In a prism, visible wavelengths spread into a spectrum because each wavelength is refracted by a different amount.
4. This separation of wavelengths is called dispersion.

Comparing materials

1. A valid comparison must keep wave intensity, angle of incidence, material thickness and detector arrangement controlled.
2. The detector must be suitable for the wavelength being tested because the human eye detects only visible light.
3. The measured incident, reflected and transmitted signals can be compared, while absorption is inferred from the missing energy.