General properties of waves

01

What travels in a wave?

A wave is a travelling disturbance that transfers energy. The material it passes through does not have to travel along with it.

An oscillation is a repeated motion about an equilibrium position. A vibrating part can move away from that position and return while passing a disturbance to its neighbours.

Follow a mark on a rope

Give one end of a stretched rope a brief up-and-down movement. A pulse travels along the rope. A small mark on the rope rises and falls as the pulse passes, then returns to its starting position.

The mark identifies the same piece of material throughout. It does not travel to the far end with the pulse. Yet the disturbance can make something at the far end move, showing that energy has been transferred.

Move the end repeatedly and a train of waves is produced. Each part oscillates about its local position as the pattern travels. This is what it means for a wave to transfer energy without transferring matter along with the disturbance. An independent bulk flow, such as a water current, is a separate effect.

Compare transverse and longitudinal motion

Transverse wave

The material vibrates perpendicular to the direction of wave travel. A wave travelling horizontally along a rope while its material moves up and down is an example. A spring can also carry a transverse disturbance if its end is moved sideways relative to its length.

Longitudinal wave

The material vibrates parallel to the direction of wave travel. Push and pull the end of a stretched spring along its length: close and wide coil spacings travel along it while a marked coil moves to and fro locally. Sound in air is another example.

The spring's close-spaced region is a compression; its more widely spaced region is a rarefaction. Those regions travel, but a particular coil does not stay in a compression and travel all the way along the spring.

Wavefronts in a ripple tank

A ripple tank is a shallow transparent tank in which disturbances on the water surface can be observed. A vibrating straight dipper can produce approximately straight wavefronts. A small point source can produce circular wavefronts spreading outwards.

A wavefront joins points at the same stage of oscillation, such as points along one crest. In the uniform region shown, the direction of wave travel is perpendicular to the local wavefront. Straight fronts travel across the tank; circular fronts spread radially away from the source.

Water surface patterns help show wavefronts and crest spacing. The rope is the simpler model for purely up-and-down transverse motion; real water-particle motion need not be an exactly vertical line.

Separate observation from explanation

Watch a marked point on a rope or spring as the disturbance passes, and compare its position before and afterwards. Describe that local motion separately from the changing position of a crest or compression. In a ripple tank, observe successive fronts and their spacing; the front itself is a pattern rather than a permanent set of particles.

Compare two directions. Transverse and longitudinal describe vibration relative to propagation. They do not mean that the material traces the whole drawn wave shape or travels with the crest.

02

Wave quantities and graphs

A snapshot shows different points at one instant. A time trace shows one point at different instants. Their horizontal axes answer different questions.

Displacement describes a point's position relative to equilibrium, with a chosen positive direction. For the rope below, upward displacement is positive and downward displacement is negative. This is the local displacement of the rope, not the distance travelled by the wave.

Five quantities to identify

Wave speed, v

The distance travelled by the wave pattern per unit time, commonly in m/s. Follow a recognisable feature such as a crest. This is different from the changing speed of an individual vibrating point.

Frequency, f

The number of complete oscillations per second, measured in hertz (Hz). A frequency of 2.5 Hz means 2.5 complete cycles each second.

Wavelength, λ

The distance between adjacent points at the same stage of the wave cycle, measured in a length unit such as m. Examples are crest to next crest, trough to next trough, or one compression centre to the next.

Period, T

The time taken for one complete oscillation, in seconds. A point returns to the same stage of its motion after one period.

Amplitude, A

The maximum displacement from equilibrium. Its unit is a length unit such as m or cm. Measure from the equilibrium line to a crest or trough, not from crest to trough.

A complete cycle links period and frequency

If a point completes five oscillations in 2.0 s, its period is 2.0/5 = 0.40 s, and its frequency is 5/2.0 = 2.5 Hz. Taking longer for each cycle means completing fewer cycles per second.

A point crosses equilibrium twice in a cycle, once in each direction. When timing from an equilibrium crossing, finish at an equivalent crossing moving in the same direction. For a longitudinal wave, a compression centre to the neighbouring rarefaction centre is half a wavelength.

Read the same rope wave in two ways

The following wave travels to the right. Its amplitude is 2.0 cm, wavelength is 0.80 m and period is 0.40 s. The first graph is a snapshot at t = 0. The second follows the marked material point at x = 0.20 m.

A graph of displacement is not a graph of wave speed

On the time trace, a rising section means that the marked point's displacement is increasing in the chosen positive direction. It does not mean that the travelling wave is speeding up. A negative displacement means the point is below equilibrium, not that the wave must travel to the left.

Read values from the axes and their scales. The graph's physical width on a phone or a printed page is not the represented wavelength or period.

Check the horizontal axis first. A distance snapshot supplies spacing. A fixed-point time trace supplies timing. They can describe the same wave while carrying different information.

03

Determine wave speed

In one period, a repeating wave pattern advances by one wavelength. Its speed is wavelength divided by period.

Speed is distance divided by time. For a periodic wave, the distance is one wavelength λ and the time is one period T. Frequency f is 1/T.

Rearrange for the unknown

A wave with speed 3.6 m/s and frequency 12 Hz has wavelength 3.6/12 = 0.30 m. Dividing frequency by speed would not give a length.

Measure several wavelengths and several cycles

For a steady ripple-tank pattern at fixed water depth, measure the separation of equivalent crests in a correctly calibrated top view. Time complete source oscillations separately. The following are supplied model readings.

One wavelength is 40.0/5 = 8.00 cm = 0.0800 m. Count the five gaps between the six crests rather than dividing by the number of crests.

The mean time is (4.90 + 5.00 + 5.10)/3 = 5.00 s. Start and finish the count at the same stage of the source's motion.

Choose improvements that address a cause

• Uncertain crest positions: measure a longer span containing several wavelengths and divide by the number of intervals. A similar endpoint-reading uncertainty then forms a smaller fraction of the full span.
• Start/stop reaction time: time several complete cycles. This reduces the fractional effect of reaction time compared with timing only one cycle.
• Incorrect length scale: calibrate the image or ruler in the observed plane and read without parallax. A projected image need not have the same physical scale as the water surface.
• Unclear pattern: keep the source steady and the water depth unchanged, and avoid an area where reflected waves overlap the incident pattern.

Repetition shows the variation in readings. It does not correct counting six intervals instead of five, using an incorrect scale, or changing the propagation conditions between the length and time measurements.

A different direct method follows the same pulse between two marked positions and uses distance divided by travel time. That travel time is not automatically the oscillation period. Identify which event starts and stops the timing.

Keep a spacing measurement separate from a cycle count. First find one wavelength and the number of cycles per second; then use v = fλ.

04

Sound, pitch and loudness

A vibrating source makes a disturbance in a material medium. In air, the disturbance travels as compressions and rarefactions.

Longitudinal motion is parallel to wave travel. Amplitude describes the size of an oscillation; frequency describes how often it repeats.

From a vibrating source to travelling sound

A tuning fork's prongs or a loudspeaker cone vibrate. As the source moves forwards, it pushes nearby air into a slightly smaller region. As it moves back, that region can become more spread out. Interactions between neighbouring regions pass the disturbance onwards.

• A compression has greater local pressure and density than the undisturbed air.
• A rarefaction has lower local pressure and density than the undisturbed air. It still contains particles; it is not a vacuum.

The repeated back-and-forth motion along the direction of travel makes sound in air longitudinal. Sound in water can be described by the same compression-and-rarefaction model. The medium carries the disturbance and energy while its particles oscillate about local positions.

Sound needs a medium

Sound can travel through gases, liquids and solids because their particles interact. It cannot propagate through an ideal vacuum, where there are no particles to pass on this mechanical disturbance.

Change amplitude and frequency separately

Under comparable conditions, a sound wave of greater amplitude is louder. A sound wave of greater frequency has a higher pitch. Increasing amplitude does not by itself require a higher frequency.

The following graphs represent the displacement of an air particle against time at the same observation position for three separate sounds. Their vertical units are micrometres (µm), where 1 µm = 0.000001 m, not microphone voltage.

Sound A: 0.20 µm amplitude, 4.0 ms period

4.0 ms = 0.0040 s, so f = 1/0.0040 = 250 Hz.

Sound B: 0.40 µm amplitude, 4.0 ms period

Its frequency is also 250 Hz, so it has the same pitch as A. Its larger amplitude makes it louder under the stated comparable conditions. Doubling amplitude does not mean that perceived loudness is exactly doubled.

Sound C: 0.20 µm amplitude, 2.0 ms period

2.0 ms = 0.0020 s, so f = 500 Hz. Its higher frequency gives a higher pitch than A. Equal displacement amplitude at a different frequency does not establish an exact equality of perceived loudness.

Read a microphone trace with its own units

A microphone and recording system can display a sound signal against time. That display usually represents the microphone's electrical output, rather than air-particle displacement measured in metres. Use the labelled vertical quantity.

To compare signal amplitudes, keep the microphone, recording gain, source distance and relevant surroundings the same. A taller trace caused by turning up the recording gain is not evidence that the source became louder. The time between repeated equivalent peaks still gives the period when the time scale is known.

Amplitude and frequency answer different questions. Taller oscillations concern amplitude; more cycles in the same time concern frequency. Neither means that the air itself travels all the way from source to receiver.

05

Echoes and distance

An echo is reflected sound received after travelling to a surface and back. The measured delay includes the complete sound path.

For approximately constant speed, distance travelled = speed x time. Before calculating, identify where the sound is emitted, where it reflects and where it is received.

Draw the outward and return paths

Place a sound source and receiver together, a distance d from a reflecting surface. A short pulse travels distance d to the surface and distance d back. If the sound speed is approximately uniform and the paths match, the total distance is 2d.

A reception time is not automatically a delay

In a separate recording, a pulse is emitted at 20 ms and its echo is received at 220 ms. The timing origin occurred before emission.

Measure the intended echo

A microphone with electronic timing or a recorded signal is useful for a short interval. Identify the emission event and the echo of that same pulse. Use a time scale with enough resolution, and account for any stated delay introduced by the instrument.

Several surfaces can produce different echoes. Select the return from the intended reflector rather than pairing the emission with an unrelated later pulse. If determining speed, measure the distance to the actual reflecting surface.

Repeating a human stopwatch measurement does not remove reaction-time limitations or guarantee that a very short echo interval has been resolved. An electronic record addresses timing; a clearer reflector arrangement addresses competing echoes. Choose the improvement that matches the problem.

The path determines whether to divide by two. A one-way transmission from a source to a separate receiver uses its one-way path. Do not divide every sound distance by two merely because sound is involved.

Ultrasound uses the same pulse-echo reasoning to measure underwater distances and locate tissue boundaries.

06 / Pure

Using ultrasound

Ultrasound is sound above the usual upper limit of human hearing, conventionally above about 20 kHz. Its echoes can locate reflecting boundaries.

1 kHz = 1000 Hz, so 20 kHz is 20000 Hz. Ultrasound is still a mechanical sound wave and needs a material medium. The high frequency does not make it electromagnetic radiation.

Underwater echo sounding

An active echo-sounding system can emit a short ultrasound pulse into water. Its transducer converts an electrical signal into sound and converts a received echo back into an electrical signal. The time between emission and reception indicates the distance along the beam when the sound speed is known.

For a vertical beam reflected from the seabed, the pulse travels down from the transducer and back up. Using an approximately uniform water sound speed, depth below the transducer = vt/2.

This example uses an active sonar system with ultrasound. Sonar more broadly uses sound: some systems listen without emitting a ranging pulse, and not every sonar frequency is ultrasonic.

Locate soft-tissue boundaries

A scanning probe sends ultrasound pulses into the body and receives returning echoes. At a boundary between tissues, some of a pulse can reflect while some continues farther. Later echoes can therefore return from deeper boundaries.

1. Emit a pulse: record its emission time.
2. Receive echoes: identify the delays of the returned signals.
3. Estimate depth: combine each round-trip delay with an appropriate assumed sound speed.
4. Build an image: combine information from many beam positions. Echo strength as well as timing contributes to the displayed image.

One echo does not give a complete photograph of an organ. The system assembles information from many reflections. In the simple model here, timing locates a boundary along the beam; a taller signal does not by itself mean a deeper boundary.

Why the probe uses coupling gel

Gel fills gaps between the probe and skin so that sound can pass into the body more effectively. An intervening air layer would cause strong reflection at the boundaries and reduce transmission. This does not mean that air cannot carry sound; the change between materials affects how much is transmitted or reflected.

Convert the time and retain the return path. Microseconds are millionths of a second. Multiplying speed by an echo delay gives the complete path; the boundary depth is half of it in this model.