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Wave Motion overview

Topic 2 of 6

Transverse graphs and phase

A displacement-position graph is one instant across many positions. A displacement-time graph follows one fixed point through time. Read the horizontal axis before interpreting a spacing or gradient.

Name the quantities

Displacement is the signed offset of a particle from equilibrium. For the transverse string here, y is positive upwards. Amplitude is the maximum magnitude of that displacement.

Wavelength λ is the distance between neighbouring points in the same phase at one instant, such as consecutive crests. Period T is the time for a complete repeated state at one fixed point. Frequency f is cycles per second, with f = 1/T.

Phase identifies the stage of an oscillation, expressed as an angle. Phase difference compares two stages. Wave speed is the rate at which a phase feature such as a crest progresses, not the particle's up-and-down speed. The oscillation quantities use the same cycle and phase meanings.

Worked space and time graphs

One wave, two different horizontal axes

The supplied right-moving transverse model has amplitude 0.0040 m = 4.0 mm, wavelength 0.80 m and period 0.20 s.

Read the horizontal axis before interpreting a curve

Both graphs have amplitude 4.0 mm = 0.0040 m. The first compares different positions at one instant. The second follows one fixed point through time.

One instant across the string: t = 0

One instant across the string: t = 0At time zero, displacement is plotted in millimetres against equilibrium position in metres. It is zero at positions zero, 0.40 and 0.80 metres, positive four millimetres at 0.20 metres and negative four at 0.60 metres. The states at zero and 0.80 metres have the same phase. Their separation, one wavelength, is 0.80 metres. This is a view of many different particles at one instant, not one particle moving along the curve.0.000.200.400.600.80-40+4λ = 0.80 mDisplacement y / mmEquilibrium position x / m

The points x = 0 and x = 0.80 m have the same phase, one wavelength apart. This graph's gradient compares displacement with position.

One fixed point through time: x = 0

One fixed point through time: x = 0At fixed equilibrium position zero, displacement is plotted in millimetres against elapsed time in seconds. The trace starts at zero and initially decreases. At times zero, 0.050, 0.100, 0.150 and 0.200 seconds, the displacements are zero, negative four, zero, positive four and zero millimetres. The first and last states have the same downward velocity, so their separation is one full period, 0.20 seconds. The phase is deliberately different from that of the time-zero spatial graph.0.000.050.100.150.20-40+4T = 0.20 sDisplacement y / mmElapsed time t / s

At x = 0, the point moves downward just after t = 0. Its displacement-time gradient is particle velocity; it is not the rightward propagation speed.

The spatial snapshot is at t = 0. The separate time trace follows the point with fixed equilibrium position x = 0, which initially moves down. The time trace therefore does not have the same starting phase as the spatial profile.

At t = 0, the snapshot has zeros at x = 0, 0.40 and 0.80 m, a positive crest at 0.20 m and a trough at 0.60 m. Consecutive crests are 0.80 m apart. The 0.40 m separation between neighbouring zeros is only half a wavelength in this sinusoidal model.

At fixed x = 0, the displacements at t = 0, 0.050, 0.100, 0.150 and 0.200 s are 0, -4.0, 0, +4.0 and 0 mm. The first and last zero crossings are both downward, so they are one full period apart. The intervening upward crossing is only half a period after the first.

The time graph's gradient gives the transverse particle velocity, in displacement per time. The spatial graph's gradient describes how displacement varies along the string. Neither is automatically the propagation speed. To measure propagation speed, track the advance of the same phase feature across position and time.

Connect phase to distance and delay

For this sinusoidal travelling wave, use positive distance or time separations to find the phase-difference magnitude:

|Δφ| = 2πΔx/λ for a spatial separation
|Δφ| = 2πΔt/T for a time separation

A separation of one wavelength or one period gives a phase change of 2π rad, returning to the same phase. Phase describes the oscillation stage, not a physical deflection angle of the drawn string.

For Δx = 0.20 m:
|Δφ| = 2π(0.20/0.80) = π/2 rad
Corresponding time delay = T/4 = 0.050 s

The time interval 0.050 s corresponds to a phase difference of π/2 rad; they are different quantities with different units. State which point reaches the corresponding motion state first.

For this right-moving wave, the downstream point lags

Solid blue: x = 0. Dashed brown: x = 0.20 m. The filled markers identify the corresponding positive peaks at 0.150 s and 0.200 s.

A point 0.20 metres downstream reaches the corresponding state 0.050 seconds laterTwo time traces have amplitude four millimetres and period 0.20 seconds. The solid trace at position zero starts at zero with downward velocity. The dashed downstream trace at position 0.20 metres starts at positive four millimetres and reaches its corresponding downward zero crossing at 0.050 seconds. Matching positive peaks are explicitly marked at 0.150 seconds for position zero and 0.200 seconds downstream, joined by a 0.050-second time bracket. The positive peak at the downstream trace's initial time belongs to the preceding cycle, not the marked event. Thus the downstream point lags by one quarter period, corresponding to a phase difference of pi over two radians.0.000.050.100.150.20-40+40.050 sDisplacement y / mmElapsed time t / s

The downstream point reaches the matching state 0.050 s later. That time lag corresponds to π/2 rad of phase. The direction of travel is needed to decide which point leads or lags.

The traces follow x = 0 and x = 0.20 m. Corresponding downward zero crossings, or the marked corresponding positive peaks, show the later arrival at the downstream point.

The downward zero crossing at x = 0 occurs at t = 0 and reaches x = 0.20 m at t = 0.050 s. Similarly, corresponding positive peaks occur at 0.150 and 0.200 s. The point to the right lags by 0.050 s in this right-moving wave.

Equal displacement at one instant does not prove equal phase: points may have opposite velocity directions. Use corresponding states and the propagation direction when deciding lead or lag.

Optional check A space snapshot has successive crests 0.80 m apart. A fixed-point time trace repeats after 0.20 s. For this right-moving wave, how does a point 0.20 m downstream compare with the first point?
A space snapshot has successive crests 0.80 m apart. A fixed-point time trace repeats after 0.20 s. For this right-moving wave, how does a point 0.20 m downstream compare with the first point?

Measure a graph with a known scale and reference

Use a calibrated spatial image for wavelength, or known sample times for a fixed-point period. Measure several wavelength or cycle intervals where possible and divide by their number. Count intervals between matching phase points, not simply the number of visible crests.

Keep the camera or observation point fixed when claiming a fixed-point time trace. Record the sample interval, spatial scale and displacement zero. A moving observation point changes which part of the wave is sampled, and a smooth-looking drawn curve does not establish fine measurement resolution.