Determining Amplitude, Wavelength, and Period from Graphs

Hello. In the previous lesson, you classified waves by the motion of the medium and by whether a material medium is required. Now we will make those waves measurable on graphs.

A curve can look similar whether it represents a wave spread across space or the motion of one point over time. The horizontal axis tells you which interpretation is correct. By the end of this lesson, you should be able to read amplitude and wavelength from a displacement–distance graph, and period from a displacement–time graph—without mixing up the quantities.

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One wave shape, two different graphs

Begin with these essential definitions:

• Amplitude, $A$, is the maximum displacement from the equilibrium, or rest, position.
• Wavelength, $\lambda$, is the distance between two neighbouring points in the same phase on successive cycles.
• Period, $T$, is the time taken for one complete oscillation.

The phrase same phase is important. A crest and the next crest are at the same stage of their motion. So are two troughs, or two equilibrium crossings where the curve is travelling upward. A crest and the nearest trough are not the same phase: they are half a cycle apart.

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Displacement–distance: a snapshot across space

A displacement–distance graph has:

• displacement on the vertical axis;
• distance or position on the horizontal axis.

It is a snapshot: at one instant, it shows the displacement of every point along the medium. The graph is not showing one particle travelling along the curved line. Each location on the horizontal axis represents a different part of the rope, water surface, or other medium.

Reading amplitude

Locate the equilibrium position first. On most test graphs, this is the horizontal line at zero displacement.

Amplitude is measured vertically, from that equilibrium line to a crest or to a trough:

$$A=\frac{y_{\max }-y_{\min }}{2}$$

If the crest is $+6.0\text{cm}$ and the trough is $-6.0\text{cm}$, then:

$$A=6.0\text{cm}$$

The full crest-to-trough height is $12.0\text{cm}$, which is twice the amplitude. This is one of the most common graph-reading errors.

If the equilibrium line is not at zero, find the midpoint between the maximum and minimum displacements. For example, if the maximum is $+7\text{cm}$ and minimum is $-3\text{cm}$, the midpoint is $+2\text{cm}$, and the amplitude is $5\text{cm}$.

Reading wavelength

Wavelength is measured horizontally. Use two adjacent points at the same stage of the cycle:

$$\lambda =x_{\text{crest 2}}-x_{\text{crest 1}}$$

The most reliable choices are:

• crest to adjacent crest;
• trough to adjacent trough;
• upward equilibrium crossing to the next upward equilibrium crossing.

Suppose neighbouring crests are at $1.5\text{m}$ and $5.5\text{m}$. Their separation is:

$$\lambda =5.5-1.5=4.0\text{m}$$

The horizontal distance from a crest to its nearest trough is only half a wavelength:

$$\frac{\lambda }{2}$$

So if the wavelength is $0.80\text{m}$, a crest and nearest trough are $0.40\text{m}$ apart.

A reliable graph-reading routine

For any displacement–distance graph:

1. Check that the horizontal axis is labelled distance, position, or length.
2. Find the equilibrium line.
3. Read amplitude vertically from equilibrium to maximum displacement.
4. Read wavelength horizontally between two matching points.
5. Include the unit printed on the relevant axis.

If several waves are visible, a longer measurement can reduce uncertainty. Measure a total distance covering $N$ complete cycles, then divide by $N$:

$$\lambda =\frac{\Delta x}{N}$$

For instance, if five complete wavelengths span $2.0\text{m}$, then each wavelength is $0.40\text{m}$.

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Displacement–time: one point oscillating

A displacement–time graph has:

• displacement on the vertical axis;
• time on the horizontal axis.

Instead of showing every point in space at one instant, it follows one fixed point in the medium as time passes. For a transverse rope wave, it may show one marked piece of rope moving up and down.

The vertical reading for amplitude works exactly as before: it is still the maximum displacement from equilibrium. But the horizontal reading has changed meaning.

On a displacement–time graph, one complete cycle is the period:

$$T=t_{\text{crest 2}}-t_{\text{crest 1}}$$

You can also measure from trough to trough, or from one upward equilibrium crossing to the next upward equilibrium crossing.

For example, if a point reaches one crest at $0.20\text{s}$ and the next crest at $0.65\text{s}$, then:

$$T=0.65-0.20=0.45\text{s}$$

The time from that crest to the next trough would be $0.225\text{s}$. That is only half a period, just as crest-to-trough distance on a displacement–distance graph is only half a wavelength.

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Why the same-looking curve means different things

The two graphs may both contain repeating curves, but their horizontal axes answer entirely different questions.

Graph type	Horizontal axis	What the curve represents	Main horizontal quantity
Displacement–distance	Distance or position	Different points in the medium at one instant	Wavelength, $\lambda$
Displacement–time	Time	One fixed point in the medium over time	Period, $T$

A useful memory rule is:

Distance graph: how far does one cycle extend? Measure wavelength.
Time graph: how long does one cycle take? Measure period.

Do not infer wavelength from a displacement–time graph unless additional information is supplied, such as wave speed. Likewise, do not infer period from a displacement–distance snapshot alone.

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Oscilloscopes and graph labels

An oscilloscope screen is usually a voltage–time graph, rather than a displacement–time graph. Nevertheless, it uses the same horizontal-cycle idea:

• vertical distance from the reference line to a peak gives the signal’s amplitude, in volts;
• horizontal distance for one full repeat gives the period, in seconds.

The important habit is to read the axis labels and scales before deciding what a measured height or width means.

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The graph traps to avoid

Treating crest-to-trough height as amplitude

Crest-to-trough is the complete vertical range, equal to $2A$. Amplitude is only the displacement from equilibrium to one extreme.

Measuring crest to trough as a wavelength

On a displacement–distance graph, this is normally $\lambda /2$, not $\lambda$. Measure between matching points such as crest-to-crest.

Reading a period from a distance axis

A repeated pattern across metres or centimetres gives a wavelength, not a period. Period requires a time axis.

Treating the curve as the path of one particle

On a displacement–distance graph, the curve is a spatial pattern at one instant. A particular particle is located at one horizontal position; it moves up and down about that point rather than travelling along the entire curve.

Forgetting units

• Amplitude and wavelength are lengths: often $\text{m}$, $\text{cm}$, or $\text{mm}$.
• Period is time: usually $\text{s}$, sometimes $\text{ms}$.

Report the unit from the graph first. Converting to SI units is a separate step when a calculation requires it.

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Key takeaways

The graph’s horizontal axis determines what one complete cycle measures. A displacement–distance graph is a snapshot across the medium: read amplitude vertically from equilibrium to an extreme and wavelength horizontally between matching points on neighbouring cycles.

A displacement–time graph tracks one fixed point as it oscillates: read period horizontally between matching stages of consecutive cycles. In both graph types, amplitude is not the full crest-to-trough height; that full height is $2A$.

Next, you will turn these graph readings into calculations, using the inverse relationship between period and frequency and the wave equation connecting speed, frequency, and wavelength.