A wave changes in both space and time, but a graph has only one horizontal axis. So there are two kinds of wave graph, and reading the wrong one is a classic mistake. Both are pictures of the same wave.
Key idea
A snapshot graph ($y$ against $x$) shows the whole wave at one instant: read the wavelength $\lambda$ between crests. A history graph ($y$ against $t$) follows one point over time: read the period $T$ between crests. Both show the amplitude $A$. Then $v = \lambda/T = f\lambda$.
Why it works
Why does the history graph look like a backward copy of the snapshot? For a wave moving to the right, the part of the wave just to the left of a point is what arrives there next. So the point's future displacements are the wave shape read from right to left. The two graphs have the same shape but different axes, with distance on one and time on the other, linked by $x = vt$.
Common mistake
Reading the "wavelength" off a $y$–$t$ graph. The distance between crests on a history graph is a time, the period $T$, measured in seconds. Check the axis label and the units: meters means wavelength, seconds means period.
Worked example: Reading two graphs
A snapshot graph of a wave shows crests $0.40\ \text{m}$ apart and a height of $3.0\ \text{cm}$ above the center line. A history graph of one point shows crests $0.050\ \text{s}$ apart. Find $A$, $\lambda$, $T$, $f$ and $v$.
- Amplitude: $A = 3.0\ \text{cm}$ (from the center line, not crest to trough).
- Wavelength from the snapshot: $\lambda = 0.40\ \text{m}$. Period from the history: $T = 0.050\ \text{s}$.
- $f = \dfrac{1}{T} = 20\ \text{Hz}$, and $v = \dfrac{\lambda}{T} = \dfrac{0.40}{0.050} = 8.0\ \text{m/s}$.
Answer: $A = 3.0\ \text{cm}$, $\lambda = 0.40\ \text{m}$, $T = 0.050\ \text{s}$, $f = 20\ \text{Hz}$, $v = 8.0\ \text{m/s}$
Worked example: Which way is it moving?
A wave moves to the right. On the snapshot graph, a point sits on the rising slope just before a crest (the crest is to its right). Is that point moving up or down right now?
- The wave moves right, so the shape just to the point's left arrives next.
- To its left the wave is lower, back down the slope.
- So the point is about to be lower: it is moving down.
Answer: Down