Physics · Ch 11 — Waves

Graphical Representation of the Wave

11.6.3

Graphical Representation of the Wave

A sinusoidal progressive wave can be visualised using two complementary types of graph, each obtained by holding one of the two independent variables (position or time) fixed while plotting displacement against the other. The space (or spatial) variation graph plots $y=A\sin(kx)$ against position $x$ at one single frozen instant of time; on this graph, the wavelength $\lambda$ appears directly as the horizontal repeat distance of the curve, since $y=A\sin(kx)$ has the same value at $x$ and at $x+\lambda$ (because $\sin(kx+k\lambda)=\sin(kx+2\pi)=\sin(kx)$ , using $k\lambda=2\pi$ ). The time (or temporal) variation graph instead plots $y=A\sin(\omega t)$ against time $t$ at one single fixed position; on this graph, the time period $T$ appears directly as the horizontal repeat interval of the curve, since the sine function similarly repeats every time $\omega t$ increases by $2\pi$ , i.e. every $T=2\pi/\omega$ seconds. These two graphs are genuinely complementary rather than redundant: the spatial graph is a single-instant photograph of the whole wave pattern in space, showing how displacement varies with position, while the te …

Figure 11.24Graph of sinusoidal function y = A sin(kx)

What this figure shows. A smooth sine curve is plotted with position x along the horizontal axis and displacement y along the vertical axis, at a single fixed instant of time, showing the wave's characteristic repeating up-and-down shape. The horizontal distance from the origin out to the point where the curve completes one full repeating cycle is marked and labelled $\lambda$ (also expressed on the axis as the angle $2\pi$ in terms of $kx$ ), directly displaying the wavelength as the spatial repeat length of the curve $y=A\sin(kx)$ . The figure is the visual definition of the "space variation" graph described in the surrounding text: by freezing time and plotting displacement purely against position, the …

Figure 11.25Graph of sinusoidal function y = A sin(ωt)

What this figure shows. A smooth sine curve is plotted with time t along the horizontal axis and displacement y along the vertical axis, for one single fixed position in the medium, again showing the familiar repeating up-and-down shape. The horizontal distance from the origin out to where the curve completes one full repeating cycle is marked and labelled T (the time period), with the axis also marked in terms of the angle $2\pi$ for $\omega t$ , directly displaying the time period as the temporal repeat length of the curve $y=A\sin(\omega t)$ . The figure is the visual definition of the complementary "time variation" graph: by instead freezing position and plotting displacement purely against time, the time period becomes immediately visible as the curve's horizontal perio …

Misc Example 11.14Wave numbers for two sine waves of different wavelength

Worked out. Two sine waves have wavelengths $\lambda_1=1\ \text{m}$ and $\lambda_2=6\ \text{m}$ , and the task is to compute the corresponding angular wave number for each. Using the direct definition $k=2\pi/\lambda$ , the first wave gives $k_1=2\pi/1=6.28\ \text{rad/m}$ , while the much longer-wavelength second wave gives a correspondingly much smaller $k_2=2\pi/6\approx1.05\ \text{rad/m}$ . The example makes concrete the inverse relationship between wave number and wavelength: a short wavelength packs many radians of phase into each metre (a large k), while a long wavelength spreads the same $2\pi$ radians of phase out over …

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