Terms and Definitions Used in Wave Motion

What this figure shows. Two sinusoidal wave curves, labelled X and Y, are drawn together on the same axes for comparison. Although both curves share the same basic smooth, repeating up-and-down sinusoidal shape, they are drawn with visibly different heights (amplitudes) and different horizontal repeat distances (wavelengths), so the two curves are clearly not identical patterns despite both being simple sine waves. The figure motivates the entire section: it poses the question of how to precisely describe and quantify the differences between two wave patterns that look superficially alike, which is exactly what the definitions of wavelength, amp …

What this figure shows. A single sinusoidal wave curve is drawn along a horizontal reference (mean-position) line, with four points O, A, B, C, D marked along the curve at its successive turning points and zero-crossings. The highest point of the curve above the reference line is labelled the crest, and the lowest point below the reference line is labelled the trough. This figure introduces the two most basic visual landmarks on any transverse wave curve, setting up the length of one complete repeating section OB (from the figure) as …

What this figure shows. The same wave curve as before is extended over two full repeating sections, with points O, A, B, C, D marked along it, and braces underneath label the section from O to B as "One wavelength = lambda" and the longer section as "Two wavelengths = 2 lambda". The figure visually establishes that a wavelength is the length of the smallest section of the wave pattern that contains no internal repetition -- here shown to be the distance OB (equivalently BD) -- and that this same length, lambda, repeats identically over and over as the pattern is extended, which is exactly the mathematical property $y(x)=y( …

What this figure shows. A transverse sinusoidal wave curve is drawn along a horizontal x-axis, with several crests visible, and the horizontal distance between two adjacent crests is marked and labelled lambda using a bracketed span underneath the curve; the same span length lambda is repeated at the next pair of adjacent crests further along, showing the measurement is identical anywhere along the wave. The figure gives the specific, transverse-wave version of the wavelength definition: for a transverse wave, wavelength is measured as the distance between any two consecutive crests (or, equivalently, any two consecut …

What this figure shows. A row of dots representing particles of a medium is drawn along a horizontal line, showing an alternating pattern of tightly bunched "Compressed" regions and widely spaced "Stretched" (rarefaction) regions, with the distance from the centre of one compressed region to the centre of the very next compressed region marked and labelled lambda. The figure supplies the longitudinal-wave counterpart of the wavelength definition given for transverse waves: for a longitudinal wave, wavelength is instead measured as the distance between two consecutive compressions (or, equivalently, two consecutive …

What this figure shows. Two snapshots of the same wave train, containing three complete wavelengths, are drawn one above the other. In panel (a), taken at time t = 0 s, the wave has just reached a fixed marked point A from the left. In panel (b), taken one second later at t = 1 s, the same wave train has advanced further to the right, and by counting the crests that have now passed point A -- exactly two full waves' worth -- the figure visually establishes the working definition of frequency used in this section: the frequency is the number of complete waves crossing a fixed point per second, giving f = 2 Hz for the specific wave train drawn here, which is then used to define the reciprocal quantity, the time period, as the time taken for one single wave …

What this figure shows. Two sinusoidal wave curves, both drawn along horizontal x-axes with vertical y-displacement, are shown side by side sharing exactly the same wavelength lambda, but one curve labelled A1 has a visibly smaller peak height above (and trough depth below) its own mean line than the other curve labelled A2, which has a noticeably taller peak and deeper trough. The figure isolates amplitude as the one wave characteristic that can differ between two waves even when their wavelength, frequency, time period and propagation speed are all identical: it visually defines amplitude as the maximum displacement of the medium away from its mean (reference) position, i.e. the …

Worked out. Three wave curves labelled (a), (b) and (c) are compared to determine which has the longest wavelength, using only their sketched shapes over the same horizontal x-axis span, without any numerical data given. By counting how many complete repeating cycles each curve fits into the same fixed span of the x-axis, the curve that completes the fewest full cycles over that span is the one whose single cycle is spread over the most horizontal distance, and is therefore the one with the largest (longest) wavelength; comparing the three sketches this way identifies waveform (c) as having the longest wavelength, since it shows the smallest number of repetitions across the same horizo …

Worked out. Three wave curves labelled (a), (b) and (c) are shown together, and the task is to rank them in ascending order first by frequency and then by wavelength, again purely from their sketched shapes rather than numerical values. Because a shorter wavelength packs more oscillations into the same length of medium, and because wave speed is common to waves in the same medium, a shorter wavelength directly corresponds to a higher frequency (since $f = v/\lambda$) -- so ranking the three curves by how tightly or loosely their cycles are packed gives the frequency order $f_c < f_a < f_b$ and, correspondingly, the exactly reversed wavelength order $\lambda_b < \lambda_a < \lambda_c$, illustrating the inverse relationship between …

Worked out. Human hearing spans frequencies from 20 Hz up to 20 kHz, and this worked example converts that audible frequency range directly into the corresponding range of wavelengths in air, taking the speed of sound as 340 m/s. Using the basic relation $\lambda = v/f$, the long-wavelength end corresponds to the lowest audible frequency, $\lambda_1 = 340/20 = 17\ \text{m}$, while the short-wavelength end corresponds to the highest audible frequency, $\lambda_2 = 340/(20\times10^3) = 0.017\ \text{m}$. The result shows that the everyday range of audible sound corresponds to a very wide physical range of wavelengths, from about 1.7 centimetres at the highest pitches human beings can hear up to 17 metres at the lowest, a span of exactly a thousandfold, p …

Worked out. A man watches a toy duck bob up and down 15 times per minute on an ocean wave whose wavelength he estimates as 1.2 m, and the task is to find the time taken for one complete up-and-down cycle (the time period) and the speed of the ocean wave. Converting the given rate to consistent SI units gives the frequency $f = 15/60 = 0.25\ \text{Hz}$, so the time period is its reciprocal, $T = 1/f = 4\ \text{s}$. The wave speed then follows directly from the fundamental wave relation, $v = f\lambda = 0.25\times1.2 = 0.3\ \text{m/s}$. The example is also used to introduce, via a short dimensional argument, why the product of wavelength and frequency must specifically be a velocity: since wavelength has dimension $[L]$ and frequency has dimension $[T^{-1}]$, their product $\lambda f$ necessarily has dimension $[LT^{-1}]$, which is exactly the dimension …

Worked out. A string fixed at one wall is shown carrying a wave pattern in two separate cases, each spanning a fixed distance of 12 m, and assuming the waves shown cross that 12 m distance in exactly one second, the task is to read off the wavelength, frequency and velocity in each case. In the first case the pattern shows two full wavelengths fitting into 12 m, giving $\lambda=6\ \text{m}$ and, since two full waves cross in one second, $f=2\ \text{Hz}$, so $v=\lambda f=6\times2=12\ \text{m/s}$. In the second case the pattern instead shows six full wavelengths fitting into the same 12 m, giving a much shorter $\lambda=2\ \text{m}$ but a correspondingly higher $f=6\ \text{Hz}$, so again $v=\lambda f=2\times6=12\ \text{m/s}$. Both cases give exactly the same velocity despite having very different wavelengths and frequencies, which is the key point of the example: the speed of a wave along a given string is a fixed constant of that string, so a higher frequency always comes packaged with a proportionally short …