Physics · Ch 8 — Sound
Common Properties of all Waves
Common Properties of all Waves
This section sets out the quantities that describe ANY wave -- illustrated here mainly for mechanical waves, though the definitions carry over to every wave type.
1. Amplitude (A). The largest displacement any particle of the medium undergoes from its own rest (equilibrium) position as the wave passes through it. SI unit: metre.
2. Wavelength (λ). The distance between two successive particles of the medium that are in exactly the same state of vibration (the same phase) at a given instant. SI unit: metre.
3. Period (T). The time taken by a single particle of the medium to complete one full vibration. SI unit: second.
4. Double periodicity. A wave repeats itself in TWO independent senses at once: at any one location, the disturbance recurs at equal intervals of TIME (period T) -- this is periodicity in time; and at any one instant, the shape of the wave recurs at equal intervals of DISTANCE (wavelength λ) -- this is periodicity in space. Because both hold simultaneously, wave motion is called a doubly periodic phenomenon.
5. Frequency (n). The number of vibrations a particle completes per second, measured in hertz (Hz), and related to the period by .
6. Velocity (v). The distance the wave itself advances per unit time. In exactly one period T, the wave covers exactly one wavelength λ, so , and combining this with gives the single most-used wave relation, . This relation shows that for a wave travelling through a GIVEN medium, v stays fixed, so raising the frequency of a wave necessarily shortens its wavelength (and vice versa) -- and crucially, when a wave crosses from one medium into another, its FREQUENCY does not change (frequency is set once and for all by whatever is generating the wave), so it is the wave's speed and wavelength that must adjust to the new medium.
Medium requirements for a mechanical wave. Three properties of a medium must all hold for it to carry a mechanical wave: (i) it must be continuous and elastic, so it springs back to its original state once a deforming force is removed; (ii) it must possess inertia, so it can store and then hand on energy in the form of the travelling wave; (iii) its frictional (resistive) losses must be small, so the oscillations are not rapidly damped out as the wave travels. …
What this figure shows. A sinusoidal graph of particle displacement y plotted against distance x along the wave, drawn for a transverse-style sine curve travelling along the +x axis, at one fixed instant of time. A sequence of labelled particles A, B, C, D, E, F sit at successive points along the curve roughly one-eighth of a wavelength apart, at successive crest/zero/trough positions of the sine shape. Particle A is at the leading edge of the disturbance where the wave has just arrived (displacement zero, phase angle taken as 0 degrees). Particle B sits at the first positive peak (positive maximum displacement, phase 90 degrees / pi/2). Particle C is back at zero displacement but moving in the opposite sense to A (phase 180 degrees / pi). Particle D is at the negative peak (trough). Particle E is again at zero displacement, having completed one full wavelength from A (phase 360 degrees / 2 pi). Particle F is at the next positive peak, one wavelength beyond B (phase 450 degrees / 5pi/2, i.e. into its second oscillation). Pairs P and Q (or E and C, or B and D) are marked as having the same displacement magnitude but oppositely directed velocities; B and F are marked as being in phase …
What this figure shows. A sinusoidal graph of the displacement y of ONE SINGLE particle of the medium plotted against time t, companion to Fig 8.1(a) which instead plots many particles' displacements against distance at one instant. This graph shows that same one particle's oscillation repeating periodically in time with period T, tracing the same up-down sine shape but now with the horizontal axis being time rather than distance, illustrating the 'periodic in time' half of a wave's double periodicity (Fig 8.1(a) …
Worked out. The speed of sound in air is given as 330 m/s and in glass as 4500 m/s. The question asks for the ratio of the wavelength of a sound of a given (common) frequency in the two media. The method uses v = nlambda separately for each medium (v_air = nlambda_air and v_glass = n*lambda_glass, with the SAME frequency n in both, since frequency is fixed by the source and does not change when a wave crosses into a new medium), then divides the two relations to eliminate n and get lambda_air/lambda_glass = v_air/v_glass = 330/4500, which works out to approximately 0.0733 -- i.e. the wavelength in air is only about 7.3% of the wavelength in the much fast …
Worked out. An in-text 'Can you recall / Activity' style exercise with two parts, illustrating how wavelength and amplitude are read off a displacement-distance graph. Part (1) asks the student to sketch, on axes of displacement versus distance, two waves A and B such that wave A has TWICE the wavelength and HALF the amplitude of wave B -- i.e. A is a long, low sine curve and B is a short, tall one, both starting from the same origin. Part (2) shows a small figure with two labelled waves P and Q already drawn on a shared displacement-distance graph and asks the student to determine the wavelength and amplitude of each directly from the picture (reading off the horizontal repeat-distance for wavelength and the peak height for amplitude), reinforcing the definitions of amplitude and wavelength from this section as purely gr …