Appendix 2 Waves and Wave Analysis

APPENDIX 2 WAVES AND WAVE ANALYSIS A2.1 Introduction wave terminology and, importantly, they can be manipulated and analysed as if they were waves recorded in the The term wave is used to describe a localised disturbance space domain. which travels through some medium. Consider a cork Waveforms are frequently encountered in geophysics, so floating in water. A ripple, i.e. a wave, travelling (propagat- it is essential to be familiar with the terminology used for ing) across the water’s surface causes the cork to bob up and describing them and the mathematical methods used for down. The movement of the cork (and of individual water manipulating them. Specific aspects of the various type of molecules) is due to the disturbance associated with the waves used in geophysical surveys, and their propagation wave. The wave can be thought of as a moving packet of in the case of moving waves, are described for each strain energy travelling through the water. Although the geophysical method in the relevant chapters of this text. water is deformed (strained) during the passage of the ripple, Here we provide only a general description of waveforms it only produces an oscillation about a point. The cork does and wave-related phenomena and the techniques used in not move laterally, demonstrating that the disturbance their analysis. associated with the wave does not represent a flow of water. In the example above, the variation in the surface height of the water could be plotted as a function of location for A2.2 Parameters defining waves different times (Fig. A2.1). Each of these plots is a spatial and waveforms series (see Section 2.2). The movement of the ripple A familiar and extremely useful waveform is a sine wave, through space is clearly apparent when the four spatial also referred to as a sinusoid. This is a graph of the sine series are compared. It is also possible to plot the height function from trigonometry, where the sine of an angle is of the water surface at some chosen point as a function of plotted against the angle. As will be demonstrated in time (Fig. A2.1b). This is a time series (see Section 2.2) and Section A2.4, other waveforms can be represented as the in the example represents the cork bobbing up and down sum of a series of different sine waves. as the ripple passes. A sine wave is continuous. A waveform is said to be A graph of the variation of some physical parameter continuous if it goes to infinity in either time or distance. related to a wave is known as a waveform. As shown in The concept of a continuous wave is fundamental to the Fig. A2.1 this can be plotted in the time domain or the space definition of a wide variety of waveforms. There also exists domain depending on the type of recording. Although these another class of waves which are said to be discontinuous. represent variations in two different physical dimensions, These have limited temporal/spatial extent, i.e. they only mathematically they can be treated identically so they can exist in a restricted area of time or space, and are known be manipulated and analysed in the same way. The seismic as wavelets. An example of a wavelet in the time domain is a and electromagnetic waves exploited in geophysical surveys single hand clap: the sound only exists in a restricted inter- travel through the subsurface and are usually analysed as val of time/space – it is a non-continuous wave. The equiva- time series recorded at a stationary recording location. lent continuous sound wave might be a continuous hum. However, graphs of the spatial variations of any of the Referring to Fig. A2.2a, a number of parameters define physical quantities measured in geophysical surveys, a wave: although not necessarily directly created by wave phenomena, have all the attributes of a waveform (see Fig. 1.1a in • The maximum variation in the sine wave, which the main text). They can be conveniently described using accounts for the positive and negative variations, is 2 Waves and wave analysis a) a) Space domain Magnitude + Direction of + propagation Phase difference (Df) Amplitude (A) Location To–¥ To¥ A t Time/location – Time = 0 + Direction of #2 B #1 propagation Wavelength (l) – or period (P ) Time/space domain Location Magnitude t Dt b) – Time = 0+ ) A ) + Direction of #1 Amplitude f Phase propagation difference #2 #2 difference #1 (Df) Phase ( Location Amplitude ( Magnitude Frequency Frequency C t Dt – Time = 0+2 Frequency domain Direction of + propagation D Figure A2.2 Continuous sine waves. The representation of two continuous sine waves in (a) the time/space and (b) the frequency domains. See text for further explanations. Location Magnitude Magnitude – Time = t +3 Dt • A sine wave repeats itself. The repeated section of the 0 Wavelength (l) Reference wave is known as a cycle. This is the section between two point consecutive equivalent points on the waveform experi- encing identical disturbance. For example, the section b) Magnitude (at reference point) between two adjacent peaks or troughs, or two zero – + cross-overs of the same slope etc., represents one cycle A t of the wave. Time = 0 B Time • In the space domain, the distance that one cycle of the Time = t + Dt 0 domain sine wave occupies is known as the wavelength (λ)of C Time = t +2D t Period (P ) 0 the wave. The time domain equivalent of wavelength is t D t the period (P) of the wave, which is the time taken for Time = 0+3 D one cycle of the wave to occur. Time • In the time domain, the number of repetitions or cycles per unit of time, i.e. cycles per second or hertz (Hz), is Figure A2.1 Waveforms. (a) Waveforms showing the magnitude of a the frequency (f) of the wave. In the space domain, the wave’s disturbance in the region of a reference point in a medium at different times (separated by time interval Δt) as a wavelet passes number of cycles per unit of distance, e.g. cycles per through the point, and (b) the disturbance at the reference point metre, is the spatial frequency or wave number (σ)of shown as a waveform varying in time. the wave. It is common for spatial frequency to be (incorrectly) referred to as frequency. • fi known as the peak value or amplitude (A) of the wave. In The various parameters de ned above are related as the water-ripple analogy, this is the height of the ripples shown in the following Eqs. (A2.1) to (A2.4): above or below the average water level. As described in σ ¼ 1 ð : Þ Section 6.3.3.1, the amplitude of a wave is a related to its λ A2 1 energy content. The term amplitude is also used in a less 1 formal sense to represent the magnitude of the waveform f ¼ ðA2:2Þ at some specific time or location. P A2.3 Wave interference 3 For a wave travelling at a velocity (V) a) Magnitude isc d g T = 2 T V n = 1 ¼ ð : Þ ti f A2 3 a t λ o T = 3 R T = 0 and so T = 4 Axis 0 1234 Time (T ) V λ ¼ ðA2:4Þ f Df • A sine wave of a particular frequency can also be b) Magnitude Phase difference described in terms of a radial line of a reference circle isc d T g = 1 n T = 0 ti rotating anticlockwise, i.e a rotating disc (Fig. A2.3a). The a t o T = 2 length of the line is the amplitude (A) of the wave R f (Fig. A2. 2a). The variation in the vertical height of the T = 3 Axis 0 1234 Time (T ) end of the amplitude line, measured above and below T the axis of the disc, as it rotates is the magnitude of the = 4 waveform with time. Df • The rotating disc analogy demonstrates clearly the con- Phase c) Magnitude cept of phase angle (ϕ), often just called the phase. In the difference (180°) isc d three parts of Fig. A2.3 the disc starts rotating with the g n ti a amplitude line in different positions. The three wave- t o f = 180 forms are shifted relative to each other, parallel to the R T = 4 T = 0 0 1234 Axis Time (T ) time axis. The amount of shift depends on the degree T = 3 T = 2 of rotation of the disc at the different starting positions. T = 1 This is why phase is an angle. • Figure A2.2a shows two sine waves of the same frequency and with different amplitudes. The offset Figure A2.3 Rotating disc analogy for the magnitude variations of a between them is their phase difference (Δϕ) (Fig. sine wave. See text for details. A2.2b). As shown, the variation represented by waveform #1 leads that represented by waveform #2 by the amount of their phase difference. Alternatively, we can amplitude as a function of frequency and known as the say that waveform #2 lags waveform #1 by the phase amplitude spectrum, and the other showing the phase difference. as a function of frequency and known as the phase • An important case is when the phase difference between spectrum. The frequency domain representation is called two waveforms is 180° (π radians), i.e. they are com- the frequency spectrum, Fourier spectrum, or simply the pletely out of phase. This is the case for the waveforms in spectrum.

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