Linear Wave Theory: Understanding Wave Behavior and Dispersion Relationships, Exams of Hydraulics

Download Linear Wave Theory: Understanding Wave Behavior and Dispersion Relationships and more Exams Hydraulics in PDF only on Docsity! Hydraulics 3 Waves: Linear Wave Theory – 1 Dr David Apsley 1. LINEAR WAVE THEORY AUTUMN 2021 1.1 Main Wave Parameters Consider a single-frequency (“monochromatic”) progressive wave on still water: η = 𝐴 cos(𝑘𝑥 − ω𝑡) We will develop a linear wave theory (or Airy1 wave theory), based on the assumption that the wave amplitude 𝐴 is small (compared with the depth ℎ and wavelength 𝐿), and, hence, we may neglect second-order and higher products and powers of wave-related perturbations. Despite the approximations embodied in the assumption of small amplitudes, linear-wave theory has proved remarkably successful at describing the behaviour of real waves, including wave transformations such as refraction, diffraction, shoaling, reflection and predicting the onset of breaking. An additional advantage of linearity is that real (multi-frequency, multi- direction) wave fields can be obtained by superposition of simple wave components. Amplitude and Height The amplitude 𝐴 is the maximum displacement from still water level (SWL). The wave height 𝐻 is the vertical distance between neighbouring crest and trough. For sinusoidal waves, 𝐴 = 𝐻 2 , or 𝐻 = 2𝐴 For non-sinusoidal waves 𝐻 is the more easily defined and measured quantity. Wavenumber and Wavelength 𝑘 is the wavenumber. Since the wave goes through a single cycle when 𝑘𝑥 changes by 2π, the wavelength 𝐿 is given by 𝐿 = 2π 𝑘 1 After George Biddell Airy, former Astronomer Royal, who first developed the theory. L h A H (x,t) SWL (z=0)x z trough crest c bed (z= -h) Hydraulics 3 Waves: Linear Wave Theory – 2 Dr David Apsley Frequency and Period ω is the wave angular frequency. Since the wave goes through a single cycle in time when ω𝑇 changes by 2π, the period 𝑇 is given by 𝑇 = 2𝜋 ω with actual frequency 𝑓 (in cycles per second, or Hertz) given by 𝑓 = 1 𝑇 = ω 2𝜋 Wave Speed (Phase Velocity, or “Celerity”) The wave profile above can also be written η = 𝐴 cos[𝑘(𝑥 − c𝑡)] where 𝑐 = ω 𝑘 η is the same for 𝑥 − 𝑐𝑡 = constant, or 𝑥 = 𝑐𝑡 + constant, so 𝑐 represents the velocity of travel of a pure sinusoidal wave (the speed at which its crest moves forward), also called the phase velocity, or celerity. It can be computed by any of 𝑐 = ω 𝑘 = 𝐿 𝑇 ( wavelength period ) = 𝑓𝐿 (frequency × wavelength) ω and 𝑘 are not independent, but are related via a dispersion relation, saying how wavelength changes with frequency. Later we shall see that for a collection of waves or wave packet the natural velocity of energy transport is not the phase velocity 𝑐 but the group velocity 𝑐𝑔 = dω d𝑘 Note that 𝑐 represents the speed of the wave form, not the individual fluid particles, whose velocity, as we shall see, is considerably smaller. 1.2 Dispersion Relationship At a given depth, waves of different frequencies (hence of different wavelengths) travel at different speeds. This phenomenon is called dispersion. For a given frequency, the wavelength, and hence wave speed, must change with depth. Hydraulics 3 Waves: Linear Wave Theory – 5 Dr David Apsley 1.3 Wave Velocity and Pressure The velocity and pressure fields for a harmonic surface displacement: η = 𝐴 cos(𝑘𝑥 − ω𝑡) are also derived in APPENDICES A1–A3. They are conveniently summarised by a velocity potential 𝜙: 𝜙 = 𝐴𝑔 ω cosh 𝑘(ℎ + 𝑧) cosh 𝑘ℎ sin (𝑘𝑥 − ω𝑡) The horizontal and vertical velocity components are: 𝑢 ≡ 𝜕𝜙 𝜕𝑥 = 𝐴𝑔𝑘 ω cosh 𝑘(ℎ + 𝑧) cosh 𝑘ℎ cos(𝑘𝑥 − ω𝑡) 𝑤 ≡ 𝜕𝜙 𝜕𝑧 = 𝐴𝑔𝑘 ω sinh𝑘(ℎ + 𝑧) cosh 𝑘ℎ sin (𝑘𝑥 − ω𝑡) The horizontal velocity component 𝑢 is in phase with the surface disturbance: it is largest underneath a wave crest. The vertical component 𝑤 is 90º out of phase with the surface disturbance: it is zero underneath a wave crest. Since sinh𝑋 < cosh𝑋, the vertical component is always smaller than the horizontal one, the more so near the bed. The pressure is 𝑝 = −ρ𝑔𝑧 − ρ ∂𝜙 𝜕𝑡 = −ρ𝑔𝑧⏟ hydrostatic + ρ𝑔𝐴 cosh 𝑘(ℎ + 𝑧) cosh 𝑘ℎ cos(𝑘𝑥 − ω𝑡) ⏟ hydrodynamic (i.e. wave) The pressure field consists of two parts: • a hydrostatic pressure – ρ𝑔𝑧, which is always present (below the SWL), regardless of whether waves exist; • a wave-induced hydrodynamic pressure −ρ𝜕𝜙/𝜕𝑡 , which exists only if waves are present. At a fixed height this varies (as might be anticipated from the amount of water above) sinusoidally from maximum positive beneath a wave crest to maximum negative beneath a wave trough. Example. A pressure sensor is located 0.6 m above the sea bed in a water depth ℎ = 12 m. The pressure fluctuates with period 15 s. A maximum gauge pressure of 124 kPa is recorded. (a) What is the wave height? (b) What are the maximum horizontal and vertical velocities at the surface? Hydraulics 3 Waves: Linear Wave Theory – 6 Dr David Apsley 1.4 Wave Energy Wave energy is of two forms: kinetic and potential. The amount of each (per unit horizontal area) can be found by integrating over the water column and averaging over a period and wavelength. This is done in APPENDIX A4. The result is (with the small-amplitude wave assumption): KE = ∫ 1 2 ρ(𝑢2 + 𝑤2) d𝑧 η 𝑧=−ℎ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ = 1 4 ρ𝑔𝐴2 PE = ∫ ρ𝑔𝑧 d𝑧 η 𝑧=−ℎ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ = 1 4 ρ𝑔𝐴2 + constant It is seen that (under linear wave theory) the average wave-related kinetic and potential energies are equal. The total energy (per unit horizontal area) is the sum of these: 𝐸 = 1 2 ρ𝑔𝐴2 or, in terms of wave height, 𝐸 = 1 8 ρ𝑔𝐻2 IMPORTANT: wave energy varies as the square of the wave height. 1.5 Group Velocity Real wave fields do not consist of individual harmonic components, but wave packets comprised of a combination of waves of different frequency. Consider first two superposed waves of similar amplitude 𝑎 but different frequencies ω± Δω and corresponding wavenumbers 𝑘 ± Δ𝑘: η = 𝑎 cos[(𝑘 + Δ𝑘)𝑥 − (ω + Δω)𝑡]⏟ component 1 + 𝑎 cos[(𝑘 − Δ𝑘)𝑥 − (ω − Δω)𝑡]⏟ component 2 Using the trigonometric formula cos α + cos β = 2 cos α+β 2 cos α−β 2 , we find η = 2𝑎 cos(𝑘𝑥 − ω𝑡) cos(Δ𝑘. 𝑥 − Δω. 𝑡) This represents (see the figure) an underlying fast-varying wave motion cos(𝑘𝑥 − ω𝑡) with the phase velocity 𝑐 = ω/𝑘 , but modulated by an envelope with time-varying amplitude: 𝐴(𝑡) = 2𝑎 cos(Δ𝑘 𝑥 − Δω 𝑡) This amplitude function represents a wave with smaller wavenumber and frequency (i.e. longer wavelength and period), travelling at speed Δω Δ𝑘 Hydraulics 3 Waves: Linear Wave Theory – 7 Dr David Apsley In general, real wave packets are formed from not just two but a continuous spread of wavenumbers. It may be shown mathematically2 that, for a wave packet strongly peaked around one wavenumber 𝑘, the packet as a whole travels with the group velocity 𝑐𝑔 = dω d𝑘 For regular waves, using the dispersion relation ω2 = 𝑔𝑘 tanh𝑘ℎ (see APPENDIX A5): 𝑐𝑔 = 1 2 [1 + 2𝑘ℎ sinh 2𝑘ℎ ] ω 𝑘 or 𝑐𝑔 = 𝑛𝑐 where 𝑛 = 1 2 [1 + 2𝑘ℎ sinh 2𝑘ℎ ] , 𝑐 = ω 𝑘 𝑛 is the ratio of group velocity to phase velocity. It is always between ½ and 1; the group velocity is always less than the phase velocity. 1.6 Energy Transfer (Wave Power) Since wave energy depends on the square of the amplitude, and the amplitude envelope travels at the group velocity, the latter also represents the velocity at which energy is transferred. • Phase velocity, 𝑐, represents the speed at which the wave form propagates. • Group velocity, 𝑐𝑔, represents the speed at which the wave transmits energy. The rate at which wave energy is transferred is called the energy flux or wave power 𝑃 and is usually expressed as the (average) wave energy passing a location per unit length of wave crest per unit time. Heuristically it may be deduced by noting that, if wave energy does travel at velocity 𝑐𝑔, then the energy in area 𝑐𝑔Δ𝑡 crosses such a line in time Δ𝑡, so 𝑃Δ𝑡 = 𝐸 × 𝑐𝑔Δ𝑡, or 𝑃 = 𝐸𝑐𝑔 where 𝐸 = 1 2 ρ𝑔𝐴2 is the wave energy per unit area of surface and 𝑐𝑔 is the group velocity. A more formal derivation (see APPENDIX A6) is to integrate the rate of working by pressure forces (i.e. pressure  area  velocity), over the depth of the water column and then average in time. Example. A sea-bed pressure transducer in 9 m of water records a sinusoidal signal with amplitude 5.9 kPa and period 7.5 s. Find the wave height, energy density and wave power per metre of crest. 2 Google “Method of stationary phase”. Hydraulics 3 Waves: Linear Wave Theory – 10 Dr David Apsley 1.8.3 Particle Motions As shown before, the trajectories of particles are ellipses with horizontal and vertical semiaxes 𝑎 = 𝐴 cosh 𝑘(ℎ + 𝑍0) sinh 𝑘ℎ , 𝑏 = 𝐴 sinh 𝑘(ℎ + 𝑍0) sinh 𝑘ℎ For deep-water waves (𝑘ℎ ≫ 1), 𝑎 = 𝑏 ≈ 𝐴e−𝑘|𝑍0| and the trajectories of particles are circles whose radius diminishes exponentially with depth over the order of half a wavelength. For shallow-water waves (𝑘ℎ ≪ 1), 𝑎 ≈ 𝐴 𝑘ℎ , 𝑏 𝑎 ≪ 1 the trajectories of particles are highly flattened ellipses; the horizontal excursion of water particles is similar at all depths and much greater than the vertical excursion. 1.8.4 Pressure The pressure is given, in general, by 𝑝 = −ρ𝑔𝑧 − ρ ∂𝜙 𝜕𝑡 = −ρ𝑔𝑧⏟ hydrostatic + ρ𝑔η cosh 𝑘(ℎ + 𝑧) cosh 𝑘ℎ⏟ hydrodynamic For deep-water waves (𝑘ℎ ≫ 1), 𝑝 ≈ −ρ𝑔𝑧 + ρ𝑔ηe−𝑘|𝑧| and the hydrodynamic pressure decays exponentially with depth below the surface over the order of half a wavelength. For shallow-water waves (𝑘ℎ ≪ 1), 𝑝 ≈ ρ𝑔(η − 𝑧) i.e. shallow-water waves are hydrostatic. This is because vertical accelerations are much smaller than 𝑔. This is the limit we have seen earlier in the Open-Channel Flow section. deep water intermediate depth shallow water Hydraulics 3 Waves: Linear Wave Theory – 11 Dr David Apsley 1.8.5 Summary of Shallow-Water/Deep-Water Approximations In shallow water: • The phase velocity and group velocity are equal and independent of wavelength ... waves on shallow water are non-dispersive. 𝑐 = √𝑔ℎ • Pressure is hydrostatic. • Velocity is predominantly horizontal and almost independent of depth; particle paths are highly-flattened ellipses. In deep water: • The phase velocity is dependent on wavelength and the group velocity is half the phase velocity ... deep-water waves are dispersive. • Phase velocity is independent of depth, as velocity and pressure perturbations do not reach the bed. • Wavelength, speed and period are connected by 𝐿 = 𝑔𝑇2 2π , 𝑐 = 𝑔𝑇 2π • Particle motions are circular and diminish in size over distance of order half a wavelength. These mean that for any body of water deeper than about half a wavelength the bed will play no influence on wave motions. Example. (a) Find the deep-water speed and wavelength of a wave of period 12 s. (b) Find the speed and wavelength of a wave of period 12 s in water of depth 3 m. Compare with the shallow-water approximation. Hydraulics 3 Waves: Linear Wave Theory – 12 Dr David Apsley 1.9 Waves on Currents Until now we have considered waves on still water. In many cases waves at sea coexist with a background steady current 𝑈. The formulae then continue to hold in a frame of reference moving with the current; i.e. 𝑥 is replaced by a value (subscript r): 𝑥𝑟 = 𝑥 − 𝑈𝑡 In particular, η = 𝐴 cos(𝑘𝑥𝑟 −ω𝑟𝑡) = 𝐴 cos[𝑘𝑥 − (ω𝑟 + 𝑘𝑈)𝑡] = 𝐴 cos[𝑘{𝑥 − ( ω𝑟 𝑘 + 𝑈)𝑡}] Hence, in an absolute frame (subscript a) which is fixed; i.e. not moving with the current: • wavenumber 𝑘 and wavelength 𝐿 are unchanged; (if you took a photograph from above, the fact that the waves are advected with the current doesn’t change their length); • the current is simply added to the wave speed (which you could probably have guessed): 𝑐𝑎 = 𝑐𝑟 + 𝑈 • the perceived frequency ω and period 𝑇 are changed: ω𝑎 = ω𝑟 + 𝑘𝑈 e.g. if waves and current are in the same direction, then the absolute frequency ω𝑎 is larger, and the absolute period 𝑇𝑎 is smaller than their “relative” counterparts (those moving with the current). This is because the current shortens the time between successive wave crests passing a point. Note, however, that it is the relative frequency ω𝑟 = ω𝑎 − 𝑘𝑈 that appears in the dispersion formula, and hence: (ω𝑎 − 𝑘𝑈) 2 = ω𝑟 2 = 𝑔𝑘 tanh 𝑘ℎ The change of apparent frequency in a fixed reference frame due to the motion of the source or transmitting medium is called a Doppler shift. Note: be careful about absolute velocities (phase or group): 𝑐𝑎 = 𝐿 𝑇𝑎 = ω𝑟 𝑘 + 𝑈 To use the second form you would need to find ω𝑟 = ω𝑎 − 𝑘𝑈 once you have solved for 𝑘. Example. An acoustic depth sounder indicates regular surface waves with apparent period 8 s in water of depth 12 m. Find the wavelength and absolute phase speed of the waves when there is: (a) no mean current; (b) a current of 3 m s–1 in the same direction as the waves; (c) a current of 3 m s–1 in the opposite direction to the waves.