Fundamentals of Fluid Dynamics: Waves in Fluids Introductory Course on Multiphysics Modelling TOMASZ G. ZIELINSKI´ (after: D.J. ACHESON’s “Elementary Fluid Dynamics”) bluebox.ippt.pan.pl/˜tzielins/ Table of Contents 1 Introduction1 1.1 The notion of wave . .1 1.2 Basic wave phenomena . .2 1.3 Mathematical description of a traveling wave . .2 2 Water waves4 2.1 Surface waves on deep water . .4 2.2 Dispersion and the group velocity . .8 2.3 Capillary waves . 11 2.4 Shallow-water finite-amplitude waves . 14 3 Sound waves 16 3.1 Introduction . 16 3.2 Acoustic wave equation . 16 3.3 The speed of sound . 17 3.4 Sub- and supersonic flow . 18 1 Introduction 1.1 The notion of wave What is a wave? A wave is the transport of a disturbance (or energy, or piece of information) in space not associated with motion of the medium occupying this space as a whole. (Except that electromagnetic waves require no medium !!!) The transport is at finite speed. 2 Fundamentals of Fluid Dynamics: Waves in Fluids ICMM lecture The shape or form of the disturbance is arbitrary. The disturbance moves with respect to the medium. Two general classes of wave motion are distinguished: 1. longitudinal waves – the disturbance moves parallel to the direction of prop- agation. Examples: sound waves, compressional elastic waves (P-waves in geophysics); 2. transverse waves – the disturbance moves perpendicular to the direction of propagation. Examples: waves on a string or membrane, shear waves (S-waves in geophysics), water waves, electromagnetic waves. 1.2 Basic wave phenomena reflection – change of wave direction from hitting a reflective surface, refraction – change of wave direction from entering a new medium, diffraction – wave circular spreading from entering a small hole (of the wavelength- comparable size), or wave bending around small obstacles, interference – superposition of two waves that come into contact with each other, dispersion – wave splitting up by frequency, rectilinear propagation – the movement of light wave in a straight line. Standing wave A standing wave, also known as a stationary wave, is a wave that remains in a constant position. This phenomenon can occur: when the medium is moving in the opposite direction to the wave, (in a stationary medium:) as a result of interference between two waves travelling in opposite directions. 1.3 Mathematical description of a traveling wave 2π 2π T = λ = ! k A A t x FIGURE 1: A simple traveling wave in time domain (left) and in space (right). ICMM lecture Fundamentals of Fluid Dynamics: Waves in Fluids 3 Traveling waves Simple wave or traveling wave, sometimes also called progressive wave, is a disturbance that varies both with time t and distance x in the following way (see Figure1): u(x; t) = A(x; t) cos k x − ! t + θ0 π = A(x; t) sin k x − ! t + θ0 ± 2 (1) | {z } θ~0 where A is the amplitude, ! and k denote the angular frequency and wavenum- ~ ber, and θ0 (or θ0) is the initial phase. Amplitude A e.g. m; Pa; V=m – a measure of the maximum disturbance in the medium during one wave cycle (the maximum distance from the highest point of the crest to the equilibrium). Phase θ = k x − ! t + θ0 [rad], where θ0 is the initial phase (shift), often ambigu- ously, called the phase. Period T [s] – the time for one complete cycle for an oscillation of a wave. Frequency f [Hz] – the number of periods per unit time. Frequency and angular frequency The frequency f [Hz] represents the number of periods per unit time 1 f = : (2) T The angular frequency ! [Hz] represents the frequency in terms of radians per second. It is related to the frequency by 2π ! = = 2π f : (3) T Wavelength λ [m] – the distance between two sequential crests (or troughs). Wavenumber and angular wavenumber The wavenumber is the spatial analogue of frequency, that is, it is the mea- surement of the number of repeating units of a propagating wave (the number of times a wave has the same phase) per unit of space. Application of a Fourier transformation on data as a function of time yields a frequency spectrum; application on data as a function of position yields a wavenumber spectrum. 4 Fundamentals of Fluid Dynamics: Waves in Fluids ICMM lecture 1 The angular wavenumber k m , often misleadingly abbreviated as “wave- number”, is defined as 2π k = : (4) λ There are two velocities that are associated with waves: 1. Phase velocity – the rate at which the wave propagates: ! c = = λ f : (5) k 2. Group velocity – the velocity at which variations in the shape of the wave’s am- plitude (known as the modulation or envelope of the wave) propagate through space: d! c = : (6) g dk This is (in most cases) the signal velocity of the waveform, that is, the rate at which information or energy is transmitted by the wave. However, if the wave is travelling through an absorptive medium, this does not always hold. 2 Water waves 2.1 Surface waves on deep water Consider two-dimensional water waves: u = u(x; y; t); v(x; y; t); 0 : @v @u Suppose that the flow is irrotational: @x − @y = 0 : Therefore, there exists a velocity potential φ(x; y; t) so that @φ @φ u = ; v = : (7) @x @y The fluid is incompressible, so by the virtue of the incompressibility condition, r · u = 0, the velocity potential φ will satisfy Laplace’s equation @2φ @2φ + = 0 : (8) @x2 @y2 Free surface The fluid motion arises from a deformation of the water surface – which is of major interest (see Figure2). The equation of this free surface is denoted by y = η(x; t) . § ¤ ¦ ¥ ICMM lecture Fundamentals of Fluid Dynamics: Waves in Fluids 5 FIGURE 2: A deformation on the free surface of water in the form of a wave packet. Kinematic condition at the free surface: Fluid particles on the surface must remain on the surface. The kinematic condition entails that F (x; y; t) = y − η(x; t) remains constant (in fact, zero) for any particular particle on the free surface which means that DF @F = + u · rF = 0 on y = η(x; t); (9) Dt @t and this is equivalent to @η @η + u = v on y = η(x; t): (10) @t @x Pressure condition at the free surface: The fluid is inviscid (by assumption), so the condition at the free surface is simply that the pressure there is equal to the atmospheric pressure p0: p = p0 on y = η(x; t): (11) Bernoulli’s equation for unsteady irrotational flow If the flow is irrotational (so u = rφ and r × u = 0), then, by integrating (over the space domain) the Euler’s momentum equation: @rφ p 1 = −∇ + u2 + χ ; (12) @t % 2 the Bernoulli’s equation is obtained @φ p 1 + + u2 + χ = G(t) : (13) @t % 2 Here, χ is the gravity potential (in the present context χ = g y where g is the gravity acceleration) and G(t) is an arbitrary function of time alone (a constant of integration). p0 Now, by choosing G(t) in a convenient manner, G(t) = % , the pressure condition may be written as: @φ 1 + u2 + v2 + g η = 0 on y = η(x; t): (14) @t 2 6 Fundamentals of Fluid Dynamics: Waves in Fluids ICMM lecture Small-amplitude waves The free surface displacement η(x; t) and the fluid velocities u, v are small (in a sense to be made precise later). Linearization of the kinematic condition @η @η @η v = + u ! v(x; η; t) = @t @x @t | {z } small Taylor @v @η −−−! v(x; 0; t) + η (x; 0; t) + ··· = series @y @t (15) | {z } small @φ v= @η @y @φ @η ! v(x; 0; t) = −−−! = on y = 0. @t @y @t Linearization of the pressure condition @φ 1 @φ + u2 + v2 +g η = 0 ! + g η = 0 on y = 0. (16) @t 2 @t | {z } small A sinusoidal travelling wave solution The free surface is of the form η = A cos(k x − ! t) ; (17) where A is the amplitude of the surface displacement, ! is the circular fre- quency, and k is the circular wavenumber. The corresponding velocity potential is φ = q(y) sin(k x − ! t) : (18) @2φ @2φ It satisfies the Laplace’s equation, @x2 + @y2 = 0. 00 2 Therefore, q(y) must satisfy q − k q = 0 ; the general solution of which is q = C exp(k y) + D exp(−k y) : (19) For deep water waves D = 0 (if k > 0 which may be assumed without loss of generality) in order that the velocity be bounded as y ! −∞. Therefore, the velocity potential for deep water waves is φ = C exp(k y) sin(k x − ! t) : (20) ICMM lecture Fundamentals of Fluid Dynamics: Waves in Fluids 7 Now, the (linearized) free surface conditions yield what follows: @φ @η 1. the kinematic condition ( @y = @t on y = 0): A! C k = A! ! §φ = k exp(k y) sin(k x − ! t) ; ¤ (21) ¦ ¥ @φ 2. the pressure condition ( @t + g η = 0 on y = 0): − C! + g A = 0 ! !2 = g k : § ¤ (dispersion relation!) (22) ¦ ¥ The fluid velocity components: u = A! exp(k y) cos(k x − ! t) ; v = A! exp(k y) sin(k x − ! t) : (23) Particle paths Any particle departs only a small amount (X; Y ) from its mean position (x; y). dX Therefore, its position as a function of time may be found by integrating u = dt dY and v = dt ; whence: X(t) = −A exp(k y) sin(k x − ! t) ;Y (t) = A exp(k y) cos(k x − ! t) : (24) Figure3 presents particle paths for a wave on deep water.