Chapter 8 Waves Abstract Chapter 8 deals with waves. The topics covered are wave equation, pro- gressive and stationary waves, vibration of strings, wave velocity in solids, liquids and gases, capillary waves and gravity waves, the Doppler effect, shock wave, rever- beration in buildings, stationary waves in pipes and intensity level. 8.1 Basic Concepts and Formulae The travelling wave: The simple harmonic progressive wave travelling in the posi- tive x-direction can be variously written as 2π y = A sin (vt − x) λ t x = A sin 2π − T λ = A sin(ωt − kx) x = A π f t − sin 2 v (8.1) where A is the amplitude, f the frequency and v the wave velocity, λ the wave- length, ω = 2π f the angular frequency and k = 2π/λ, the wave number. Similarly, the wave in the negative x-direction can be written as 2π y = A (vt + x) sin λ (8.2) and so on. The superposition principle states that when two or more waves traverse the same region independently, the displacement of any particle at a given time is given by the vector addition of the displacement due to the individual waves. Interference of waves: Interference is the physical effect caused by the superposi- tion of two or more wave trains crossing the same region simultaneously. The wave trains must have a constant phase difference. 339 340 8Waves Vibrating strings: Stationary waves are formed by the superposition of two similar progressive waves travelling in the opposite direction over a taut string clamped by rigid supports. Wave equation: ∂2 y F ∂2 y = (8.3) ∂t2 μ ∂x2 Wave velocity: v = F/μ (8.4) where F is the tension in the string and μ is the linear density, i.e. mass per unit length. The general solution of (8.3)is y = f1(vt − x) + f2(vt + x) (8.5) Harmonic solution: y = 2A sin kx cos ωt (8.6) When the displacement in the y-direction is maximum (antinode) the amplitude is 2A, the antinodes are located at x = λ/4, 3λ/4, 5λ/4 ... and are spaced half a wavelength apart. The amplitude has a minimum value of zero (nodes). The nodes are located at x = 0, λ/2, λ... and are also spaced half a wavelength apart. Ends of the strings are always nodes. Neighbouring nodes and antinodes are spaced one- quarter wavelength apart. The frequency of vibration is given by v N F N F f = = = (8.7) λ 2L μ 2L ρ A where ρ is the density and A the cross-sectional area of the string and N = 1, 2, 3,.... Vibration with N = 1 is called the fundamental or the first harmonic, N = 2 is called the first overtone or the second harmonic, etc. Power: The energy per unit length of the string is given by = 1 μ 2 E 2 V0 (8.8) where V0 is the velocity amplitude of any particle on the string. Since the wave is travelling with velocity v, the power (P) is given by 1 1 P = Ev = μV 2v = V 2 Fμ (8.9) av 2 0 2 0 8.1 Basic Concepts and Formulae 341 Waves in Solids In solids, compressional and shearing forces are readily transmitted. (i) Transverse waves in wires/strings in which the elastic properties of the material are disregarded: v = F/μ (8.4) (ii) Transverse waves in bars/wires 1 V ∝ Y/ρ λ (8.10) where Y is Young’s modulus of elasticity. (iii) Longitudinal waves in wires and bars V = Y/ρ (8.11) (iv) Torsional vibrations in wires/bars V = η/ρ (8.12) where η is the shear modulus of elasticity. In all these cases the material of restricted dimension is considered. Waves in Liquids The wave motion through liquids is influenced by the gravity and the characteristics of the medium such as the depth and surface tension. Canal Waves If the wavelength is large compared with the wave amplitude, surface tension effect is small. The controlling factors are then basically gravity (g) and the boundary con- ditions. Furthermore, if the surface is sufficiently extensive so that the wall effects are negligible then the depth (h) alone is the main boundary condition. The velocity (v) of the canal waves is given by V = gh (8.13) Surface Waves These are the waves found on relatively deep water. The velocity of deep water waves is given by V = gλ/2π (8.14) 342 8Waves For long waves in shallow water V = gh (8.15) Capillary Waves Surface waves are modified by surface tension S.Ifh is large compared with λ 2πs gλ V 2 = + (8.16) ρλ 2π The minimum value of λ is given by minimizing (8.16) s λ = 2π (8.17) min gρ If λ is sufficiently large the second term dominates and the controlling factor being mainly gravity. Thus, the velocity of the gravity waves is given by V = gλ/2π (8.18) If λ is very small, the first term in (8.16) dominates and the motion is mainly con- trolled by capillarity and 2πs V = ρλ (8.19) Acoustic Waves ∂2ξ ∂2ξ = V 2 (plane wave equation for displacement) (8.20) ∂t2 ∂x2 ∂2 P ∂2 P = V 2 (plane wave equation for pressure) (8.21) ∂t2 ∂x2 where V = B/ρ0 (8.22) B being the bulk modulus of elasticity. Sound Velocity in a Gas γ P V = ρ (Laplace formula) (8.23) 8.1 Basic Concepts and Formulae 343 Sound Velocity in a Liquid γ Bτ V = ρ (8.24) where BT is the isothermal bulk modulus. The energy in length λ is given by 1 2 2 Eλ = ρ ω A λ (8.25) 2 0 The energy density 1 2 2 E = Eλ/λ = ρ ω A (8.26) 2 0 The intensity, i.e. the time rate of flow of energy per unit area of the wave front 1 I = ρVA2ω2 (8.27) 2 Intensity Level (IL):Decibel Scale IL = 10 log(I/I0) (8.28) where log is logarithmic to base 10, I0 is the reference intensity (the zero of the scale) and IL is expressed in decibels. Stationary Waves in Pipes (i) Closed pipe (pipe closed at one end and opened at the other) f1 = V/4L, f2 = 3V/4L, f3 = 5V/4L ... (ii) Open pipe_(pipe opened at both ends) f1 = V/2L, f2 = V/L, f3 = 3V/2L ... Doppler effect is the apparent change in frequency of a wave motion when there is relative motion between the source and the observer. (a) Moving Source but Stationary Observer If the source of waves of frequency f moves with velocity v and if vs is the sound velocity in still air then the apparent frequency f would be / f v f = (8.29) v ± vs where the minus sign is for approach and plus sign for separation. 344 8Waves (b) Source is At Rest, Observer in Motion Let the observer be moving with speed v0. Then (v ± v ) f = f 0 v (8.30) where the plus sign is for motion towards the source and the minus sign for motion away from the source. (c) Both Source and Observer in Motion (v ± v ) f = f 0 (8.31) (v ∓ vs) (d) If the medium moves with velocity W relative to the ground along the line join- ing source and observer, (v + W ± v ) f = f 0 (8.32) (v + W ∓ vs) Shock waves are emitted when the observer’s velocity or the source velocity exceeds the sound velocity and Doppler’s formulae break down. The wave front assumes the shape of a cone with the moving body at the apex. The surface of the cone makes an angle with the line of flight of the source such that sin θ = v/vs (8.33) The ratio vs/v is called Mach number. An example of shock waves is the wave resulting from a bow boat speeding on water, a second example is a jet-plane or missile moving at the supersonic speed, a third example is the emission of Cerenkov radiation when a charged particle moves through a transparent medium with a speed exceeding that of the phase velocity of light in that medium. Echo is defined as direct reflection of short duration sound from the surface of a large area. If d is the distance of the reflector, V the speed of sound then the time interval between the direct and reflected waves is T = 2d/v (8.34) Reverberation: A sound once produced in a room will get reflected repeatedly from the walls and become so feeble that it will not be heard. The time t taken for the steady intensity level to reach the inaudible level is called the time of reverberation: TR = 0.16V/KS (Sabine law) (8.35) 8.2 Problems 345 where V is in cubic metres and S in square metre for the volume and surface area of the room, respectively, and K is the absorption coefficient of the material of the floor, ceiling, walls, etc. summed over these components. Beats: When two wave trains of slightly different frequencies travel through the same region, a regular swelling and fading of the sound is heard, a phenomenon called beats. At a given point let the displacements produced by the two waves be y = A sin ω1t (8.36) y = A sin ω2t (8.37) By the superposition principle, the resultant displacement is given by y = y1 + y2 =[2A cos 2π( f1 − f2)t/2] sin 2π( f1 + f2)t/2 (8.38) The resulting vibration has a frequency f = ( f1 + f2)/2 (8.39) and an amplitude given by the expression in the square bracket of (8.38).