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Unit 1 WAVES Contents WAVES ............................................................................................... 3 Definitions......................................................................................... 3 Speed of waves in various media:..................................................... 9 Reflection, transmission and absorption ......................................... 12 Energy (power) carried by waves ................................................... 15 Interference of waves ...................................................................... 16 Standing waves................................................................................ 21 Resonance ....................................................................................... 22 Unit 1 Waves 2 WAVES Why study waves? • Everything around us can be described in terms of particles and waves! We need to learn how to deal with waves. • Examples of waves: sound, light, water waves, earthquakes, ‘Mexican’ wave, etc. • Later you will learn about other type of waves, in which “fields” do the “waving”. What is common in all these different waves? What is common in sound, light, water waves, earthquake, etc? Answer: The math! The same type of equations can describe all these waves. • The math in all cases describes a disturbance of the given medium that propagates from one region to another. • While the disturbance propagates, the material stays near its equilibrium position. What propagates with the wave is energy and momentum. Definitions • Wave pulse: eg transverse wave-pulse set up in a taut rope • It is possible to generate a train of wave pulses: eg hammering, periodically throwing pebbles into water, etc • In many cases, the train of wave pulses is extremely long, with large number of repetitions. In fact, in most of our calculations we shall assume that the wave train is infinitely long. We’ll say it is perfectly periodic. • Transverse and longitudinal waves. Unit 1 Waves 3 Examples: http://www.physics.nwu.edu/ugrad/vpl/waves/wavetypes.html (local copy) http://members.aol.com/nicholashl/waves/movingwaves.html (local copy) http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html (local copy) Other assumptions: • We’ll only deal with one-dimensional waves (waves that travel along only one axis) • The function describing the wave depends on the coordinate (say x) and time (t): f(x,t) (Often the Greek letter R(x,t) is used to describe the wave). • Waves that do not change their shape as they travel What do we know about the function, f(x,t), that describes the wave? • It is a periodic function of space: f(x,t) = f(x+8, t) where 8 is called the “wavelength”. X Unit 1 Waves 4 Later on we shall often use the term “2B/8”. Let’s define a term: k = 2B/8 for shorthand. We’ll call this the angular wave number. Its units are [k] = rad/m • f(x,t) is also a periodic function of time: f(x,t) = f(x, t+T) where T is called the “period” of the wave; units of period: [T] = second • In the math that follows, we shall often use the term ‘1/T’. We define a new “shorthand’ term that we call the frequency, f: f = 1/T Units: [f] = 1/s = Hz (Hertz). (Sometimes, instead of ‘f’ we use the Greek letter ‘ <‘ for frequency.) • We also define a term, which is analogous to frequency, but measures the angular change per time. We call this the angular frequency: T (omega) = 2Bf Units [T] = radians/sec • In one period, the wave travels one wavelength, so the speed of the wave can be defined as: v = 8/T which is the same as v = 8f = (8/ 2B) (2B f) = T /k Units of speed [v] =m/s. • The wave speed … speed of particles in the wave! What about the amplitude of a wave? • It depends on the type of wave we are talking about. • In sound waves it can be the pressure, in light waves it is the magnitude of the electric field, in some quantum phenomena it can Unit 1 Waves 5 be the probability amplitude. So for each type of wave the amplitude of the wave is different. What about f(x,t)? What kind of function can describe a wave? • ‘Looks like’ sin(x,t) or cos(x,t). Does that make sense? • Yes, cos or sin functions can describe waves. • More precise answer: Is it always OK to use harmonic functions to describe waves because (according to the Fourier theorem) any periodic function can be written as the sum of harmonic functions: f(x) = 'ansin(knx) For example: http://www.phy.ntnu.edu.tw/java/sound/sound.html (no local copy) From what we have learned so far, we can write the equation of a wave as: f(x,t) = Asin(ax+bt) where ‘x’ is distance, ‘t’ is time, ‘A’ is the amplitude of the wave, and ‘a’ and ‘b’ are yet to be determined. What do ‘a’ and ‘b’ represent? • Assume that t = 0 (ie. we take a snapshot of the wave at t=0): f(x,0) = f(x+8,0) Asin(ax,0) = Asin(ax+a8,0) ax = ax + a8 "2B ˆ a = 2B/8 Therefore a = k = 2B/8 angular wave number. • Assume now that at x = 0: f(0,t) = f(0,t+T) Unit 1 Waves 6 Asin(0,bt) = Asin(0, bt+bT) bt = bt + bT "2B ˆb = 2B/T Therefore b = T = 2Bf angular frequency. Now we are ready to specify the general equation of a wave: f(x,t) = Asin(kx + Tt + N) where A = amplitude, (kx + wt + M) is called the phase of the wave, and M is the phase constant. Other forms of the same wave: f(x,t) = Asin(2Bx/8 +2Bf t+ N) f(x,t) = Asin2B(x/8 +f t+ N) f(x,t) = Asin2B/8(x +2Bv t+ N) The next question we have to answer is: If the above equations describe a wave that is moving along the x-axis, how do we know from these equations which direction the wave is travelling? The answer is simple: from the sign of the ‘Tt’ term. Let’s see why: • We know that the actual shape of the wave does not change as the wave moves. That means that the phase of the wave is constant: kx - Tt = constant Unit 1 Waves 7 which means that the phase at a given time and given position is the same as the phase at a later time and position: kx’- Tt’ = kx - Tt Let’s denote x’ = x + )x and t’ = t+)t. If the wave is moving towards the positive x-direction, )x is positive. Thus for the phase to stay constant, the sign of the ‘Tt’ has to be negative. That is, a wave travelling toward larger x values (positive x) is describe by the function f(x,t) = Asin(kx - Tt) If we change the sign in front of the ‘Tt’ term, the function will describe a wave travelling in the negative x direction: f(x,t) = Asin(kx + Tt) (Note the ‘+’ sign in front of the Tt term.) Unit 1 Waves 8 Speed of waves in various media: Waves on a string: Waves travel with different speeds in different media. Let’s calculate the speed of travelling waves on a tightly stretched string, and learn some general concepts from it. a. Assume that the wave moves with a constant speed v to the right b. The string under constant tension Ftension c. The mass density of the string is : kg/m d. The amplitude of the wave is small e. We’ll consider the string in the wave frame of reference; that is we are travelling along with the wave. If the wave is moving to the right, from the wave frame of reference, the string will move from left to right: R A B (textbook p:454) Unit 1 Waves 9 • We’ll look at a small segment of the string AB that makes an angle )2 . o We shall approximate this segment as an arc of circle of radius R. o The length of the segment is R)2 o The mass of this segment is given by the mass density, :, times the length of the segment: Mab = R)2: o Looking at the string from the frame of reference of the wave, the tiny segment of the string is essentially revolving around point C with speed v o Since it is moving at a constant speed around a radius ‘R’, the net force acting on this segment is equal to the centripetal force acting on the string, which must come from the y component of the tension in the string: 2(Ftension)y = Mabv2/R o For small angles, sin2 . 2, hence: 2Ftsin(½ )2) = Ft)2 = Mabv2/R = (R)2:)v2/R = )2:v2 o Rearranging this we get: Ft V = µ Note: • Speed depends on the tension, Ft, in the string, and the mass density (mass per unit length) of the string. • Speed does not depend on the frequency of the wave How about the speed of waves in other media? Things like sound waves in air or a fluid? Unit 1 Waves 10 Speed of compression waves Compression waves: imagine a number of ping-pong balls attached by small springs all placed on a table: If we start the first ball on the left to vibrate in simple harmonic motion, the other balls will follow suit. Propagation of compression waves occurs along the direction in which the balls oscillate, and is characterised by a series of alternating high and low density of balls. This models is OK for atoms in an elastics substance and/or a liquid. See for example: http://www.physics.nwu.edu/ugrad/vpl/waves/wavetypes.html (local copy) It can be shown (see textbook p:459) that the speed of sound for compression waves in liquids is: B V = ρ where B is the Bulk Modulus defined as B = -)P/()V/V) and D is the density of the liquid. This expression may look different from the one we derived for a wave on a string, but it is not. If you look at the units B (kg/m.s2) and that of D (kg/m3) , you find that it is the same as that for the speed of a wave on a string, that is (force/density)1/2 . We’ll look at example, when we talk about sound waves in a later unit. Unit 1 Waves 11 Reflection, transmission and absorption When waves arrive at the interface between two media, they undergo reflection, transmission and/or absorption. For example, when light waves arrive from air to a window, part of the light wave will be reflected, part will be transmitted, and a (small) part will be absorbed. Let’s try to model these phenomena with mechanical waves, eg. a wave-pulse travelling on a rope (pictures from textbook pp: 456-457): Unit 1 Waves 12 (see for example, http://www.kettering.edu/~drussell/Demos/reflect/reflect.html Unit 1 Waves 13 Unit 1 Waves 14 Energy (power) carried by waves Question: do waves carry energy? Light waves do (eg. solar cells), sound waves do (our eardrums move), etc. How about waves running on a string? Let’s shake a piece of string up and down. How much energy travels in the string to the other end? Let’s calculate it: As we move our hands up and down at a point ‘x’ with force Fy in the y- direction, we do work. The power involved is: P = Fy vy where ‘vy’ is the speed of the string in the y-direction, and Fy is the force in the y-direction: Fy = Fsin2 = F2 (for small angles) Since we know the function that describes the wave, we can calculate the speed,vy, and the angle, 2, at any point on the string: y(x,t) = Acos(kx-Tt) vy = dy/dt = ATsin(kx-Tt) 2 . tan2 = dy/dx = -k Asin(kx-Tt) So the power travelling in the string is P = Fy vy = F2 vy = F (k Asin(kx-Tt)) (ATsin(kx-Tt)) = A2kFTsin 2(kx-Tt) Using v = T/k and F = :v2 we find that the power flowing past the point ‘x’ at a given time, t, is: P = A2:v T2 sin 2(kx-Tt) This is the instantaneous power. We are often more interested in the average power. So we calculate the power in one cycle (integrate over one cycle): Paverage = ½ A2:v T2 Unit 1 Waves 15 This is an important result: the power carried by a wave is proportional to the amplitude squared! This is true for other type of waves, not only waves on a string. This is something we’ll discuss when we start with quantum physics. Interference of waves Question: What happens when two or more waves pass through the same region at the same time? In the case of particles, they “see” each other and scatter (eg. billiard balls). Do waves behave similarly? Do they change each other’s path? Answer: Waves do not interact with each other! When two or more waves overlap, each wave “does it’s own thing”. The resultant wave is the algebraic sum of the primary waves. This is called the Superposition Principle. (textbook p: 465) Unit 1 Waves 16 Let’s look at specific examples: a. Two waves: y1 and y2 have identical amplitude, frequency and wavelength but different phase constant: y1 = Acos(kx+Tt) and y2 = Acos(kx+Tt +N) The new wave will be the algebraic sum of these two: yT = y1 + y2 = Acos(kx+Tt) + Acos(kx+Tt +N) Using cos" + cos $ = 2cos [("+$)/2] cos[("-$)/2] we get: yT = 2Acos(kx+Tt+N/2)cos(N/2) = A*cos(kx+Tt+N/2) where A* = 2A cos(N/2) What this means: The new wave, yT, has the same frequency and wavelength as the original waves, but the amplitude of the new wave, A*, depends on the phase constant difference of the original two waves, N . For example: if N = 0 A* = 2A if N = B A* = 0 In general when 0 # N # B , the new amplitude is 2A # A* # 0 We call this phenomenon wave interference, or just interference. When A* < A => destructive interference When A* > A => constructive interference See, for example http://www.gmi.edu/~drussell/Demos/superposition/superposition.html (local copy) Unit 1 Waves 17 (textbook :464) (b) Now let’s add two waves that have the same amplitude, phase constant but slightly different frequency: yT = y1 + y2 = Acos(kx + T1t) + Acos(kx + T2t) where: T2= T1 + )T and )Tis very small. yT = Acos(kx + T1t) + Acos(kx + (T1 + )T) t) To make things simpler, let’s assume x = 0: yT = A[cos(T1t) + cos(T1 + )T) t] Using (again) Unit 1 Waves 18 cos" + cos $ = 2cos [("+$)/2] cos [("-$)/2] , we find: yT = 2Acos[(T1 + T2)t/2]cos()Tt/2) which can be written as yT = A*cos(ft) where f is the average frequency: f = (T1 + T2)/2 What this means is that by adding two waves of slightly different frequencies, we get: (i) a new wave whose frequency is the average frequency of the starting two waves (ii) the new amplitude, A*, that varies in time We call this the beating of two waves. Easy to observe for sound, but it is also true for all type of waves. The beat frequency is defined as: fbeat = f1 -f 2 (see for example, http://webphysics.ph.msstate.edu/jc/library/15-11/index.html (local copy) or http://www.gmi.edu/~drussell/Demos/superposition/superposition.html (local copy)) (textbook p:465) (c) Two identical waves travelling in the opposite direction: Unit 1 Waves 19 y1 = Asin(kx - Tt) y2 = Asin(kx + Tt) The sum of the two waves results in yT = y1 + y2 = A[sin(kx-Tt) + sin(kx+Tt)]= yT = 2A sin(kx)cos(Tt) Please note that since wave phase is not (kx " Tt) type, this is not a travelling wave. If we plot this wave: This is not a travelling wave, it is a standing wave. Unit 1 Waves 20 Standing waves Definitions: • Nodes: positions on a standing wave where the amplitude is always 0. Nodes occur when y(x) = Asin(kx) = 0 ˆ kx=nB where n =0,1,2, ... That is 2Bx/8 = nB which means nodes occur at positions: xn = n8/2 • Anti-Nodes: positions where the amplitude is maximum. Anti- nodes occur when sin(kx) = 1 ˆ kx=(n+1/2)B 2Bx/8 = (n+1/2)B That is, anti-nodes occur when: xn = (n+1/2)8/2 = (2n +1) 8/4 http://www.gmi.edu/~drussell/Demos/superposition/superposition.html (local copy) Unit 1 Waves 21 Resonance Assume a string with both ends fixed. If we generate waves on this string, waves will bounce back and forth, thus very soon we’ll get standing waves emerging. Since both ends are fixed, the end points are nodes. Only certain waves satisfy this ‘boundary condition’. The wavelengths of the waves that can exist on a string with both ends tied down are: 8n = 2L/n where n = 1,2,3,...and L is the length of the string (textbook p:480) Unit 1 Waves 22 The frequencies is these waves are: fn = v/8n = nv/2L where n is an integer. Since we know how to calculate the speed of a wave on a string, the frequencies can be also written as: n Ftension fn = 2L µ The lowest frequency (longest wavelength) is called the fundamental mode or first harmonic. The next allowed frequency is the second harmonic, followed by the 3rd harmonic, etc Summary: • Standing waves with boundary conditions result in discrete frequencies! (This is true for all type of waves, not only waves on a string) • Discrete frequencies => frequencies ‘quantized’ Question: What happens if we try to generate all type of waves on a string with 2 ends fixed? Answer: We find that only the ‘allowed’ frequencies will be excited. The amplitude of the other vibrations will stay very low: Amplitude fn fn+1 fn+2 frequency These ‘special’ frequencies are called resonant frequencies. Unit 1 Waves 23