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Today’s Lecture Waves on a String Wave Power Interference Another Example The figure shows a harmonic wave as a function of x at t=0, and the same wave as a function of t at x =λ/2. Write a mathematical description of the wave including the sign of the velocity. The resulting wave form is: Wave on a String Any way to calculate the wave speed? What is it likely to depend on? Amplitude of the wave? Wave length? Mechanical properties of the string? All of those options are plausible, but it turns out the wave speed only depends on mass of the string (rope) and its tension. Wave on a String Waves on a string resemble harmonic oscillations of a mass on a spring. Tension provides the restoring force, which wants to make the oscillations more frequent. Mass of the string provides the inertia, which slows down oscillations and wave propagation. Wave Equation for a Wave on a String F is the tension in the string x+dx x μ is the mass per unit length Summing forces in the y direction Assume that θ <<1 so that Wave Equation for wave on a string: Speed and Frequency of a Wave on a String In general From the wave equation The frequency of oscillations is: f = v λ = 1 F λ μ Increasing tension by a factor of 2? http://phet.colorado.edu/simulations/sims.php?sim=Wave_on_a_String Speed and Frequency of a Wave on a String A mass m1 is attached to a wire run over a pulley. The wave speed is v = 20m/s. If a second mass, m2, is added to the first the wave speed becomes v = 45m/s. (a) Find the ratio of the second mass to the first. Since v2 = F/μ, we have: (b) Find the frequency if the wavelength is 2.25m. Frequency a Wave on a Guitar String f = v λ = 1 F λ μ Wavelength, λ = 2L, where L is the length of the string. If the string has a mass m, μ = m/L, and we have 1 F 1 F f = = 2 L m / L 2 Lm Thick and heavy strings - long piano strings – low pitch. Tight strings – high pitch. Compare: a mass on a spring. Spring constant (restoring force) F 4L vs. Frequency of a Wave on a Piano Wire A 4.5g piano wire is under 680N tension. Waves propagate at 320m/sec. (a) What is the mass/length of the wire? (b) What is the length of the wire? (c) If the fundamental frequency has a wave length of λ=2L, what is this frequency? A propagating wave communicates motion and carries away energy. We can define wave power as amount of energy carried away from the source (and transported through the medium) per unit time Wave Power Defining u to be the vertical velocity of small region of the string, the rate that work is done by the tension in the string is: For a harmonic wave propagating along a string: Substituting into the equation for the power, P: Wave Power P = FωkA sin (kx − ωt ) 2 2 If we average P over time, at any position, x, we find that < sin (kx − ωt ) > = 1 / 2 2 Therefore the power averaged over time We can plug in 1 2 P = FωkA 2 k = ω / v and F = μv 2 1 2 2 P = μω A v 2 to get Let’s take a closer look at the power equation. What kind of sense does it make? 1 2 2 P = μω A v 2 μ is the mass per unit length; ω is the angular frequency of oscillations in the wave A is the amplitude of the oscillations v is the wave speed ωA is the amplitude of velocity of the oscillations of the material of the string The power is proportional to the mass per unit length, to the square of the amplitude of oscillations of the velocity of string material and to the wave speed. Wave front is a continuous line or a surface connecting nearby wave crests. Plane waves Wave fronts are flat surfaces for well directed light, radar or sound beams, which propagate in one direction without spreading Wave fronts are straight lines for ripples on water surface at shore line. Why do we draw the lines through troughs rather than crests? It does not really matter. Only shape matters. “Straight” waves wave fronts Wave front is a continuous line or a surface connecting nearby wave crests. Wave fronts are spherical surfaces for spherical waves originating from a point source and propagating in 3D space. Wave fronts are circles for waves on water surface originating from a point source. Spherical and circular waves becomes less intense as they travel further away from their source, because the same power emitted by the source is spread over larger area (circumference). Wave intensity is the wave power per unit area P I= A Measured in J W or 2 2 s⋅m m For a plane wave the intensity remains constant. For a spherical wave it decreases with the distance, r, from the source, like P I= 2 4πr Does it mean that the power is lost as the wave propagates? Power P of the source is 1000 watts. The energy spreads outward through the air. At a distance r from the source the area A of the spherical front is 4πr2 Intensity = number of joules per second per square meter = power/area = P/A Example: r = 10 m 1000 Joules per second of sound energy is generated. Spherical wave fronts spread outward from sound source at the center. P 1000 W 2 I= = ≈ 0 . 8 W / m 4πr 2 4π ⋅ 100 m 2 Is this a rock concert or a whisper? From the table we see that .8W/m2 is similar to that for a loud rock band. Is the light coming from the Sun a spherical or a plane wave? At the Earth surface the radius of curvature of the wave front is 150,000,000km, The intensity is the same everywhere, and the wave is practically flat. It is a spherical wave on the scale of the Solar system, though. At the Earth surface the radius of curvature of the wave front is 150,000,000km, The intensity is the same everywhere, and the wave is practically flat. It is a spherical wave on the scale of the Solar system, though. Remember: P I= 4πr 2 A light bulb is a source of a spherical wave. If you are 5ft away from the bulb and walk another 5ft away the light intensity decreases by a factor of 22 = 4. In terms of intensity of sunlight, it is about the same as going from Venus to mars (a) If Intensity (flux) of the energy from the Sun is 1.3kW/m2 at the radius of the Earth, what is the power output of the Sun? At the Earth surface the radius of curvature of the wave front is 150,000,000km, (b) At what distance is the intensity of a 40W bulb the same as that for a 75W bulb at 1.9m? If there are big waves and ripples on the water surface, which are going to run faster? For some kinds of waves the wave speed of a simple harmonic wave depends on the wave length. This phenomenon is called dispersion. Example – for large waves on the surface of deep water λg v= 2π When two (or more) waves of the same kind propagate through the same region produce a composite wave. This phenomenon is called interference. It is constructive, when the waves reinforce each other. It is destructive, when they reduce each other’s amplitude. Usually the disturbances (displacements) the waves produce are added algebraically. This is called superposition principle. Superposition of pulses: Constructive Interference Superposition of pulses: Destructive Interference For a moment it the string becomes a straight line – no disturbance is seen. Does the energy of wave motion disappear? Where does it go? How can the two waves go on after that? Two kinds of energy in the wave motion: potential – depends on deflection of the string from the straight line; kinetic – depends on velocity of motion of the string. In the very moment, when the string becomes straight, it is actually moving very fast - high K. Reflection of a wave on string. Full or partial reflection always occurs at a boundary the wave cannot cross For example, with a medium, which does not support the waves of that kind. Reflection depends a lot on the boundary conditions: a clamped string vs. a freely sliding string. http://phet.colorado.edu/simulations/sims.php?sim=Wave_on_a_String Reflection of a wave on string. A partial reflection always occurs if properties of the medium change abruptly. Abruptly compared to what? Examples: reflection of sound off a rock, light off a glass window… Two connected strings. What is different between them, F? μ? How do the wave speeds in them compare? What about the amplitudes? v= F μ 1 1 2 2 2 2 P = μω A v = Fω μ A 2 2 When there are two interfering waves with close but different frequency the result of the interference are beats. They are perceived as a wave with an average frequency, but with a slowly oscillating amplitude. y1 (t ) = A cos(ω1t ) y2 (t ) = A cos(ω2t ) Resulting composite wave: y (t ) = y1 (t ) + y2 (t ) = A cos(ω1t ) + A cos(ω2t ) After a little trigonometry or making use of Euler’s formula : 1 1 y (t ) = 2 A cos[ (ω1 − ω2 )t ] ⋅ cos[ (ω1 + ω2 )t ] 2 2 When there are two interfering waves with close but different frequency the result of the interference are beats. They are perceived as a wave with an mean frequency, but with a slowly oscillating amplitude. 1 1 y (t ) = 2 A cos[ (ω1 − ω2 )t ] ⋅ cos[ (ω1 + ω2 )t ] 2 2 slowly oscillating amplitude mean frequency When there are two interfering waves with close but different frequency the result of the interference are beats, with beat frequency ω = ω1 –ω2. They are perceived as a wave with an average frequency, but with a slowly oscillating amplitude. http://www.kettering.edu/~drussell/Demos/superposition/superposition.html