Mechanical Waves. 3: Mechanical Waves (Chapter 16) Waves: Space and Time

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1 3: Mechanical Waves (Chapter 6) Phys3, A Dr. Robert MacDonald Mechanical Waves A mechanical wave is a travelling disturbance in a medium (like water, string, earth, Slinky, etc). Move some part of the medium out of equilibrium, and that motion travels (or propagates) from one place in the medium to another. Since it took energy to disturb the medium from equlibrium, that energy propagates with the disturbance. (Different parts of the medium are moving as the wave moves; that s kinetic energy.) Waves on a string are a good basic model to use to explore this. Waves: Space and Time - Snapshot at time t = : m/s t = History of the string at x = 8 m: x = 8 m t (s) Snapshot at t= s of the below wave: t = s History at x = 6 m, for a wave travelling right at m/s: x = 6 m t (s)

2 Snapshot at t=3 s of a wave travelling right at m/s: Delta y - m/s t = 3s History of the string at x = m: - - Snapshot at t= s of the below wave: t = s History at x = 4 m, for a wave travelling left at m/s: t (s) t (s) Equlibrium position Wave pulse Snapshot: Snapshot of a Longitudinal Wave Delta x (cm) Δx is displacement from equlibrium x (cm) Periodic Waves Up to now we ve been playing with pulses. Any wave with a repeating shape is a periodic wave. You can have square waves, triangular waves, sinusoidal waves, etc. Sinusoidal waves are the most common. Also the most important: it turns out you can represent any wave as a combination of sinusoidal waves. (This is called Fourier analysis.) Sinusoidal waves are generated by moving something in simple harmonic motion. 8

3 Snapshot at t= Delta y(x, t=) (mm) v = m/s Snapshot at t= with wavelength λ halved: Delta y(x, t=) (mm) Snapshot at t=t/4 Delta y(x, t=t/4) (mm) Snapshot at t= with frequency f halved but speed unchanged: Delta y(x, t=) (mm) Snapshot at t=t/ Delta y(x, t=t/) (mm) Longitudinal Waves Periodic: Pulse: Speaker Equlibrium position Wave pulse Compression (higher density) Rarefaction (lower density) Compression (higher density) v Rarefaction (lower density) wavelength (λ) Rare (lower Compression (higher density What Waves Are A wave is a disturbance from equlibrium, propagating through some medium (like a string). It takes energy to disturb a particle from equlibrium, so energy must be travelling through the medium causing the disturbance. Waves transport energy, not matter. We ll be using waves on a string as our basic model. Waves on a String Simulation:

4 Particle on a String In a transverse sinusoidal wave, each bit of string is moving vertically in simple harmonic motion. Particles in a fluid move back and forth in SHM as well during a longitudinal sinusoidal wave. Wave Speed By the time a particle has completed one cycle of SHM (e.g. moved down and back up), the wave has moved forward by one wavelength (λ). The time it takes for the particle to do this is one period (T). So, since the wave speed v is constant, v = λ/t or v = λf. 3 Fig Fig. 6.4 Example: Tsunami An earthquake off the coast of Sumatra on 6 December, 6, sent a tsunami smashing into southeast Asia. Satellites measured the wavelength to be about 8 km (!), and a period between waves of one hour. What was the wave speed, in km/h and m/s? Example: Ultrasound Ultrasound sound with frequency too high for humans to hear (above about Hz) is a useful tool for medical imaging. Send the sound waves into the body, and listen to the echoes off of various tissues and bones etc. The speed of sound in body tissue is typically about 5 m/s. To get a clear image, considering the typical size of what the doctors are looking at, the wavelength of the sound should be about mm or less. What is the minimum frequency they should use? 5 6

5 Describing wave shape We ve already described the way a particle in SHM moves over time (history graph!) back in the previous chapter:! y(t) = A cos(ωt + ϕ). Each point on the string (or whatever s waving) is moving up and down with the same period, but with different phase. One point is at a peak, another at a trough, another moving through the equilibrium position, etc. In other words, each point on the string has a different initial phase ϕ, so ϕ depends on x. We can write this as ϕ(x). So we need to translate a position in space (x) into a phase angle so we can use cosine. 7 If x = λ is one wavelength from the origin, then x/λ is the fraction of a wavelength we are away from the origin. There are π radians in one cycle. So πx/λ should give the correct phase! For convenience, define the wavenumber: This is not the spring constant! So in space, then: y(x) = A cos(kx + ϕ) Also useful:! v = λf = π ω, so v = ω/k. k π 9 What about the shape of a wave in space at a given moment (snapshot graph!)? It s still sinusoidal, so we can use cosine (or sine). In time (history graph), we had a phase that looks like ωt + ϕ. A phase like that gave us a nice cosine curve with the right period, and it let us choose t = to be at any point in the cycle. Let s try something similar with position. We can t use ω since the units are wrong (time, not space), so we have to come up with something else... 8 Waves in Time and Space Now we d like to put it all together, to get a wavefunction that describes the wave s behaviour in time and in space. Each bead or bit of string is moving in simple harmonic motion, with a different phase. The phase, in other words, depends on x. So we can describe this by! y(t) = A cos(ωt + ϕ(x)). If you move over a distance x, the phase will be different by kx = πx/λ. Exactly what phase you end up with depends on what phase you had at the origin, just like with time, so we need a phase constant ϕ(x=) = ϕ. Then! ϕ(x) = πx/λ + ϕ.

6 But what about the sign? Should we use y(x, t) = A cos(kx ωt) or y(x, t) = A cos(kx + ωt)? It s not obvious! Consider the first point that starts on axis at t= in the diagram to the right. Its phase at this time is 3π/. Increasing the phase would move this bit of string up at first. But instead, as the wave moves to the right this bit goes down! In order to describe this, we need to use ωt; this way the phase decreases as time increases. So a sinusoidal wave moving to the right can be described by:! y(x, t) = A cos(kx ωt + ϕ) This is called a wavefunction. Fig. 6.4 More on Phase We ve put back the phase constant ϕ:! y(x, t) = A cos(kx ωt + ϕ) ϕ represents the phase at (x, t) = (, ). cos(theta) graph of cos(θ) Phase (theta), radians increasing phase! The value (kx ωt + ϕ) gives the phase at any point in space and time. For a crest (y = +A), the phase can be, π, 4π, π. 4π, etc. For a trough (y = A), the phase can be π, 3π, 5π, π, 3π, etc. v = ω/k (or λf) is often called the phase velocity vp. 3 Going the Other Way So far we ve been describing waves moving to the right (positive x direction). What about waves moving to the left? The behaviour of a bit of string as time proceeds is now just the opposite of what it was before. So we just need to flip the sign on the time-dependent term in our phase i.e. replace ωt with +ωt. So depending on direction, our wavefunction is:! y(x, t) = A cos(kx ωt + ϕ)! y(x, t) = A cos(kx + ωt + ϕ) Sinusoidal wave moving in +x direction. Sinusoidal wave moving in x direction. Example: Birds on a String Two tiny birds are sitting 3. m apart on a long, heavy rope. Some joker comes along and starts shaking one end of the rope in SHM, with a frequency of. Hz and an amplitude of.75 m. The speed of the resulting wave is. m/s. At time t= the person s hand is at the top of its motion (maximum positive displacement). What are the amplitude, angular frequency, period, wavelength, and wave number of the wave? What is the difference in phase between the two birds? 4

7 So now you know: what waves are. what actually travels along a wave (energy!). how a particle moves as a wave passes through. how the frequency, wavelength, and speed of a wave are related. how to describe a wave mathematically. Aside: Partial Derivatives A partial derivative is the type of derivative you need to use when you have a function of more than one variable, such as the wavefunction y(x, t). It works the same as a regular derivative except you replace d with (a sort of curly d ). It s basically a derivative that winks at you ;-) and says, We both know this is a function of x and t, but I ll just pretend it s only a function of one of them for now. You take the derivative with respect to, say, x, treating t as just another constant (or vice versa). 5 6 The Wave Equation Pretty much all waves follow something called the wave equation. (Don t confuse this with the wave function, which is y(x,t) and describes a specific wave!) The wave equation looks like this: It s a relationship between the curvature of the wave (at some location, at some time) and the acceleration of the particle in the wave at that same place and time. (We ll be using it to find v (the wave speed) for some specific types of waves.) 7 (You don t need this on your formula sheet.) Vertical displacement of the string at x: Vertical velocity of the string at x: These are the same SHM equations of motion we already know, but the initial phase (the phase when t=) at position x is kx + ϕ. (Compare with A cos(ωt + ϕ).) We can also study the shape of the string (or whatever s waving) at some specific snapshot in time, by looking at how the displacement y varies with x... 8

8 Slope of the string: The Wave Equation Solve both the particle acceleration and the string curvature for y and equate them: Curvature of the string: Now this is interesting... compare to the accelleration: Remember that wave speed v = ω/k, so: Wave speed, not particle speed! Both are a constant times y. And they re similar constants: (π/λ) vs (π/t). 9 3 The Wave Equation Applying the Wave Equation To get a wave equation for a particular type of wave, you need these ingredients: A restoring force (F net). Newton s second law (F net = ma). Lots of linearization and simplification (calculus!). Our purpose: once we re done, we ll be able to read v (the wave speed) directly from the resulting wave equation. 3

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