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Wave Equations and Functions Transverse Waves Summarized Longitudinal Waves (eg. stretched string) (eg. vibrating rod) 2 y 1 2 y 2 1 2 x 2 v 2 t 2 x 2 v 2 t 2 T Y v wave speed v normal mode normal mode solutions solutions (standing (standing waves) waves) y x, t f x cos t general x, t f x cost solution Boundary both ends fixed Boundary one end fixed Conditions: Conditions: one end free n x n x y x, t An sin cos n t x, t An sin cos nt v v where 1 1 n T 2 n 1 Y 2n 11 2 n n1 n 2 L L Also (in Hz) 1 1 fn n T 2 fn n 1 Y 2 2 1 2 2 L 2L Week 8 Lecture 1: Problems 47, 48, F2000Q3a Travelling wave solutions to the wave equation Recall the Plan….. Earlier we derived the wave equation for both longitudinal and transverse waves. There are lots of variations on this but here is an example: y y 2 2 x 2 T t 2 Remember that this describes the general behaviour for a wave on a string. However, if we specifically want to know where (in y) some point x is along a string at a given time t, we have to get a solution to the wave equation We said there were two types of solutions Normal modes Travelling waves aka aka (stationary waves) Progressive waves (standing waves) We have been working Now need to look on these for the last few at this type of 2 lectures solution Traveling Wave (Progressive Wave) Solutions to the Wave Equation (French pg 202-209) So far we have considered only the stationary wave (normal mode) solution to the wave equation. Now we will look at the traveling wave solution. In fact – a stationary wave is just a traveling wave which gets reflected from one end of a medium. The incident and reflected waves superimpose, and if conditions are right (i.e. appropriate frequencies and wavelengths then a standing wave ( stationary wave or normal mode) develops. Traveling Wave Solution to the Wave Equation y y0 sinkx t For a transverse wave Note: 1. There is no phase constant in this equation as written (but you could add one). 2. k is not the spring constant but the wave number (rad/m). 3. Since a stationary vibration is actually created from a travelling wave, you should be able to show that these 2 solutions are the same and you can (see French pg 203-204). 3 What exactly is the Wave Number? “k” in a y vs x plot is the same as “” in a y vs t plot! Explanation: If you plot the y displacement of a single point x on a string as a function of time you get: y T = time for 1 complete oscillation on a y vs t plot t and T = angular frequency = 2/T (rad/s) Now, if we look at a single snapshot in time you are essentially plotting y as a function of x: y = wavelength = distance for 1 complete oscillation on a y vs x plot x k = wave number = 2/ (rad/m) 4 Note here: on pg 214 French the wave number is defined as k = 1/, but then French corrects for the missing 2 by multiplying it through later. We will use k = 2/ but be aware of the little catch (note: most other books use 2/) 2 2 so and k T traveling wave solution is y y0 sinkx t subbing in and k gives x t y y0 sin 2 T what fraction of a what fraction of wavelength is this position? a period has passed? 5 Wave Speed relationships The wave speed is the speed at which the wave travels along the medium (in m/s). put the traveling wave solution back into the wave equation into 2 y 2 y y y0 sin kx t v2 2 t 2 x gives y 0 2 sinkx t v 2 k 2 y 0 sinkx t v k 2 2 2 or v k units are = rad/s and k = rad/m So there IS a relationship between and k – they are related by the wave velocity 6 How everything fits in… 22 v , k , k T 2 f f is v T 2 T frequency in Hz Keep handy - will use this v f over and over k T 7 One more thing to clean up… Recall that earlier we stated that We stated this when T we wrote down the Wave speed = v wave equation but we didn’t derive it Mass/length Can we derive this? (of course, or I wouldn’t be asking..) Consider the following travelling wave pulse on a rope: If the pulse is small enough then we can assume the tension T in the rope doesn’t change, and we can consider Speed v the wave a circular arc. s R Sum forces (tension): Horizontal: SFh=0 s Vertical (also the net radial force): ½ T T SFv=2Tsin½ 2T ½ T R Mass of segment m = s=R Apply Newton’s 2nd law (radial) v2 T R R T v 8 Voilla!! Remember this slide?? General Wave Equation (1D): 2 y 2 y v2 2 t 2 x where v is the wave speed Can write this for all sorts of situations! stretched string transverse displacement (transverse 2 y T 2 y T tension v oscillations): t 2 x 2 mass/ length longitudinal displacement longitudinal waves Y 2 2 Y Young’s v modulus in a solid: t 2 x 2 density transverse (shear) 2 y Y 2 y n shear v modulus t 2 x 2 waves in a solid: 2 B 2 B bulk longitudinal waves v modulus t 2 x 2 in a gas or liquid: The velocity terms are here. We derived the top one and could9 derive the others if we wanted…. Naw… Particle Velocity and Pressure So far the only parameter we have looked at is particle displacement y y0 sinkx t (transverse wave) 0 sin kx t (longitudinal wave) However, other parameters such as particle velocity and pressure (stress) are also important. Considering longitudinal waves in a thin rod (so Y is the appropriate modulus) (you could also do for transverse) particle displacement: 0 sin kx t particle velocity: 0 coskx t (NOT same as wave speed) t pressure: strain 0 k coskx t x stress Y Y0 k coskx t x pressure P compressiv stress e Y0 k coskx t 10 OR if you have a gas modulus B: P B0 k coskx t Note this is analogous to Specific Acoustic Impedance refractive index in (Courseware pg 72-73) E/M waves Very important property of materials that are transmitting sound waves (but “impedance” generally is applicable to any wave in any medium). pressure P Defined as z particle velocity (Effectively a resistance term – high zs mean that it takes a large pressure to induce a given particle velocity high “stiffness”). (since Y is a Y0 k coskx t Yk plug in equations for pressure and stiffness this z 0 coskx t fits!) Now we really need z in a form that we can associate with material properties, so recall some previous relationships… then z Y since v k v v Y v 2 also or Y v z 2 v v z v units kg m 3 kg 2 m s m s So materials with higher zs (stiffer) have higher wave velocities. P So particle speed also note z v is inversely proportional to wave speed v 11 See table of z and v values 2 pages along! Traveling Waves: Summary Page Wave Velocities T Y transverse v longitudinal oscillation v oscillation (string) (solid rod) transverse n longitudinal oscillation B v v oscillation (solid) (gas or liquid) 2 2 wave number k T All of these v f equations appear on the formula k T sheet of my exams Longitudinal Wave: (solid) particle displacement: 0 sinkx t But Not these particle velocity: 0 coskx t pressure: P Y Y0k coskx t x Specific Acoustic Impedance: P Yk Y z v v Intensity: 2 Note we haven’t I zm 2 m 2 1 1P watts done intensity yet – 2 2 z m next lecture I decibels dB 10 log10 12 dB 12 10 Calculated from long. velocity 13

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