Chapter 13 Wave Motion

Various waves Water wave Sound Earthquake Light Radio wave Microwave Vibration Waves

Conditions of mechanical wave Wave motion is a propagation of oscillation Mechanical waves governed by Newton’s laws: 1) A wave source (pulse or oscillation) 2) Medium (in which the wave travels) Wave pattern (not particles) travels in medium 3

Particle motion ⊥ direction of wave traveling Wave types Transverse wave * Particle motion ⊥ direction of wave traveling Longitudinal wave * Particle motion ∥ direction of wave traveling Other waves: Water wave 4

Characteristics of wave motion Source at SHM → medium particles at SHM And the shape of wave is also sinusoidal Amplitude A Wavelength λ Crest A Space Periodicity Frequency f Period T Angular frequency ω Trough Determined by the source Time Periodicity 5

Velocity of wave pattern → phase velocity Wave velocity Velocity of wave pattern → phase velocity It travels one wavelength in one period v is different from the velocity of particles and it depends on the properties of medium From dynamic analysis of the medium, We have 6

Velocity of different waves 1) Transverse Wave on a String: FT: tension in the string μ: mass per unit length 2) Longitudinal Waves in a Solid body E: elastic modulus ρ: mass density 3) Longitudinal Waves in a liquid or air B: bulk modulus ρ: mass density 7

How to obtain a frequency of 2f ? Playing a guitar Example1: A guitar string (L=65cm, m=1.3g) is stretched by a tension FT=80N. Determine: a) the velocity of waves on the string; b) the frequency if the wave length is 2L. Solution: a) b) How to obtain a frequency of 2f ? 8

Communication faster than light? Push a long rod Example2: Transmission of force takes time. If you push on a long rod (L=100m, ρ=1g/cm3, E=5×108 N/m2) , a pulse travels in the rod with wave speed. When does the force come to the other end? Solution: Sound speed in the rod Communication faster than light? 9

Source at SHM, and shape of wave is sinusoidal Plane harmonic wave Wave surface Ray Plane wave Spherical wave Source at SHM, and shape of wave is sinusoidal Medium: uniform, isotropy, no absorption Plane harmonic wave → 1-D wave Represented by a wave traveling along a line 10

Wave motion is a propagation of oscillation Representation of PHW Wave motion is a propagation of oscillation Motional function of any particle: wave function v x: equilibrium position y: displacement v: wave velocity If we know the motional function of point O: How about other points, such as point P? 11

If the wave travels along negative x-direction Wave function v Same A, ω Phase difference Point P: If the wave travels along negative x-direction 12

Equivalent forms “-”: travels in +x “+”: travels in -x 4 key factors: A, ω, k, φ 2 variables: x, t Phase of wave: Wave number Phase velocity 13

Time and space periodicity 1) If x is determined ( x=x0): T It describes the motion of a particle at x=x0 as a function of time 2) If t is determined ( t=t0): It describes the shape of wave at t=t0 14

Traveling wave on string Example3: A wave traveling on a string is shown by y=0.1cos(6x + 80t + π/3) (SI). Determine (a) the wavelength, frequency, wave velocity and amplitude; (b) maximum speed of particles in the string. Solution: a) b) comparing with v 15

Motional function of P: Oscillation and wave Example4: A wave travels along -x with v=10m/s and A=0.04m, point P moves from trough to crest when t=0→2s. a) motion of P; b) wave function. x 5m P v y o Solution: a) Period T = 4s Initial phase of P: Motional function of P: b) Wave function 16

Shapes of wave Example5: A wave traveling toward +x, its shapes at t=0 and t=0.5s are shown in the graph (T>1s). Solve: a) wave function; b) motional function of point P. Solution: a) o 1 2 3 4 x(m) y(m) 0.2 t=0.5 t=0 P A=0.2, =4m T= /v=2s，=  =/2 b) Point P: x=2 17

The wave equation 2-variable function Wave equation Comparing with 18

* Transverse Wave on a String Consider the small amplitude case, so that each particle can be assumed to move only vertically. 19

Energy transported by waves Wave motion is also a propagation of energy. Energy is transferred as vibrational energy from particle to particle in the medium. dx dm= Sdx v S For a SHM: For infinitesimal: Energy transported by wave 20

Average power: (rate of energy transferred) Intensity dx=vdt v S Average power: (rate of energy transferred) Intensity of a wave: average power transferred across unit area ⊥ the direction of energy flow 21

Energy of spherical wave Example6: The intensity of earthquake wave is 2.2106 W/m2 at a distance of 100 km from the source. What was the intensity when it passed a point only 4.0 km from the source? Solution: It is a spherical wave Total power: 22

Principle of superposition In a superposition, the actual displacement is the vector or algebraic sum of the separate displacements. 1) Amplitude not too large Hooke’s law still holds 2) After passing by, continue to move independently. * 3) Composite wave Fourier’ theorem 23

2) same oscillation direction coherent waves Interference 1) same frequency 2) same oscillation direction coherent waves Stable picture that does not vary with time Constructive & destructive interference 24

out of phase, destructive In phase & out of phase where S2 S1 r1 r2 P in phase, constructive out of phase, destructive 25

Interference from two sources Example7: Point P stays at rest in the interference. The motion of S1 is y1s=Acos(2πt+0.5π), what is the motion of S2? (S1P=1.5λ, S2P=1.2λ) Solution: Oscillation from S1 to P: Destructive interference 26

What about area from S1 to S2? Interference on a line Example8: Two coherent waves from S1 and S2 are traveling on a line. Same A, , determine the positions of constructive interference. Solution: Left of S1: constructive Right of S2: destructive What about area from S1 to S2? S1 S2 3/4 a x x c x b standing wave 27

A special case of interference Standing waves A special case of interference same A, but travel in opposite directions Standing wave node antinode A standing wave is not a traveling wave It has a certain wave-pattern, but it does not move (to left or right) 28

Mathematical representation Two traveling waves Superposition: This is the equation of standing wave 1) x and t are separated, not a traveling wave 2) Amplitude varying oscillation 29

3) Nodes: minimum amplitude = 0 Nodes & antinodes 3) Nodes: minimum amplitude = 0 Antinodes: maximum amplitude = 2A Distance between neighboring nodes → λ/2 A method to measure the wavelength 30

cos k x > 0 → phase of particle: Phase and energy 4) Nodes: cos k x = 0 cos k x > 0 → phase of particle: cos k x < 0 → phase of particle: Particles between neighboring nodes: in phase Particles in different sides of a node: out of phase 5) A standing wave does not transfer energy local flow of energy, average rate is 0 31

Solution: a) Wave function of y2 can be written as: Make a standing wave Example9: Waves y1 and y2 make a standing wave, where and there is a node at x=2. Determine: a) wave function of y2 ; b) equation of standing wave; c) position of antinodes. Solution: a) Wave function of y2 can be written as: A node at x=2: 32

b) Equation of standing wave: c) Position of antinodes: maximum amplitude where n = 0, 1, 2, … How about nodes?

Reflection and transmission 1) wave from light section to heavy section x standing wave transmission node reflection phase change of  ( fixed end ) 2) wave from heavy section to light section x antinode no phase change ( free end ) 34

Standing waves on a string On a string: incoming wave + reflection wave Fixed ends → nodes natural frequencies harmonics n=1: fundamental frequency n>1: overtones 35

Solution: Reflection wave Reflect at a free end Example10: A wave y1=Acos(ωt -kx) is traveling on a string, and it is reflected at a free end (x=L). Determine the standing wave on the string. x=L Solution: Reflection wave No phase change at a free end (antinode): 36

Reflect at a fixed end Question: A wave is traveling on a string, and it is reflected at a fix end (x=5.2). Determine the standing wave on the string. x=5.2 37

*Refraction & Diffraction Transmission with an angle → refraction Waves bend around an obstacle → diffraction wavelength ~ size of obstacle 38