Waves. In short: 1. A disturbance or variation which travels through a medium 2. Must transfer energy from one location to another.

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1 Waves What is a wave? A disturbance or variation that transfers energy progressively from point to point in a medium and that may take the form of an elastic deformation or of a variation of pressure, electric or magnetic intensity, electric potential, or temperature. In short: 1. A disturbance or variation which travels through a medium 2. Must transfer energy from one location to another Types of waves: Mechanical waves A wave which needs a medium in order to propagates itself. Sound waves, waves in a Slinky, and water waves are all examples of this. Sound waves need air molecules in order to exist; the Slinky waves need the Slinky, and the waves in the ocean need the water It follows, then, that mechanical waves cannot exist in a vacuum. This is the factor that distinguishes them from electromagnetic waves Electromagnetic Waves Radio signals, light rays, x-rays, and cosmic rays 1

2 Mechanical Waves Two basic types of mechanical waves Longitudinal Waves: In a longitudinal wave the particle displacement is parallel to the direction of wave propagation. The figure below shows a onedimensional longitudinal plane wave propagating through air. The particles do not move down the tube with the wave; they simply oscillate back and forth about their individual equilibrium positions Transverse Waves: In a transverse wave the particle displacement is perpendicular to the direction of wave propagation. The animation below shows a one-dimensional transverse plane wave propagating from left to right. The particles do not move along with the wave; they simply oscillate up and down about their individual equilibrium positions as the wave passes by. Pick a single particle and watch its motion. 2

3 Quantitative description of waves Periodic waves normal understanding of waves - The disturbance repeats over and over - Can be generated by simple harmonic resulting in SHM of the elements of the medium The wave is characterized by Amplitude A: The maximum displacement of the elements of a medium Frequency f: Number of crests or dips that pass by a point per unit of time Phase φ: Where is the wave at time = 0? Propagation speed v From there we can derive: Period T: How long it takes for an entire cycle to pass by T = 1/f Wavelength λ : The distance from crest to crest or the length of one complete cycle v = f λ 3

4 Mathematical description A complete mathematical description will give the displacement at any position x with any instant time t y = Asin( 2πft ± +/- for direction of motion 2πx / λ) substitute: λ = y = Asin(2πft v / f ± 2πxf / v) SinusoidalWavefunction y = Asin(2πf ( t ± x / v)) Examples 4

5 Velocity of the waves Velocity is not arbitrary! Depends on characteristics of the medium through which waves move in case of EM radiation in vacuum it is constant So: The frequency and the wavelength are not independent of each other. Usually the frequency is fixed by the source. Waves on a string (transverse wave) Depends on Tension & mass of string Higher tension (F) leads to a higher velocity wave will travel more rapidly More mass or better mass per length = m/l leads to a higher inertia and to lower velocity Speed of a wave on a string v = F ( m / L) 5

6 Sound (longitudinal wave) A sound wave is similar in nature to a slinky wave for a variety of reasons. First, there is a medium which carries the disturbance from one location to another. Typically, this medium is air; though it could be any material such as water or steel. Speed of sound in Air v 343 m/s 770 mi/h at atmospheric pressure and room temperature Material Speed (m/s) Aluminum Granite Steel Pyrex glass Copper Plastic Fresh water (20 ºC) Fresh water (0 ºC) Hydrogen (0 ºC) 1284 Helium (0 ºC) Air (20 ºC) Air (0 ºC) Frequency human ears can detect/hear are in the range Hz 6

7 Sound Intensity Notice that sound waves carry energy. We define the intensity I as the rate at which energy E flows through a unit area A perpendicular to the direction of travel of the wave. Intensity = Power / Area I = P / A = E / (At) For a point source, energy spreads out in all directions Area of a sphere A = 4 π r 2 Intensity with distance from a point source P I = 2 4πr 7

8 Human Perception of Sound The loudness of a sound depends on its intensity. A sound perceived as roughly twice as loud as another actually has an intensity that is 10 times greater. We measure loudness β by a logarithmic scale of the intensity level of a wave: β = 10 log (I/I 0 ), where I 0 = W/m 2 β is dimensionless but we give it a name, decibels (db). 3dB is a factor 2 change in intensity Every 10dB is a factor 10 change in intensity 20 db is a factor 100 change in intensity 8

9 Doppler Effect Heard an ambulance go by recently? Remember how the siren's pitch changed as the vehicle raced towards, then away from you? First the pitch became higher, then lower. This change in pitch results from a shift in the frequency of the sound waves and is known as the Doppler Effect + means that the observer is moving toward from the source u 1 ± f ' = u 1m observer source v v f u observer/source = speed of observer/source + means that the source is moving away from the observer As a rule: When the motion of detector and source is toward the other, there is an upward shift in the frequency. When the motion of pf detector and source is away from each other there is a downward shift in frequency. Three cases: Moving Observer, Moving Source or both observer and source are in motion 9

10 Moving Observer We know: v = λ f If observer moves towards the source with speed u more compressions move past the observer per time (higher f). f ' = v /λ = (v+u)/ λ = (v+u)/ (v/f) = f (1+u/v) If observer moves away less compressions move past the observer: v = (v - u) and f = f (1 - u/v) Doppler Effect for moving observer: f = (1 ± u/v) f 10

11 Moving Source Towards the observer: λ' = vt ut = (v - u)t Away from the observer: λ' = vt + ut = (v + u)t And therefore with v = λ' f f ' = v/λ = v/(v-u)t = v/(v-u) (1/f) = f (1/(1-u/v)) and f ' = v/λ = v/(v+u)t = v/(v+u) (1/f) = f (1/(1+u/v)) Doppler Effect for moving source 1 f ' = u 1 m source v f 11

12 Doppler-shifted frequency versus speed (400 Hz source) 12

13 Superposition &Interference If two or more waves are moving through a medium, the resultant wave is found by adding together the displacements of the individual waves, point by point. As a result of superposition, waves can interfere. Interference does not mean that waves are destroyed; they will pass through each other. Construcitve Interference Destructive Interference 13

14 Standing Waves A standing wave, also known as a stationary wave, is a wave that remains in a constant position (and therefore v=0). This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions. Wire (Instrument) Transverse wave Fundamental mode (lowest frequency) or First harmonic λ = 2L f 1 = v/λ = v/(2l) Node Anti-node Second harmonic λ = L f 2 = v/λ = v/l = 2 f 1 Ends are tight up so the wave stays in place Third harmonic λ = 3/2 L f 2 = v/λ = v/(2/3 L) = 3 v/(2l)= 3f 1 n th harmonic: λ n = 2L/n and f n = n v/(2l) (n: 1,2,3,..) 14

15 Vibrating Columns of Air (longitudinal waves) a) λ = 4L b) λ = 4/3 L c) λ = 4/5 L Standing wave in a bottle or column closed at one end f 1 = f 2 = Anti-node f 3 = f n = Important Example: Node Human ear canal 15

16 Standing wave in a pipe that is open a) λ = b) λ = c) λ = Standing wave in a bottle or column closed at one end f 1 = f 2 = f 3 = f n = 16

17 Beats Overlapping of waves of same frequency leads to constructive and destructive interference and the amplitude is constant. If the frequency of the waves is slightly different we have the phenomenon of beats. The amplitude is no longer constant and the sound level alternately rises and falls The number of times per second that the loudness rises and falls is the beat frequency and is the difference between the two sound frequencies f beat = f 1 f 2 17