Waves. Wave: A traveling disturbance consisting of coordinated vibrations that transmit energy with no net movement of the matter.

Transcription

1 Waves Wave: A traveling disturbance consisting of coordinated vibrations that transmit energy with no net movement of the matter. Source: some kind of disturbance from the state of equilibrium. Propagation: due to some properties of the medium such as elasticity.

2 Quick movement of hand up & down What happens? Flick of hand produces a disturbance. This disturbance travels down the rope. What happens if we continue to move hand up & down? Disturbances travel in form of vibrations But no matter is moved along the rope, just energy

3 Examples of waves: Mechanical waves Ripples on water, surface waves on water Vibrating strings Seismic waves (during earthquakes) Sound Electromagnetic waves Radio waves, Microwave, Infrared, Light, UV, X, γ And there are more to come

4 Transverse and longitudinal waves TRANSVERSE WAVE LONGITUDINAL WAVE Transverse: disturbances are perpendicular to propagation Longitudinal: disturbances are parallel to propagation

5 Some properties of waves: QUESTION: I generate waves on a rope. As I increase the displacement of my hand 1. The wave will travel faster. 2. There will be no difference. 3. The wave will be bigger. The disturbance will be bigger: The AMPLITUDE will be larger. Amplitude: The maximum displacement of points in a wave. Measured from zero (equilibrium position). Amplitude depends on the source, but may change in the medium (damping).

6 QUESTION: If I swing my hand up & down faster 1. The wave will travel faster. 2. The wave will have larger amplitude. 3. None of the above. Answer 3: The speed of the wave is unaffected by how fast I wiggle my hand up & down. What is affected is the FREQUENCY. The number of oscillations per second in a wave at any point along the wave. Frequency is a property of the SOURCE only.

7 Speed of the wave speed of the wave propagation Speed of the wave is a property of the medium. In mechanical waves it depends on the elasticity of the medium. 1. Waves in the string: v = F, ρ where ρ = m l linear mass density 2. Waves in the air (sound). v = p D, where p - pressure, D - density for air v = 201. T, wheret is in o K 3. Seismic waves: Elastic property 4. In general: Inertial property v v = = B D, where B - bulk modulus, D - density

8 Example: If the string is 1 m long and its mass is 5g, The tension is 100 N. Find the speed of wave propagation: v = F ρ, ρ = m l = 0.005kg 1m = kg m v = = m s

9 How long does it take for each point along the wave wiggle up & down once? PERIOD How far has the wave moved in one PERIOD? ANSWER: One Wavelength

10 WAVELENGTH, λ: The distance between two equivalent points in a wave, e.g. the distance between two adjacent peaks or two adjacent valleys.

11 Amplitude and Wavelength: Which wave has the largest amplitude? Which wave has the largest wavelength?

12 QUESTION: I increase the frequency of a wave traveling on a rope (keeping everything else unchanged). What happened to the wavelength and the speed of the wave? 1. The wavelength increased, but the speed stayed the same. 2. The wavelength stayed the same, but the speed increased. 3. The wavelength decreased, but the speed stayed the same. 4. Both the wavelength and the speed stayed the same.

13 So, if the wave travels one wavelength, λ, in a time equal to the period, T, how fast is it going? v = d/t = λ/t Wavelength is a function of speed and frequency

14 Frequency = f = 1/T Relationship between frequency, wavelength and the speed of the wave: Speed of a wave: v = f λ The speed of a wave depends on things like forces, mass per unit length etc., but not typically on the frequency. If the frequency changes, the wavelength must change to keep the velocity the same.

15 Example: A wave on a rope travels at 2 m/s. I wiggle one end of the rope at a frequency of 1 Hz. What is the wavelength of the resulting wave? 1. 2 m v = f λ λ = v/f

16 Wave Propagation: How waves move REFLECTION DIFFRACTION INTERFERENCE

17 Waves are reflected when they encounter a sudden change in the medium through which they travel. Examples: Mirror Satellite dish Echo

18 What happens when two waves meet?

19 INTERFERENCE! Constructive Interference Destructive Interference destructive constructive

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22 DIFFRACTION An opening in a room acts as a point source sending waves in all directions causing the wave to bend.

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24 The Doppler effect: dependence of the wavelength on the motion of the source relative to the observer

25 Finalize general properties of waves: 1. Main characteristics of waves: f, v, λ, A, T. 2. Frequency f is a property of a source. 3. Wave speed is a property of the medium. 4. Every point perturbed by the wave becomes a source for further wave 5. All perturbations that we discuss are linear. Therefore, we have the superposition principle. Wave properties or features as interference are the consequence of this principle.

26 Sound SOUND is a (longitudinal) pressure wave in air, any gas, liquid, or solid. The amplitude of sound waves are very small compared to the atmospheric pressure: P/P ~

27 SPEED OF SOUND: Depends on elasticity (forces between atoms or molecules) and density For ideal gases depends on the mass of atom or molecule and temperature. Speed of sound in air is about = 20.1 In air at room temperature ~ 343 m/s (768 mph) T Example: If the time between the lightning and a thunder is 3 s, how far is the T-storm? The air temperature 75 o F. 1. Convert to Kelvins: T 2. Calculate the speed of 3. Calculate the distance : d = sound : v = = v t 5 9 (75 32) = = 297 o = ,040m K m s

28 Waveform = graph of the pressure variations in air versus time Pure tone: sound with a sinusoidal waveform, single λ. Complex tone: repeats itself (periodic), but multiple λ s a superposition of pure tones Noise irregular

29 0.025 sec What is the frequency of this sound wave? 1. It cannot be determined Hz Hz Hz Hz What is the wavelength of this sound wave (T=293 o K)? v = 20.1 λ = v f = = m s = 17.2m Period T = 0.05 sec f = 1/T = 20 Hz

30 Generation of a complex wave using 4 harmonics

31 Production of sound: Frequency Amplitude Complexity (spectrum) Perception of sound: Pitch Loudness Tone quality How do these perceptions correspond to physical attributes of sound?

32 How is sound produced? Any object vibrating in air will produce pressure waves: Vibrating strings, plates, membrane Standing waves in wind instruments Frequency depends on length of string or pipe etc. = resonance frequency. Higher modes can be generated by blowing harder.

33 Production of the sound: vibration + amplification A string fixed on both ends can only have such and higher harmonics, the rest are damped. First harmonics fundamental λ = L/2, v is determined by the string and its tension, f = v/λ , 2 3, 2 2, , 3 2,, L v f L v f L v f L v f v f L L L L F v = = = = = = = = = = λ λ λ λ λ ρ Guitar: 1. Different strings different ρ s. 2. Changing tension tune v, and hence f 1

34 Amplification : resonator boxes Select (filter) and amplify certain harmonics to produce a complex (or pure) sound.

35 Resonance in open pipes: Both ends open One end open: organ and wind instruments

36 How do we hear? Nerve sends electrical signals to brain for further analysis Frequency analyzes sound and converts into electrical signals Sound excites eardrum These guys amplify the sound mechanically by about 20x.

37 Loudness: Which physical quantity is responsible for how loud a sound is? 1. Frequency 2. Wavelength 3. Amplitude 4. Speed 5. Waveform Loudness is mostly determined by the amplitude of the sound wave.

38 Sound level: Measured in decibel (db) If amplitude is 10x bigger, the decibel level is increased by 20 db. 90 db sound has 10x the amplitude of 70 db sound.

39 Loudness = Perception of loudness 0 db threshold of hearing 1 db minimal loudness change detectable by ears 10 db difference = sounds twice as loud 120 db - threshold of pain If you add two sounds of the same loudness, the resultant loudness increases by 3 db (sound of 50 db) + (sound of 50 db) = (sound of 53 db) Loudness also depends on frequency. Ears are most sensitive around Hz.

40 Sound Propagation Sound becomes weaker further away from a sound source. By how much? Energy is conserved, but sound is spreading over an area ~ d 2. Energy ~ Amplitude 2 Energy per unit area ~ 1/d 2 Therefore: Amplitude ~ 1/d

41 PITCH How high a low a sound is Which physical quantity is pitch due to? 1. Amplitude 2. Frequency 3. Speed of sound 4. Shape of waveform

42 PITCH: A sound with a higher frequency will sound higher, a sound with a lower frequency will sound lower.

43 Why do CD players have a range of 20 20,000 Hz, when musical instruments range from Hz only? Range of human hearing: Maximum 20 20,000 Hz Tone quality is determined by the actual waveform of the sound. For accurate representation of the sound higher frequencies are needed. High fidelity

44 Sounds of instruments etc. are due to Complex waveforms = combination of waves of several frequencies. Lowest frequency = Fundamental, determines pitch Higher frequencies = Harmonics, determine tone quality. Frequencies of harmonics are whole number multiples of the fundamental frequency.

45 =tone quality

46 QUESTION: You are a designer for a new concert hall. Which wave effects do you have to take into account? 1. Reflection 2. Diffraction 3. Interference 4. All of the above 5. None of the above 6. These things have nothing to do with sound.

47 Answer: All of the above Example: Reflection Which path takes the most time? 1. Path 1 2. Path 2 3. Path 3 4. Path D 5. They all take the same amount of time. The goal: deliver the same complex sound to all places

48 Multiple Reflection lead to reverberation. Reverberation good and bad: Amplifies the sound Makes it more uniform Can make echo or multiple sounds