Schedule for the remainder of class

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Byron Wilkinson
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1 Schedule for the remainder of class 04/25 (today): Regular class - Sound and the Doppler Effect 04/27: Cover any remaining new material, then Problem Solving/Review (ALL chapters) 04/29: Problem Solving/Review (ALL chapters) - highly relevant for final exam! 04/26 and 04/28 (3:30 pm - 5 pm): OPTIONAL recitations in White /05 (7 pm - 9 pm): Final exam, room TBD Students who need a makeup final exam MUST contact me ASAP.

2 Recap Hooke s Law: F s = kx In SHM the acceleration is not constant: a = k m x Spring potential energy: PE s = 1 2 kx2 Period of SHM: T =2 r m k Position, velocity, acceleration as a function of time: x = A cos (2 ft) v = A! sin (2 ft) a = A! 2 cos (2 ft) s L Period of a pendulum: T =2 g

3 What is a wave? Until now we discussed oscillations, i.e. simple harmonic motion around a static equilibrium point. Waves are moving oscillations, i.e. the equilibrium point moves and is no longer static. Examples for important wave phenomena: Sound waves Light waves Seismic waves radio waves water waves A mechanical wave requires: (i) A source of disturbance (ii) A medium that can be disturbed The medium itself does not move - the disturbance moves.

4 Different types of waves Transverse waves: Each element that is disturbed moves in a direction perpendicular to the wave motion. Example: Wave on a rope. Longitudinal waves: The elements of the medium undergo displacements parallel to the motion of the wave. Example: Sound (density wave)

5 Wave motion The brown curve is a snapshot of the wave at some instant in time The blue curve is later in time The high points are crests of the wave The low points are troughs of the wave

6 Wavelength and wave speed The distance between two successive points that behave identically is called the wavelength, λ. A wave advances a distance of one wavelength during one period: v = x t = T = f The speed of a wave propagating on a string stretched under some tension, F is: Here, µ is called the linear density. The speed depends only upon the properties of the medium through which the disturbance travels

7 Example problem: Radio Waves A certain FM radio station broadcasts music at a frequency of MHz. Find the wave s period and its wavelength. (Radio waves are electromagnetic waves that travel at the speed of light, 3.00x10 8 m/s).

9 Interference of Waves Two traveling waves can meet and pass through each other without being destroyed or even altered. Waves obey the Superposition Principle: When two or more traveling waves encounter each other while moving through a medium, the resulting wave is found by adding together the displacements of the individual waves point by point. Actually only true for waves with small amplitudes The plot shows two waves of identical frequency and amplitudes traveling through the same region. If crest meets crest and trough meets trough, there is constructive superposition (the waves are in phase) and the resulting wave has twice the amplitude.

10 Constructive Interference

11 Destructive Interference If crest meets trough, there will be destructive interference, i.e. the amplitude of the superposition is lower.

12 Wave reflection - fixed end Whenever a traveling wave reaches a boundary, some or all of the wave is reflected When it is reflected from a fixed end, the wave is inverted, but the shape remains the same This is caused by Newton s third law (action = - reaction), i.e. the fixed end exerts a force in opposite direction on the rope.

13 Wave reflection - free end Whenever a traveling wave reaches a boundary, some or all of the wave is reflected. When it is reflected from a free end, the wave is not inverted.

14 Production of Sound Waves A sound wave is a longitudinal wave, i.e. the elements of the medium undergo displacements parallel to the motion of the wave (compression wave). The medium, that is being compressed, is air. A sound wave can be represented by a sine curve illustrating the periodic density variations of air (compression).

As the tine swings to the right, it forces the air molecules on the right to move together.

15 Production of Sound Waves Any sound wave is produced by a vibrating object that periodically compresses the surrounding air. An example for such an object is a tuning fork. As the tine swings to the right, it forces the air molecules on the right to move together. This produces a high density area in the air (compression). As the tine swings to the left, it forces the air molecules on the right to spread out. This produces a low density area in the air (rarefaction).

16 Production of Sound Waves

17 Sound Wave Spectrum v = f

18 The speed of sound in different media In a fluid the speed of sound is: s B v = Here, B is the bulk modulus of the fluid and ρ is its density. In a solid rod the speed of sound is: s Y v = Here, Y is Young s modulus. The speed of sound is generally higher in solids compared to fluids, since the atoms interact more strongly with each other in a solid.

19 The speed of sound depends on temperature The speed of sound also depends on temperature. In air: v = (331 m/s) r T 373 K Generally, the speed of sound is higher at higher temperatures.

20 Intensity of Sound Waves The average intensity, I, of a wave on a given surface is defined as the rate at which the energy flows through the surface, ΔE /Δt divided by the surface area, A: SI unit: W/m 2 The direction of energy flow is perpendicular to the surface at every point. The rate of energy transfer is the power, P. The sensation of loudness is logarithmic to a human ear. β is the intensity level or the decibel level (in db) of the sound: I0 = 1 x W/m 2 is the threshold of hearing, i.e. the faintest sound most humans can hear.

21 Intensity level in Decibels Threshold of pain: loudest sound most humans can tolerate; about 1 W/m 2 The ear is a very sensitive detector of sound waves. It can detect pressure fluctuations of about 3 parts in Decibel means one-tenth of a bel (inventor of the telephone).

22 Examples: Decibel scale Threshold of hearing: = 10 log W/m W/m 2 = 10 log(1) = 0 db 0 db does not correspond to zero sound intensity. 10 times higher intensity: 100 times higher intensity: W/m 2 = 10 log W/m 2 = 10 log(10) = 10 db W/m 2 = 10 log W/m 2 = 10 log(100) = 20 db Increasing the sound intensity by a factor of 10 adds 10 to the decibel level.

23 Spherical sound waves Sound waves are spherical waves. At each radial distance from the source the energy is spread of the surface area of a sphere (A = 4ᴨr 2 ): Moving farer away from the source, therefore, reduces the sound intensity based on an inversesquare law. The intensities at two different radii can then be compared:

Homework #4 Reminder Due Wed. 10/6

Homework #4 Reminder Due Wed. 10/6 Homework #4 Reminder Chap. 6 Concept: 36 Problems 14, 18 Chap. 8 Concept: 8, 12, 30, 34 Problems 2, 10 Due Wed. 10/6 Chapter 8: Wave Motion A wave is a sort of motion But unlike motion of particles A propagating