Physics 116 Lecture 6 Sound Oct 7, 2011 R. J. Wilkes Email: - - PowerPoint PPT Presentation

• R. J. Wilkes

Email: ph116@u.washington.edu

Physics 116

Lecture 6

Sound

Oct 7, 2011

• Guest lecturer today: Michael Dziomba
• Wilkes will be back at 2:45 today, for office hour

(until 3:15)

Announcements

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Today

Lecture Schedule

(up to exam 1)

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Waves on strings

• End effects:

– If end of rope is fixed in wall and can’t move,

• reaction force from wall (Newton’s 3rd Law) opposes motion: upward pulse coming

toward wall becomes downward pulse leaving wall

• Pulse is inverted

– If end of rope is unconstrained and can move vertically,

• no reaction force
• pulse is reflected without inversion
• Wave speed on a rope or string depends on

– Tension in string : if F=0 wave does not propagate – Mass of string : really, mass per unit length

• Speed of wave on a string

Justify this result with dimensional analysis:

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Example

• Rope has 92 N tension and is 12 m long
• Pulse takes 0.45 s to travel length of rope
• What is total mass of rope?

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We’ve been through this already with oscillations…

• We could stand at one place and watch wave move past us vs time
• Graph of displacement vs time
• Period T = time for one cycle ("wavelength" in time units) to go past
• Frequency f = cycles passing per second (hertz, Hz) = 1/T

– This wave has 1 cycle in 1 s, so T = 1 s – Amplitude is 2 meters

T= 1 s A = 2 m

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Space and time pictures of waves

time, seconds (Here: period T=1sec)

variation with time at a fixed point in space

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• Previous picture was graph of displacement vs time at one location
• Here: Picture of rope at one instant of time (say, t=0):

– We see rope’s displacement vs position along rope (y vs x)

• Wavelength ! = length of one full cycle (distance between peaks)
• Amplitude A = maximum displacement (height)

! = 1 m A = 2 m snapshot = picture of rope, frozen at one instant of time: configuration in space at a fixed point in time

At one instant: a snapshot in time

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(Here: wavelength ! =1 m)

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Harmonic functions to describe wave motion

• To describe waves, we need to give {disturbance} = y (x, t)
• From our “snapshot” plot we see
• From our “standing in one place” plot, we see that the position of

the peak that was at x=0 at t=0, has moved to the right after time t, by a distance

• So the value of the wave function at time t is equal to the value at

time t=0 for any combination of x and t such that

• So we can describe the wave with the “harmonic function” *

* The circular functions sin/cos are “harmonic” because they can describe sound waves that are multiples of some base frequency – next time…

x t T = 0

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Example

• A transverse wave has ! = 2.6 m, and moves in the + x direction

(“to the right”) with speed 14.3 m/s

• Its amplitude is 0.11 m and it has y=0.11 m at t=0
• Give an equation describing y(x,t) for this wave

General form for a wave moving in +x direction* is (for y=A at t=0 *) For this wave, So * What if it were moving to the left (-x)? what if it had y=0 at t=0, with y increasing at that time? what if it had y= -1 at t=0, with y increasing?

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• Sound waves are an example of longitudinal waves

– Disturbance consists of periodic changes in density of the medium – At any point, material is alternately compressed and rarified – Compression peaks propagate through the medium

• Sound = compression wave in material medium (air, water, iron)
• Sound speed depends on material properties and density (so,

temperature, humidity etc)

www.kettering.edu/~drussell/Demos/waves/wavemotion.html

Sound waves

Last time:

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Speed of sound

• Speed of sound c is about 343 m/s in air (depends on air density,

temperature and humidity) = 1235 km/hr = 770 mph

– So sound travels 1 mile in about 5 sec (lightning and thunder) At 0°C, c = 331 m/s At 15°C, c = 340 m/s At 20°C, c = 343 m/s At 25°C, c = 346 m/s

• Speed is faster in denser materials:
• Example:

Pirate sees cannon on pursuing ship flash, and counts 8 seconds before he hears the boom – how far away is the Royal Navy?

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Sound frequency; pitch and musical tones

• Frequency ranges

– Audible – nominally 20 Hz to 20 kHz (actual range is closer to 50Hz-15kHz) – Infrasonic - below audible (below about 0.1 Hz we call it “vibration” !) – Ultrasonic - Above 20 kHz

• Speed of sound does not vary much with f

– If v depended on f, sound signals would change significantly depending upon how far away you are

• This is called “dispersion”

– Small f dependence can be observed, for example in undersea sound transmission

• A pulse with many frequencies in it will spread out in time as it travels
• Pitch will vary – pulse becomes a “chirp”
• Perception of sound

– Pitch = perceived frequency of sound – Associated with musical tones by our brain – JND = “just noticeable difference” in frequency ~0.4 Hz – Harmonic scales: eg in western music, “A above middle C” = 440 Hz, next A (one octave higher pitch) = 880 Hz - octave = doubling of base frequency) – “Equal temperament” scale: 12 tones per octave, each is 1.06 f of previous (factor = 12th root of 2)

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