Transcript sound and music

Music
Physics 202
Professor Vogel
(Professor Carkner’s notes, ed)
Lecture 9
Music
A musical instrument is a device for setting up
standing waves of known frequency
A standing wave oscillates with large amplitude and so
is loud
We shall consider an generalized instrument
consisting of a pipe which may be open at one or
both ends
Like a pipe organ or a saxophone
There will always be a node at the closed end
and an anti-node at the open end
Can have other nodes or antinodes in between,
but this rule must be followed
Closed end is like a tied end of string, open end is like a
string end fixed to a freely moving ring
Sound Waves in a Tube
Harmonics
Pipe open at both ends
For resonance need a integer number of ½
wavelengths to fit in the pipe
Antinode at both ends
L=½ln
v = lf
f = nv/2L
n = 1,2,3,4 …
Pipe open at one end
For resonance need an integer number of ¼
wavelengths to fit in the pipe
Node at one end, antinode at other
L = ¼l n
v = lf
f = nv/4L
n = 1,3,5,7 … (only have odd harmonics)
Harmonics
in Closed
and Open
Tubes
Adding Sound Waves
If two sound waves exist at the same
place at the same time, the law of
superposition holds.
This is true generally, but two special
cases give interesting results:
Adding harmonics
Adding waves of nearly the same
frequency
Adding Harmonics
Superposition of two or more sound
waves
that are all harmonics of the same
fundamental frequency
one may be the fundamental
The sum is more complicated than a
sine wave
but the resultant wave oscillates at the
frequency of the fundamental
simulation link
Beat Frequency
You generally cannot tell the difference
between 2 sounds of similar frequency
If you listen to them simultaneously
you hear variations in the sound at a
frequency equal to the difference in
frequency of the original two sounds
called beats
fbeat = |f1 –f2|
Beats
Beats and Tuning
The beat phenomenon can be used to tune
instruments
Compare the instrument to a standard
frequency and adjust so that the frequency of
the beats decrease and then disappear
Orchestras generally tune from “A” (440 Hz)
acquired from the lead oboe or a tuning fork
The Doppler Effect
Consider a source of sound (like a car)
and a receiver of sound (like you)
If there is any relative motion between
the two, the frequency of sound
detected will differ from the frequency
of sound emitted
Example: the change in frequency of a
car’s engine as it passes you
Stationary Source
Moving Source
How Does the Frequency
Change?
If the source and the detector are moving
closer together the frequency increases
The wavelengths are squeezed together and get
smaller, so the frequency gets larger
If the source and the detector are moving
further apart the frequency decreases
The wavelengths are stretched out and get larger
so the frequency gets smaller
Doppler Effect
Doppler Effect and Velocity
The degree to which the frequency changes
depends on the velocity
The greater the change the larger the velocity
This is how police radar and Doppler weather
radar work
Let us consider separately the situations
where either the source or the detector is
moving and the other is not
Stationary Source, Moving
Detector
 In general f = v/l but if the detector is
moving then the effective velocity is v+vD
and the new frequency is:
f’ = v+vD/l
but l=v/f so,
f’ = f (v+vD / v)
If the detector is moving away from the
source than the sign is negative
f’ = f (v  vD /v)
Moving Source, Stationary
Detector
In general l = v/f but if the source is moving
the wavelengths are smaller by vS/f
f’ = v/ l’
l’ = v/f - vS /f
f’ = v / (v/f - vS/f)
f’ = f (v/v-vS)
The the source is moving away from the
detector then the sign is positive
f’ = f (v/v vS)
General Doppler Effect
We can combine the last two equations and
produce the general Doppler effect formula:
f’ = f ( v±vD / v±vS )
What sign should be used?
Pretend one of the two is fixed in place and
determine if the other is moving towards or away
from it
For motion toward the sign should be chosen
to increase f’
For motion away the sign should be chosen to
decrease f’
Remember that the speed of sound (v) will often
be 343 m/s
The Sound Barrier
 A moving source of sound will produce wavefronts
that are closer together than normal
 The wavefronts get closer and closer together as the
source moves faster and faster
 At the speed of sound the wavefronts are all pushed
together and form a shockwave called the Mach cone
 In 1947 Chuck Yeager flew the X-1 faster than the
speed of sound (~760 mph)
 This is dangerous because passing through the shockwave
makes the plane hard to control
 In 1997 the Thrust SSC broke the sound barrier on
land
Bell X-1
Thrust SSC