17.4 Beats

Two overlapping waves with slightly different frequencies gives rise to the phenomena of beats.

1 Traveling waves with slightly different frequencies

The two displacements at a particular location vs. time

http://www.kettering.edu/physics/drussell/Demos/superposition/super4.gif

http://www.kettering.edu/physics/drussell/Demos/superposition/super4.gif

17.4 Beats

The beat frequency is the difference between the two sound frequencies. 2

The math of beats At a particular location:

)2sin()2sin()()()( 2121 tfAtfAtytyty ππ +=+=

We now make use of a trigonometric identity:

[ ]

( )tftfAy

tfftffA

tfAtfAty

2sin2

2cos2

22sin

22cos2

)2sin()2sin()(

2121

21

ππ

ππ

ππ

∆=⇒

+

−=

+=An identity is an equation that is always true regardless of the value(s) of the arguments involved.

Result is a wave with a frequency equal to the average of f1 and f2, but modulated by a sinusoidal envelope of frequency (f1- f2)/2. But the sound is loudest when the envelope function is either maximum or minimum.

3

Multi-Concept Example: Two speakers are set up along the x-axis 25 m apart, both playing a 512 Hz pure tone. Batman is walking at speed u away from one speaker and toward the second. At half way between the speakers, he hears a beat frequency between the two speakers at 4.0 Hz. The speed of sound is 343 m/s. At what speed u is Batman walking?

17.4 Beats

1 2 u

4

Multi-Concept Example: Two speakers are set up along the x-axis 25 m apart, both playing a 512 Hz pure tone. Batman is walking at speed u away from one speaker and toward the second. At half way between the speakers, he hears a beat frequency between the two speakers at 4.0 Hz. The speed of sound is 343 m/s. At what speed u is Batman walking?

17.4 Beats

Solution: Batman represents a moving observer away from stationary source 1, so he hears frequency f1 from speaker 1:

−=

vuff s 11

And: Batman represents a moving observer toward stationary source 2, so he hears frequency f2 from speaker 2:

+=

vuff s 12

The sound from the two speakers will interfere, and produce beats of frequency

Hz)2(512 Hz)m/s)(4.0 343(

22

2

1121

==→=→

−=−−−=

+−

−=−=∆=

s

BsB

sssss

ssB

fvfuf

vuf

fvuf

vuff

vuf

vuf

vufffff

m/s 34.1=⇒ u

1 2 u

5

17.5 Transverse Standing Waves

Two waves with the same amplitude and frequency traveling in opposite directions on the SAME medium interfere in such a way as to produce a stationary oscillating pattern This is called a Standing Wave (e.g. the line segment between the two bobs in the ripple tank)

yAgain: the locations where you always have destructive interference are called “nodes”. The locations with maximum constructive interference are “anti-nodes”

[ ])/22sin()/22sin(),(),(),()/22sin(),( ),/22sin(),(

λππλππλππλππ

xftxftAtxytxytxyxftAtxyxftAtxy++−=+=

+=−=

−+

−+

The math of Standing Waves: the rightward and leftward waves are given, respectively:

We again use the identity:

[ ] )2sin()/2cos(2),()2sin()/2cos(2),(ftxAtxy

ftxAtxyπλπ

πλπ=⇒

−=

http://www.kettering.edu/physics/drussell/Demos/superposition/super3.gif

http://www.kettering.edu/physics/drussell/Demos/superposition/super3.gif

6

17.5 Transverse Standing Waves

It is possible to produce self-sustaining standing waves on nearly ideal (very little energy loss as the wave propagates) by taking advantage of reflections There are two special cases of reflections: (1) Reflection from a fixed end (where y=0 always)

Click to start movie http://www.youtube.com/watch?v=LTWHxZ6Jvjs

http://www.kettering.edu/physics/drussell/Demos/reflect/hard.gif

http://www.youtube.com/watch?v=LTWHxZ6Jvjshttp://www.kettering.edu/physics/drussell/Demos/reflect/hard.gif

7

17.5 Transverse Standing Waves

(2) Reflection from a free end (where the slope of the tangent ∆y/∆x=0 always)

Click to start movie

http://www.youtube.com/watch?v=aVCqq5AkePI

http://www.kettering.edu/physics/drussell/Demos/reflect/soft.gif

http://www.youtube.com/watch?v=aVCqq5AkePIhttp://www.kettering.edu/physics/drussell/Demos/reflect/soft.gif

17.5 Transverse Standing Waves

8

By providing small impulses at one end, one can generate a large standing wave from the reflections (the hand in this case is very nearly a node), provided that the frequency is right , such that the end (boundary) conditions are satisfied The boundary conditions depend on the type of ends: (1) Fixed end = node (y=0 always) (2) Free end = anti-node (and ∆y/∆x=0 always) Case A: two fixed ends (all string musical instruments are made this way)

When you get the right frequencies where small stimuli maintains a large standing wave: we have what is called resonance All musical instruments depends on resonance to generate sound og a definite pitch/frequency.

http://www.youtube.com/watch?v=jovIXzvFOXo

http://www.youtube.com/watch?v=jovIXzvFOXo

17.5 Transverse Standing Waves

,4,3,2,1 2

=

= n

LvnfnString fixed at both ends 9

f1 f2=2 f1 f3=3 f1

First Harmonic n = 1

Second Harmonic n = 2

Third Harmonic n = 3

==→=

LvvfL

21

2 11

1

λλ

==→=

LvvfL

22

22

12

2

λλ

==→=

LvvfL

23

23

13

3

λλ

L=length of string The progression of harmonics adds ONE node at a time

http://www.physicsclassroom.com/class/waves/u10l4eani1.gif

http://www.physicsclassroom.com/class/waves/u10l4eani1.gif

17.5 Transverse Standing Waves

,4,3,2,1

2

=

=

nLvnfn

10

Guitar players shorten the “open” string to produce sound of a higher frequency than the open string: as L decreases, f increases

Conceptual Example 5 The Frets on a Guitar Frets allow a the player to produce a complete sequence of musical notes on a single string. Starting with the fret at the top of the neck, each successive fret shows where the player should press to get the next note in the sequence. Musicians call the sequence the chromatic scale, and every thirteenth note in it corresponds to one octave, or a doubling of the sound frequency. The spacing between the frets is greatest at the top of the neck and decreases with each additional fret further on down. Why does the spacing decrease going down the neck?

17.6 Longitudinal Standing Waves

11

Longitudinal standing waves can also be generated using a speaker and a spring This set up gives roughly the equivalent situation of two fixed ends (except the bottom end is not quite a node) 2nd harmonic: ~32 Hz: one node in the middle between ends 4th harmonic: ~64 Hz three nodes between ends 6th harmonic: ~96 Hz five nodes between ends

http://www.youtube.com/watch?v=12pjjPIE2IQ

http://www.youtube.com/watch?v=12pjjPIE2IQ

17.6 Longitudinal Standing Waves

,4,3,2,1 2

=

= n

LvnfnTube open at both ends 12

Most wind instruments use sound waves resonating longitudinally in columns of air. 2 cases: (1) Two open (free) ends e.g. a flute The movie clip shows the equivalent transverse harmonic

=

=

Lvf

L

2

2

1

1λ

=

=

Lvf

L

22

22

2

2λ

http://www.youtube.com/watch?v=7_GeW73SGnc

http://www.youtube.com/watch?v=7_GeW73SGnc

17.6 Longitudinal Standing Waves

Example: Playing a Flute

When all the holes are closed on one type of flute, the lowest note it can sound is middle C (261.6 Hz). If the speed of sound is 343 m/s, and the flute is assumed to be a cylinder open at both ends, determine the distance L.

13

17.6 Longitudinal Standing Waves

Example: Playing a Flute When all the holes are closed on one type of flute, the lowest note it can sound is middle C (261.6 Hz). If the speed of sound is 343 m/s, and the flute is assumed to be a cylinder open at both ends, determine the distance L.

14

,4,3,2,1 2

=

= n

Lvnfn

( )( ) m 656.0Hz 261.62

sm34312

===nf

nvL

Method 2: memorize the appropriate formula (of the three cases), and apply it

Method 1: look at the physical situation: open ends anti-nodes at both ends. Lowest frequency: only one node between the ends (and only one node total)

( )( ) m 656.0 Hz261.62

sm3432

22/

221

===

===→=∴

fvL

fvfvLL λλ

Here n=1 for lowest frequency

17.6 Longitudinal Standing Waves

,5,3,1 4

=

= n

LvnfnTube open at one end 15

(2) One open (free) end e.g. a medieval trumpet The movie clip shows the equivalent transverse harmonics, starting at the 2nd harmonic (n=3). **N=1 is too difficult

=

=

Lvf

L

4

4

1

1λ

=

=

Lvf

L

43

43

3

3λ

Even n modes do not exist

http://www.youtube.com/watch?v=DWhCdlPM19M

http://www.youtube.com/watch?v=DWhCdlPM19M

17.4 Beats17.4 BeatsSlide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 717.5 Transverse Standing Waves17.5 Transverse Standing Waves17.5 Transverse Standing Waves17.6 Longitudinal Standing Waves17.6 Longitudinal Standing Waves17.6 Longitudinal Standing Waves17.6 Longitudinal Standing Waves17.6 Longitudinal Standing Waves