standing waves in strings - memorial university of...

Standing Waves in Strings

1

The two ends must be nodes (because they are fixed).

Nodes (N) and anti-nodes (A) are separated by λ/4

For the nth normal mode: L=n�

2We call this value :�n

�n=2L

n f n=v

�n=n

2Lv=

n

2L � T�=nf 1 n=1,2,3,...


2

Example 14.3

A middle C string on a piano has a fundamental frequency of 262 Hz and the A note has a frequency of 440 Hz.

[A] Calculate the frequencies of the next two harmonics of the C string.

f n=nf 1

f 2=2f 1=524Hz

f 3=3f 1=786Hz


3

Example 14.3


[B] If the strings for the A and the C notes are assumed to have the same mass per unit length and the same length,

determine the ratio of the tensions in the two strings.

f 1A=vA

2L=

1

2L � T A� f 1C=vC

2L=

1

2L � T C�Note:

Velocity of the transverse wave for the A note is NOT the same as the velocity for the C note.

The wavelength of the fundamental is 2L in both cases.


4

Example 14.3


[B] If the strings for the A and the C notes are assumed to have the same mass per unit length and the same length,

determine the ratio of the tensions in the two strings.

f 1A

f 1C

=� T AT C

Key: If one takes ratios of frequencies the common constants � and L get divided out.

So: and:T A

T C=f 1A

2

f 1C

2=

4402

2622=2.82

To tune it to A you will need to make the string ~ 3 times as taut.

Differences with Standing Waves in Strings and Pipes

5

Standing waves in strings are transverse waves

Standing waves in pipes are longitudinal sound waves

Note: You can only �hear� longitudinal waves, because transverse waves do not propagate through air or liquid, only through solid.

To �hear� the string, the transverse waves on the string have to excite longitudinal sound waves in the air which then reach your

ear drum �

Differences with Standing Waves in Strings and Pipes

6

Strings: Ends are always nodes

Pipes: � Closed ends are nodes� Open ends are anti-nodes

DidgeridooOne open end & one closed end

Bamboo fluteTwo open ends

Image from grapevineroad.orgImage from i235.photobucket.com

Standing Waves in Pipes (Air Columns)

7

http://www.youtube.com/watch?v=QSXjpWUDvO4

Wikipedia: For humans, hearing is normally limited to frequencies between about 12 Hz and 20,000 Hz (20 kHz),

although these limits are not definite.

The didgeridoo fundamental frequency is closer to the bottom end � I estimate the one we heard to be about 80-90 Hz.

After hearing this, you decide to make yourself a didgeridoo with a PVC pipe from the hardware store.

How long should this pipe be in order to have a fundamental frequency of 80 Hz?

Standing Waves in Open Pipes

8

The two ends must be anti-nodes.


For the 1st normal mode: L=�

2We call this value :�1

�1=2L

NAA

L


9

The second normal mode has 1 anti-node in the centre as well.


For the 2nd normal mode: L=�

We call this value :�2

�2=L

N NAA A

L


10

The third normal mode has 2 anti-nodes in the centre as well.


For the 3rd normal mode:L=3

�


�3=2L

3

N NA AN AA

L


11

For the nth normal mode, there will be n-1 antinodes in between.


For the nth normal mode: L=n�


�n=2L

nf n=

v

�n=n

v

2L=nf 1 n=1,2,3,...

NA

NA

NA

NA

NAA

L

Notation

12

Standing Waves in (both-end) Open Pipes

First Harmonic: n = 1Second Harmonic: n = 2

Third Harmonic: n = 3

f n=v

�n=n

v

2L=nf 1 n=1,2,3,4,...

Standing Waves in Pipes:One end open, one end closed

13

Open end: anti-nodeClosed end: node


For the 1st normal mode: L=�


�1=4L

AN


14



For the 2nd normal mode: L=3�


�3=4L

3

AN

AN


15



For the 3rd normal mode: L=5�


�5=4L /5

AN

A ANN


16



In general for n = 1, 3, 5, 7 ....L=n�


�n=4L /n

AN

A ANN

f n=v

�n=n

v

4L=nf 1 n=1,3,5,...

AN

Notation

17

Standing Waves in (both-end) Open Pipes

First Harmonic: n = 1Second Harmonic: n = 2

Third Harmonic: n = 3

f n=v

�n=n

v

2L=nf 1 n=1,2,3,4,...

Standing Waves in one-end-Open one-end-Closed Pipes

f n=v

�n=n

v

4L=nf 1 n=1,3,5,7,...

First Harmonic: n = 1Third Harmonic: n = 3Fifth Harmonic: n = 5

Only odd harmonics are present !

The PVC Didgeridoo

18

We want the fundamental frequency to be 80 Hz.

One end of the pipe is effectively closed (at your mouth)

f n=v

�n=n

v

4L=nf 1 n=1,3,5...

L=v

4f 1

=343m.s

�1

320 s�1

=1.1m

A PVC middle-C Flute

19

We want the fundamental frequency to be middle-C : 262 Hz.

You blow across the hole: so the hole is actually open to the atmosphere. So this would be a two-end-open pipe.

f 1=v

2L

L=v

2f 1

=343m.s

�1

2�262 s�1=0.65m

�Beats�: The Sum of Two Sine Waves of Slightly Different Frequency

y1�t�=Asin �kx�1 t�

1=2 f 1 2=2 f 2 �=1�2=2� f 1� f 2�

y2 �t �=Asin �kx�2 t �

=A sin�kx��t �


21

�


22

y1�t�=Asin �kx�1 t�y2 �t �=A sin �kx�2 t�= A sin �kx��

1��t �

0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0

� 1 0

� 5

5

1 0

0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0

� 1 0

� 5

5

1 0

+


23

y1�t�=Asin �kx�1 t�y2 �t �=A sin �kx�2 t�= A sin �kx��

1��t �

0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0

� 2 0

� 1 0

1 0

2 0

=

How about Square Waves (instead of sines and cosines)?

24

25

�0.09 A1 sin2�5f 1�t�0.0027 A1 sin2�7f 1� t

Constructing a Square Wave from a sum of sines...

26

Harmonics :f 1 ,3 f 1 y �t �=A1 sin2 f 1t�0.3�A1 sin2�3f 1�t


27

Harmonics :f 1 ,3 f 1, 5f 1

y �t �=A1 sin2 f 1t�0.3�A1 sin2�3f 1�t

�0.09 A1 sin2�5f 1�t


28

Harmonics :f 1 ,3 f 1, 5f 1

y �t �=A1 sin2 f 1t�0.3�A1 sin2�3f 1�t

�0.09 A1 sin2�5f 1�t�0.0027 A1 sin2�7f 1� t

standing waves in strings - memorial university of...

Documents