l21 ch21 standing wavesdepartment of physics and applied physics 95.144 danylov lecture 21 standing...

Department of Physics and Applied Physics95.144 Danylov Lecture 21

Lecture 21

Chapter 21

Standing Waves

Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsII

Lecture Capture: http://echo360.uml.edu/danylov201415/physics2spring.html

Physics II


Standing Waves ( long string)

We’ve introduced traveling waves. Now, let’s consider two waves traveling in opposite directions on the same infinite string with the same amplitude (a), frequency (), and wavenumber (k)

Since, they are on the same string, let’s add them:

2 2 2Let’s use a trig identity:

, 2 )

Look, now the timing info is separated from the spatial.

We can treat 2 as an amplitude which changes with x.

, ) This is an equation of a standing wave

Let’s plot it.


Plotting Standing Waves ( long string)

First, let’s find where the amplitude becomes zero: 2 0

where m=0,1,2,3,…m m 2These points stay always still at y=0.They are called NODES

Nodes

x

t=T/2t=T/4

t=02a

‐2a

, 2 )Second, let’s look at the amplitude at different moments of time (T is a period)

t=0

t=T/4

t=T/2

, 0 2

, /4 2 ) 22

40

, /2 22

22

AntiNodes

λ

2

,2


Standing Waves (string with boundaries)

So, the theory for an infinitely long string with two sources at infinity (same amplitude, frequency) gives us standing waves at any frequency.

However, the theory for an infinitely long string is not very practical since most of the strings are finite and fixed at the ends (like in a violin, guitar, etc)

So, we need to adjust the theory to include boundaries (boundary conditions)

What happens at the boundaries (walls)?

Assume there is a wave traveling to a boundary. After reflection from the wall, it preserves its amplitude and frequency.So, now there are two waves (incident and reflected) traveling in the opposite directions. And that is what we had in our theory, but we need to impose the fact that the ends are tied (boundary conditions)


Resonant Frequencies (string with boundaries)

The frequencies at which standing waves are produced are called resonant frequencies.Let’s find them now. Consider a string of length L tied at the ends to walls.

Apply boundary conditions (ends are tied)1 0 02 0

, 2 )

Condition 1 is automatically satisfied.

We need to impose them on our equation

, 02 ) Since it must be true for any time (t), then

0 , where m=1,2,3,…

2

or since 2

Thus, only at these wavelength/frequencies standing waves can exist.

resonant frequencies

Let’s apply the second one:

2

2 2


The first four possible standing waves on a string

These possible standing waves are called the normal modes of the string.

Each mode, numbered by the integer m, has a unique wavelength and frequency.

The lowest allowed frequencyfundamental frequency (mode 1).

The fundamental frequency f1 can be found as the difference between the frequencies of any two adjacent modes: f1 = Δf = fm+1 – fm.

22

, where m=1,2,3,…

m=121

12

m=222

2 2

m=323

3 2

The second harmonic (mode 2).

The third harmonic (mode 3).

m=424

4 2The forth harmonic (mode 4).

2

3

4


Standing Wave Generation (Demo)

There are nodes, where the amplitude is always zero, and antinodes, where the amplitude varies from zero to the maximum value.

What is the mode number of this standing wave? A) 4

B) 5C) 6D) Can’t say without knowing what kind of wave it is.

Number of antinodes = mode number

ConcepTest Standing Wave


A guitar/ViolinWhen you pluck a string of a guitar, that is exposing the string to a whole set of frequencies.And so the string, now, decides which frequencies it like to oscillate in.And so it selects these resonance frequencies.And so if the string has a fundamental of 400 hertz, then it would start to resonate at 400, but simultaneously, it will be very happy with 800 hertz, and with 1200 hertz.

/

Wave velocities of a string

Different mass

12

You can make strings out of different material ‐‐ different mass per unit length ‐‐ and so that gives you, then, difference velocities ‐‐ you can also fool around with the tension ‐‐ and so the six strings, then, have all six different fundamental frequencies.

All that is left over is L, that's the only thing you can change, and that's what a player is doing.

Goes with the finger, back and forth over the strings, make them shorter, pitch goes up, frequency goes up, makes them longer, frequency goes down.

2


A guitar/Violin

Stringed instruments would not be very loud if they relied on the vibrating strings to produce the sound waves since the strings are too thin to compress and expand much air.

Therefore, they make use of a kind of mechanical amplifier known as a sounding board (piano) or sounding box (guitar, violin)

You may think it is much easier to play the piano that to play a violin, because you don't have to change L all the time, and be exactly at the right length.Well, that is true, of course, but given the fact that you have 88 keys, you can imagine you can hit occasionally the wrong key, and that's not what you want.

Standing waves can sometimes be destructive:

Tacoma Narrows bridge was collapsed because of standing waves.

https://www.youtube.com/watch?v=j‐zczJXSxnw

It collapsed the morning of November 7, 1940, under high wind conditions 4 month after its construction.

At a football game, the “wave” might circulate through the stands and move around the stadium. In this wave motion, people stand up and sit down as the wave passes. What type of wave would this be characterized as?

A) polarized waveB) longitudinal waveC) lateral waveD) transverse waveE) soliton wave

The people are moving up and down, and the wave is traveling around the stadium. Thus, the motion of the wave is perpendicular to the oscillation direction of the people, and so this is a transverse wave.

ConcepTest The Wave


What you should readChapter 21 (Knight)

Sections 21.5 21.6 21.7


Thank youSee you on Friday

l21 ch21 standing wavesdepartment of physics and applied physics 95.144 danylov lecture 21 standing...

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