7/05
Standing Waves
The musical scale used today was first discovered by the Greek mathematician
Pythagoras in the 6th century BC. While playing the lyre, he noticed that he could create the different harmonics by placing his finger at fractional points on the string. Where he placed his finger corresponded to the nodes of each harmonic.
http://www.physics.uiowa.edu/~umallik/adventure/music.htm
About this lab The physical basis of the octave (factor of two frequency ratio) can easily be seen in terms of linear (one dimensional) systems bounded at two ends to establish standing waves with either one quarter, three-quarter, five-quarter etc., or one half, two half, three half etc. wavelengths etc. A fixed wave velocity in the medium (air, water, string etc.) then implies corresponding frequencies with simple integer ratios. Such systems probably occurred naturally
(reeds, bowstrings, etc.), and were later developed into instruments specifically for music. Air sound systems do not afford control of wave velocity, but the wave velocity in strings can be tuned by selection of linear density (i.e., material) and of tension.
Bounded systems in higher dimensions, e.g,. a drumhead in two dimensions, a bell in three, also establish standing wave patterns, but without the simple numerical frequency ratios of the onedimensional systems.
The complete musical system might also include a sound box mechanically coupled to the string
(to couple sound efficiently into the air), an acoustically designed hall and the auditory system of the recipient, human or animal. The box, the hall and the ear cannot be sharply tuned, since they must reflect and absorb many frequencies.

Physical terms related to “pitch” and “loudness” would be frequency and intensity. As usual with waves, intensity is proportional to net amplitude squared (thus has half the amplitude period), where the net amplitude can exhibit interference effects between different waves.
References: Physics: Cutnell & Johnson , 6 th
Physics: , Serway & Beichner , 6
Ed., Chapters 16, 17 th Ed v.1, Chapters 16, 17, 18 (Harcourt, 2001)
Introduction
Figure 1 Experimental standing wave apparatus: String with mechanical driver and tube with speaker driver at one end. The string tension T is provided by a weight hanging over a pulley. The long white rod at the near tube end has a small microphone mounted at the far end, providing pressure wave amplitude pickup in the tube interior for oscilloscope or FFT computer display. The string standing wave patterns are detected visually.
The tube is shown in a closed-open configuration (different boundary conditions) at the two reflecting ends. The string has fixed-fixed boundaries (similar boundary conditions at the two reflecting ends). The tube end cap can be removed, converting to open-open
(similar) boundary conditions. String boundary conditions cannot be altered.
The length L of the string can be altered by sliding the slotted driver along it; the length of the tube cannot be altered, and the effective length may not be the physical length because of the coupling to the speaker driver.
The mike is connected into the lower input of the other speaker; the string driver into the upper (earphone) input. To use the mike, remove the string driver connection. Replace at end of lab, for next group.

Traveling waves have the form f(x-vt) or f(x+vt) where f stands for some function such as a sinusoid, e.g.
> sin(x-vt) or sin(x+vt).
The (x-vt) version represents a wave (state of affairs) traveling to the right (+ x direction is to the right), and the (x+vt) version represents a wave traveling to the left since (x-vt) stays the same when x increases while t does, and vise versa for (x+vt).
More generally, a sinusoidal wave could be written as
A sin ( kx  t ) where A is the amplitude, k is the “wave number” (# radians per meter at fixed time t) and ω is the angular frequency (# radians per second at fixed position x). Equivalently, in terms of the temporal periodicity f (frequency in cycles/s or hertz (Hz) and spatial periodicity λ (wavelength in meters) y = A sin (
2  x

2  ft ) and f  = v (wave propagation velocity).
For a string under tension, v =

T

(T = tension, ρ = string mass/meter)
In an ideal gas, v sound
is proportional to sqrt(Temp/m) (Temp = temperature K).
We will assume 20 0 C, for which v sound
= 343 meters/second.
Standing waves are a phenomenon of interference between two traveling waves of the same wavelength moving in opposite directions . Typically, a traveling wave is generated at one end of a linear system, and the oppositely moving traveling wave is obtained by reflection at the other end, only to be reversed again by reflection when it reaches the sending end. The standing wave pattern depends on the relative boundary conditions at the two ends: similar , producing integer number of half waves) or different , (producing odd-integer number of quarter waves, where the wavelength is the wave velocity divided by the frequency of sinusoidal excitation:
 = f
For a string under tension, similar boundary conditions would be fixed-fixed (our experimental case) or free-free; different boundary conditions might be fixed-free or free-fixed. Waves in a string are transverse (involve particle motion perpendicular to direction of traveling wave propagation) might be either vertical or horizontal (two polarization directions). Waves in a fluid are longitudinal (individual particle motion in direction of traveling wave propagation.)
For a tube, similar boundary conditions would be open-open (o-o) or closed-closed (c-c); different boundary conditions would be open-closed (o-c) or closed-open (c-o).
For similar boundary conditions, possible standing wave patterns satisfy

L = n

2
( half waves, n = 1,2,3,4,------
Equation 1: λ similar
= (2L)/n and f similar
) or
= (nv)/(2L) where L is the length of the bounded system.
For different boundary conditions
L = ( 2n  1 )

4 ( odd # of quarter waves, n = 1, 3, 5, --- or
Equation 2: λ different
= (4L)/(2n+1) and f different
= (2n+1)/(4L) .
Nomenclature: The lowest resonant frequency (longest allowed standing wavelength) is referred to as the fundamental frequency. The next higher resonant frequency (next shortest allowed wavelength) is call the second harmonic, etc.
This can lead to a little confusion. Where allowed involve half wave fits (similar boundary conditions at the two ends), the relevant expression (Equation 1) involves n = 1 2, 3, etc. and the fundamental has n = 1, the 2 nd harmonic has n = 2 etc.
But, where only odd (2n+1) quarter wave fits are allowed (different boundary conditions at the two ends), the relevant expression (Equation 2) involves n = 0, 1, 2, 3, etc. and the fundamental has n = 0, the 2 nd harmonic has n = 1, the 3 rd harmonic has n = 2 etc.
Some aspects of the production of standing waves by reflection may be seen in the following from http://www.kettering.edu/~drussell/Demos/reflect/reflect.html
(If no animation, click in center of page and wait a little.)
Figure 2 A sharp discontinuity in the wave medium typically produces both reflection and transmission.

Figure 3 “Hard” Boundary. Reflection with 180 0 phase change. Reflection coefficient is 1 and the transmission coefficient is 0.
Figure 4 “Soft” Boundary. Reflection without phase change; R coefficient = 1.

Figure 5 The blue source wave is moving right (sin(kxω t)), the green wave of the same wavelength, produced by reflection of the blue wave at the right boundary, is moving left (sin(kx+ ω t)). The reflection has only recently started; full overlap in length L has not yet been obtained.

Figure 6 Longitudinal wave in a fluid:Pressure vs. position along a tube for the longest possible standing wavelength pattern, corresponding to fundamental frequency f
1.
For different boundary conditions: one-quarter wave (¼ λ
0
) fits in length L, or λ
0
= 4L.
Faint curves are the envelope of the longitudinal pressure wave.
Since v = f λ , the frequency required for this standing wave pattern (different bc's) is f
0 diff
= v/ λ
0
= v/(4L) .
http://www.physics.smu.edu/~olness/www/03fall1320/applet/pipe-waves.html

Figure 7 Longitudinal wave in a fluid: First pressure wave overtone (second harmonic of the fundamental frequency f) given by L = three quarter-wavelengths:
L = (3 λ
1
)/4 = (3v)/(4f
1 harmonic #.
) (the subscript is the n of the (2n+1) coefficient; it is not the
Different boundary conditions at the two ends require an odd number of quarter wavelengths:
L = (1 λ
0
)/4, (3 λ
1
)/4, (5 λ
2
)/4, ----, [(2n+1) λ n
]/4 where the subscripts are n = 0, 1, 2, ---, n and each coefficient is (2n+1).
The faint curve is the envelope of the standing wave vibration produced by interference of two traveling waves moving in opposite directions. One wave is produced by reflection of the other.

Figure 8 Fundamental frequency pressure pattern for similar boundary conditions at the two ends. An integer number of half waves must fit into in length L. The same condition holds for two closed ends, though the nodes N and anti-nodes A interchange positions.
The wavelength for the fundamental frequency f
1 sim
is 2L and λ = v/f
1
, so the fundamental
frequency for similar bc's is given by f
1
sim = v/ λ
1
= v/(2L). Here again, the subscript is n,
and (for similar boundary conditions) the coefficient is n, whereas it was (2n+1) for different boundary conditions.
Standing Wave Observations
Part A Transverse Waves
String 1 Equation 1 applies (same boundary conditions at each end):
Equation 1: λ similar
= (2L)/n and f similar
= (nv)/(2L)
Briefly, you will first produce the fundamental standing wave pattern with tension T = 0.250*g , estimating the initial trial driving frequency from known tension (hanging mass*g) and assumed linear string density ρ ( 0.0036 kg/m ). Then tune f (up/down arrows: click-click = 1 Hz, click =
0.1; page up/down: 100/10 Hz)) for maximum amplitude, to find f
Measure the wavelength λ
1
and calculate the corresponding wave velocity v string. (Hanger has 50 gram mass!) Don't forget g = 9.8!
1
(fundamental frequency).
1
in the tensioned

Note: The string is elastic. If your drive amplitude is too large, the string will set the pulley rotating as the hanging mass oscillates up and down. In this case, reduce the drive amplitude (controlled by the speaker volume control).
Then estimate the resonant driving frequencies f
2
, f
3
etc. of higher harmonics, tune f to maximize amplitude, measure wavelengths and calculate corresponding v's.
For these subsequent estimates of initial trial drive frequencies, you could scale from the relation: f
2
λ
2
= f
1
λ
1
etc. (same v's).
Calculate the average wave velocity v the expression v = sqrt(T/ ρ ).
av
and use this to calculate the string linear density ρ from
String 2
For fixed L, change the properties of the wave medium (string) by increasing the hanging mass to
350 grams and repeat the previous observations as directed. The resonant f's should differ from those of string 1 because the wave propagation velocity has changed for two reasons: Increased tension T and somewhat decreased linear density ρ as the hanging mas increases (you will probably observe the string to stretch with the additional load).
Part B Longitudinal Waves
Tube 1 Equation 2 applies (different boundary conditions at the two ends):
Equation 2: λ different
= (4L)/(2n+1) >>> f diff
= (2n+1)/(4L)
The mike responds to pressure, not to longitudinal displacement, so the closed end should involve a maximum (antinode, A) and you should expect an odd number (2n+1) of quarter waves in the tube length for the resonant frequencies.
Unplug the string drive leads from the upper speaker input. Follow the same general procedure as for strings, estimating fundamental frequency from v air
= 343 sqrt[(1+T
C
/273)] and assuming  fundamental
= 4L tube
λ fundamental
= 4L tube
(four quarter waves). This time, you cannot visually observe the pressure amplitudes in order to tune. Insert the mike part way into the tube and observe the variation in height of the drive frequency peak with FFTscope in FFT mode. (Box low frequencies to zoom). First maximize the FFT peak by varying drive f, then fine tune to minimize base with of the FFT peak. IGNORE 60 Hz LINE FREQUENCY
PEAK.
Double click the FFT graph to re-expand the frequency scale.
Predict harmonic f's. For each, find resonant drive frequency as specified above. Then slide the microphone alone the tube and locate all internal nodes (record protruding portion of mike wand, take differences). Successive differences are half wave lengths ; record positions on scrap and average successive differences to determine the observed half wave length. Calculate f * λ = v for each harmonic.

Note on harmonic choice: It is easier to see the A's and N's for higher harmonics than for the fundamental or second. You might want to record #'s 3, 4, 5, 6 or try 4,5,7,8. It might be interesting to see how high you can go in harmonic number. When you start each harmonic, place the mike at a position of large amplitude and adjust the drive frequency for maximum amplitude, and then for minimum width at base. Keep the sound level as low as you can work well with. Silence FG when not in use.
Notes on FFT use: For better resolution, you must run longer (i.e., give the FFT more information about the frequency spectrum of the signal). But, with a time window of 0.1 second, looking for FFT peak base narrowing after maximizing amplitude works quite well.
Remember that the mike senses mechanical vibrations, so jerky movements will result in jittery
FFT peaks. Try to move the mike wand smoothly and slowly.
Tube 2 Equation 1 applies (same boundary conditions at each end):
Equation 1: λ similar
= (2L)/n >>> f sim
= (nv)/(2L)
Remove the end cap and repeat as in Tube 1. Here, you should expect the ends to be nodes (N) of pressure, and half waves to fit the tube length L.
FFTScope Appendix
This will provide a brief introduction to the principal capabilities of this program, which uses the computer processing capabilities and sound card to provide versatile signal generation and analysis capabilities.
Study the icons at the top and the various menus.

Time analysis of waveform
Note: Don't have two FFTScope programs open at once – confusion may result.
Open FFTSCOPE-scope mode. Turn on your speaker volume switch. Note that the horizontal axis is set for time (in milliseconds). This is appropriate for waveform display. Select the
Function Generator (FG) menu (should be silent). Select sine. Go (Enter). Reduce amplitude
(speaker volume control) to a reasonable level. Hold the microphone near the speaker. Start data acquisition, then auto scale (see above). You should hear and see the default 440 Hz (cycles per second) signal. Hold the microphone at various distances from the speaker and observe the sinusoidal waveform, possibly modulated by 60 or 120 Hz noise. Air flow mechanical noise will also show up. This is less important at higher frequencies. Observe the effect of varying the speaker volume.
Stop acquisition (Enter or icon). In FG, select Frequency and set to another value. (You must delete the former value, and you can't just Enter the new frequency - you must click OK.) Repeat observations.
Rapid changes can be accomplished with up/down arrows: (click-click = 1 Hz, click = 0.1; page up/down, 100and 10Hz.
Stop acquisition. In FG, listen to triangle, square, white noise, sweeping sine, etc. Start and observe visual representations, auto scaling as needed.
Frequency analysis of waveform
Open FFTSCOPE- FFT mode. Note that the horizontal axis is set for frequency. Select sine,
Start. Auto scale. Observe as with the waveform. You will need to expand the display to show only the low frequency portion – hold down the left mouse button, box the desired region and release. Note that the frequency analysis separates the frequency of interest from background noise at other frequencies.
Note autoscale arrows.
To re-expand the frequency scale, double click the graph.