University of Florida  - Department of Physics

 

PHY2464 Exam 2 – Hall, Ch. 6 - 11      11 08 07      Name________SOLUTIONS_____

 

For this exam you are permitted the use of a calculator and one 8.5Ó x 11Ó sheet of notes (both sides). All cell phones off, please!     Thank you for following the instructions carefully.

 

                                Useful (?) constants:  2 1/12 = 1.05946 ;   vsound = 344 m/s

Part I.  Multiple-choice questions (5 points each):  Write the letter designating the answer of your choice in the blank to the left of the question.

 

__D__1. The sone is a unit of:    A. intensity    B. intensity level     

             C. loudness level       D. loudness          E. sound pressure

 

__C__2. .The normal vibrational modes of a  string stretched between two points:

            A. have an antinode at one of the extreme ends    B.  donÕt depend on its mass   

            C. have a node at both ends     D. are always exactly harmonic     E.  none of these

 

__D__3.  Violin strings radiate sound to the listener mainly through

             A. the strings themselves     B. the tuning pegs     C.  the neck     D. the violin body    

              E. the tailpiece

 

__E__4. The  mechanistic model of a musical instrument that we often use in our PowerPoint 

             presentations has these four Ôprimary subsystemsÕ or principal components:

             A. Energy source, player, damper, antenna       B.  player, resonator, antenna, body

             C. bridge, player, energy source, damper          D.  soundboard, tailpiece, bridge, resonator

             E. energy source, frequency generator, filter(s), antenna

 

__C__5.  The principal function of the ÔbridgeÕ on a string instrument is to

            A.  keep the strings from touching each other 

            B.  move the strings away from the soundboard                      

            C.  convert lateral string motion into vertical soundboard motion

            D.  give the player better access to the strings

            E. none of these

 

_E___6.  The musical (pitch) interval most readily recognized by humans is the

            A.  fifth    B.  fourth    C. minor third    D. major third     E. none of these

 

__D__7.  The ÒpsychophysicalÓ entities of loudness, pitch, and timbre most closely match the

           ÒphysicalÓ quantities, respectively, of:

            A.   intensity level, spectrum, frequency      B.   duration, pressure level, intensity 

            C. sonic pattern, duration, waveform            D.  intensity, frequency, waveform

            E.  loudness level, nodal persistence, damping factor

 

_A___8. The ÔpinnaÕ of the ear helps primarily with determining: 

            A.  directionality of sounds     B.  pitches      C.  loudness     D.  duration     E. none of these

 

__B__9. "Vibrato" refers to the modulation or variation in a tone's

             A.  sound pressure level      B.  frequency  C.   relaxation time   D. Pythagorean temperament 

             E. none of these

 

__A__10.   More harmonics  present in an audible tone result in:

                A, a more complex waveform      B.  increased loudness      C.  fewer Fourier components

                D.  decreased loudness       E. none of these

 

PHY2464  Exam 2 110807 p.2

 

Part II. For these questions, please write your answers and show your work (calculations) in the space provided with final answer (including proper units) circled to obtain full credit.  Points are given in (parentheses). If you need more space, use the back of the page, clearly indicating that you have done so.

 

11. (5) Explain briefly why the modern piano employs three strings per note over most of its range.

 

1) Mainly the two 'extra' strings, tuned in unison with the first, provide additional sound-energy and thus increase the loudness.  This is especially important at the upper end of the keyboard [higher frequencies] where the shorter strings  don't have a large vibrational amplitude.

 

2)  The 3 strings, via the bridge and soundboard,  exchange  vibrational energy and thus produce a collective tone of longer duration.  Also, they eventually get slightly out of tune with each other and the resulting slow beats add to the 'singing' tone.

 

 

 

 

 

12. (6) Briefly describe the advantages and disadvantages of the Ôequal-temperedÕ tuning scale.

 

Advantages: (1) Any and all keys can be played interchangeably. (2)  Thus the composer or player is free to modulate wherever they wish with no glaring discords due to keys being 'out of tune' with each other.  (3) Various instruments and families of instruments, tuned to equal temperament, can play ensemble music without discords.

 

Disadvantage: No musical intervals except the octave are 'perfect'.  Thus the intervals in any and all chords (except octaves) 'beat' with each other.

 

 

 

 

13. Use the ÔFletcher-MunsonÕ/ Ôequal loudnessÕ  curves provided on the attached sheet to answer these questions.  For these, you may take (sound pressure level = sound intensity level):

 

(a) (4).  What is the loudness level of an 80-Hz tone at 70 dB?

 

                        60 phons

 

 

 (b) (4).  For a 9000 Hz tone at the 30 phon loudness level, what is the intensity level?

 

                        40 dB

 

 

  (c) (4)  What is the intensity of a 4000 Hz tone of equal loudness level to a 60 Hz tone at 90 dB?

 

                        60 Hz @ 90 dB  -->  80 phon loudness level;

                        80 phon level @ 4 kHz--> ~70 dB

 

 

 

 

 

PHY2464  Exam 2 110807 p.3

 

14. (a) (6) The lowest note on the piano, A0 has a frequency of  27.5 Hz.  If this string on a grand piano is 2 m long and is stretched to a tension of 500 N, what is its mass in grams?

 

fn = n/(2L) Ã(T/ µ)   or for the fundamental (n=1), f = 1/(2L) Ã(T/ µ .  Hence  f2 = 1/(4L2) (T/µ )

 

and therefore µ = T/(4L2 f2)  = 500 kg m/s2 /(4 x 4m2  x 756.25 /s2 = 4.13 x 10-2 kg/m

 

4.13 x 10-2 kg/m x 2m = 8.26 x 10-2 kg,  or 82.6 grams

 

 

 

(b) (5) Is this string likely to be a solid wire?  If not, why not(?), and  briefly describe its structure.

 

No, strings for low frequencies that are strong enough to bear the tension and with enough mass to vibrate at the lower frequency tend to be thick and stiff.

Thus their higher normal modes (say, above n = 2 or 3) will be inharmonic and the string's sound will not match that of the upper notes. 

            Thus the lower strings are fairly thin, but wound with one or more layers of flexible wire (such as copper) to increase the mass without impairing the flexibility.

 

 

 

 

15. (a) (6) A  string vibrates with a third harmonic at 300 Hz.  Its tension is 30 N and its mass

             per unit length is  0.001  kg/m.  How long is the string?

 

fn = n/(2L) Ã(T/ µ) or for n = 3,  f = 3(/2L) Ã(T/ µ ; thus L = 3/(2f) Ã(T/ µ

 

= 3/ (2 x 300/s x Ã[(30 kgm/s2)/0.001 kg/m]  =3/ 2( 300 x Ã(30000 m2/s2 )

 

=3/ ( 600/s) x 173.2 m/s) =  0.866 m  or 86.6 cm

 

 

     (b) (5)  If the above string were bowed, how would you encourage production of the fifth

            harmonic?

 

It's easier to figure this out if you sketch the standing waves for this situation.

5th harmonic (n=5) means that we want 5 antinodes along the string [note that this means there are 7 nodes: one at either end and 4 along the string].  Thus each antinode's length is L/5 and the nodes are L/4 apart.   To best excite this mode , we want the excitation to be at an antinode's midpoint, or at L/10 from one end.

 

Or, bow it in the middle and touch the string at a nodal point, i. e.,  L/4 from the end.

 

 

16. (5) Explain what is meant by 'inharmonicity".  Give an example of a situation in which it might be observed.

 

Inharmonicity refers to the  fact that the normal modes of some vibrating systems are not all integral multiples of the fundamental frequency.  For instance if the fundamental frequency f1were, say, 100 Hz,  'consonant' harmonics would be 200 Hz, 300Hz, 400Hz,  etc. , i.e., fn = nf1 Inharmonicity  results when the 'overtones' do not follow this rule.

An example is the vibration of metal bars, or stiff strings.  A very stiff string might have its second mode frequency as much as 6% above that of the octave of the fundamental---close to a semitone!