Lesson 15: Standing waves, Sound ( )

Size: px

Start display at page:

Download "Lesson 15: Standing waves, Sound ( )"

Flora Parsons
7 years ago
Views:

1 esso 5: Stadig waes, Soud (.-.) tererece tererece is a cosequece o the priciple o superpositio. The waes eed to be coheret same requecy ad maitai a costat phase relatioship. (For icoheret waes the phase relatio aries radomly.) Costructie itererece occurs whe the waes are i phase with each other. The amplitude o the resultig wae is the sum o the amplitudes o the two waes, A + A Destructie itererece occurs whe the waes are 8 o out o phase with each other. The amplitude o the resultig wae is the dierece o the amplitudes o the two waes, A A Otherwise the wae has amplitude betwee A A ad (A + A). The two rods ibrate ad dow i phase ad produce circular water waes. the waes trael the same distace to a poit they arrie i phase with each other ad iterere costructiely. At other poits, the phase dierece is proportioal to the path dierece. Sice oe waelegth o path dierece correspods to a phase dierece o radias, d d phasedierece rad esso 5, page

2 esso 5: Stadig waes, Soud (.-.) the path dierece d d = ( is ay iteger) the phase dierece is rad ad costructie itererece occur at P. the path dierece d d =, etc., the phase dierece is,, 5, etc. ad destructie itererece occurs. Whe coheret waes iterere, the amplitudes add or costructie itererece ad subtract or destructie itererece. Sice itesity is proportioal to the square o the amplitude, you caot simply add or subtract the itesities o the coheret waes whe they iterere. For icoheret waes, there is o ixed phase relatio. The total itesity is the sum o the itesities o the idiidual waes. Diractio Diractio is the spreadig o waes aroud a obstacle. The obstacle must be similar i size to the waelegth o the wae or the eect to be oticeable. May aimatios are o the web. Stadig Waes Stadig waes occur whe a wae is relected at a boudary ad the relected wae itereres with the icidet wae so that the wae appears ot to propagate. A wae propagatig i the +xdirectio is described by The ierted relected wae is y( x, t) Asi( t kx) y( x, t) Asi( t kx) (Why are there sies ad ot cosies?) The waes iterere ad This looks like y( x, t) Acos t si kx The places that are statioary are called odes. Midway betwee the odes are atiodes. Suppose a strig is held at both eds. esso 5, page

3 esso 5: Stadig waes, Soud (.-.) The irst our possible patters are gie. Higher orders are possible, but become less importat. (Bedig the strig takes eergy. More bedig requires more eergy.) For the top patter = ad the requecy is The secod patter = The third patter.5 = ad.5 The possible requecies are multiples o the lowest requecy which is called the udametal requecy. These are the atural requecies or resoat requecies o the strig. We will id a similar situatio whe discussig stadig soud waes i a pipe. To begi our discussio o music gie you the amazig Vi Hart: Chapter Soud Waes We study the properties ad detectio o a particular type o wae soud waes. A speaker geerates soud. The desity o the air chages as the wae propagates. esso 5, page

4 esso 5: Stadig waes, Soud (.-.) Notice that the displacemet maxima ad miima occur where the pressure ariatio is zero ad the pressure ariatio maxima ad miima occur where the displacemet is zero. The rage o requecies that ca be heard by humas is typically take to be betwee Hz ad, Hz. Most people struggle to hear the highest requecies ad that ability lesses with age. Speed o Soud Recall Restorig Force ertia luids B The speed o the wae i a luid (especially air) depeds o temperature. solids esso 5, page

5 esso 5: Stadig waes, Soud (.-.) Y Amplitude ad tesity o Soud Waes A soud wae ca be described either by talkig about pressure or displacemet. Sice the displacemet creates the pressure chage, there is a relatioship betwee the amplitude o the pressure p ad the amplitude o the displacemet s. For a harmoic soud wae the relatio is p s A larger amplitude wae appears louder, but the relatio betwee amplitude ad loudess is ery complicated. oudess is subjectie ad depeds o the respose o the ear ad the brai. Usually the itesity ad ot the amplitude is used or loudess. Agai or a harmoic wae p The most importat thig to remember is that itesity is proportioal to the amplitude squared, which is true or all waes, ot just soud. (p 7) Decibel Scale The perceptio o hearig is roughly proportioal to the logarithm o the itesity. The lowest itesity o soud that ca be heard by most people is. W/m esso 5, page 5

6 esso 5: Stadig waes, Soud (.-.) is called the threshold o hearig. t is used as the reerece leel or measurig soud itesity. The soud itesity leel i decibels is deied as ( db)log (Be sure to practice with the decibel scale. ogarithms ca be tricky.) A itesity leel o db correspods to the threshold o hearig. For icoheret soud waes with itesities ad, the total itesity is the soud waes are coheret, the waes ca iterere ad the itesity is betwee ad +, depedig o the phase relatioship betwee the two waes. Decibels ca be used i a relatie sese. The dierece i two db readigs ( db)log ( db)log ( db)log esso 5, page 6

7 esso 5: Stadig waes, Soud (.-.) is related to the ratio o the itesities. Stadig Soud Waes Recall that a stadig wae is the superpositio o two traelig waes. The wae relects at the boudary o the wae. Pipe ope at Both Eds The boudary coditios are the same at both eds. Sice the ed is ope to the atmosphere, the pressure at the eds ca ot deiate much rom atmospheric pressure. The eds are pressure odes. Pressure odes are displacemet atiodes. From the diagram, the waelegths satisy The requecies The idex is a iteger ad it ca ary rom,, etc. Pipe Ope at Oe Ed The situatio is dieret rom the pipe opeed at both eds. The closed ed is a pressure atiode. The air at the closed ed is isolated rom the atmosphere ad the pressure ca deiate ar rom atmospheric. The air at the closed ed is a displacemet ode sice the rigid wall preets the air rom moig. esso 5, page 7

8 esso 5: Stadig waes, Soud (.-.) From the diagram, the waelegths satisy The requecies This time has odd alues oly (,, 5, etc.) Problem 5 Two tuig orks A ad B, excite the ext-to-lowest resoat requecies i two air colums o the same legth, but A s colum is closed at oe ed ad B s colum is ope at both eds. What is the ratio o A s requecy to B s requecy. Sice A excites the pipe ope at oe ed, oly the odd harmoics are possible esso 5, page 8

9 esso 5: Stadig waes, Soud (.-.) esso 5, page 9 Where =,, 5, etc. Next to lowest resoat requecy reers to the secod requecy. Here that mea = ad A For B, all the harmoics are possible sice it is excitig a pipe ope at both eds. =,,, etc. Next to lowest i this sequece correspods to =, B Formig a ratio B A

Partial Di erential Equations

Partial Di erential Equations Partial Di eretial Equatios Partial Di eretial Equatios Much of moder sciece, egieerig, ad mathematics is based o the study of partial di eretial equatios, where a partial di eretial equatio is a equatio