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phy131studiof15:lectures:chapter18 [2015/07/22 11:34]
mdawber [Principle of Superposition]
phy131studiof15:lectures:chapter18 [2015/11/09 08:51]
mdawber [Interference between two speakers]
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 A one dimensional pulse on a string which reaches the end of string will be reflected. The direction of the pulses displacement depends on the boundary condition where the reflection takes place, ie. whether the string is fixed or free.  A one dimensional pulse on a string which reaches the end of string will be reflected. The direction of the pulses displacement depends on the boundary condition where the reflection takes place, ie. whether the string is fixed or free. 
  
-This can be demonstrated on the Shive wave machine.+This can be [[http://​www.animations.physics.unsw.edu.au/​jw/​waves_superposition_reflection.htm|demonstrated on the Shive wave machine.]]
  
 When a wave encounters a change in medium there will be some partial reflection with a phase change that depends on whether it is being reflected from a more or less resistive medium ​ . When a wave encounters a change in medium there will be some partial reflection with a phase change that depends on whether it is being reflected from a more or less resistive medium ​ .
  
-We can also look at reflection in 2 dimensions, in a [[http://​www.falstad.com/​ripple/​|virtual ripple tank]]. 
  
-Here we find that the angle of reflection ​is equal to the angle of incidence.+ 
 +=====Spatial Interference ===== 
 + 
 +The equation for a traveling wave 
 + 
 +$D(x,​t)=A\sin(\frac{2\pi}{\lambda}x-\frac{2\pi t}{T})=A\sin(kx-\omega t)$ 
 + 
 +tells us that if we are standing a distance $r_{1}(x)$ from a one dimensional wave the wave displacement at a time $t$ will be 
 + 
 +$D_{1}(x,​t)=A\sin(kr_{1}(x)-\omega t)$ 
 + 
 +if a second identical source is a distance $r_{2}(x)$ away 
 + 
 +$D_{2}(x,​t)=A\sin(kr_{2}(x)-\omega t)$ 
 + 
 +and the total wave displacement is 
 + 
 +$D_{1+2}(x,​t)=A\sin(kr_{1}(x)-\omega t)+A\sin(kr_{1}(x)-\omega t)$ 
 + 
 +Using $\sin\theta_{1}+\sin\theta_{2}=2\sin\frac{1}{2}(\theta_{1}+\theta_{2})\cos\frac{1}{2}(\theta_{1}-\theta_{2})$ 
 + 
 +$D_{1+2}(x,​t)=2A\sin(\frac{k}{2}(r_{1}(x)+r_{2}(x))-\omega t)\cos(\frac{k}{2}(r_{1}(x)-r_{2}(x)))$ 
 + 
 +We thus hear a wave that has the same wavelength and frequency as that coming from the source, but the amplitude will depend on the distance between the sources, being maximum ($2A$) when $\frac{k}{2}(r_{1}(x)-r_{2}(x))$ is 0 or a whole number multiple ​of $\pi$ which corresponds to the waves at that point being in phase. The amplitude ​is minimum ($0$) when $\frac{k}{2}(r_{1}(x)-r_{2}(x))$ is a multiple of $\frac{\pi}{2}$ which corresponds ​to the waves at that point being out of phase. 
 + 
 +===== Interference between two speakers ===== 
 + 
 + 
 +If the waves propagate from the source in all 3 dimensions then we need to take in to account that  as we showed in [[phy131studiof15:​lectures:​chapter17|chapter 17]], $A\propto\frac{1}{r}$. To determine [[phy131studiof15:​lectures:​chapter17&#​loudness_and_decibels|perceived loudness]] we need to remember that it depends logarithmically on intensity. I have factored these considerations in the following calculations I performed in Maple. The patterns are for two speakers separated by 1m. 
 + 
 +{{speakerinteference.png}}\\ 
 + <​html>​ 
 +<audio controls>​ 
 + <​source src="​sounds/​490Hz30s.ogg">​ 
 +  <source src="​sounds/​490Hz30s.mp3">​ 
 +</​audio>​ 
 +</​html>​ 
 +**490Hz**\\ 
 + <​html>​ 
 +<audio controls>​ 
 + <​source src="​sounds/​686Hz30s.ogg">​ 
 +  <source src="​sounds/​686Hz30s.mp3">​ 
 +</​audio>​ 
 +</​html>​**686Hz**\\ 
 +<​html>​ 
 +<audio controls>​ 
 + <​source src="​sounds/​1143Hz30s.ogg">​ 
 +  <source src="​sounds/​1143Hz30s.mp3">​ 
 +</​audio>​ 
 +</​html>​**1143Hz**\\ 
 +<​html>​ 
 +<audio controls>​ 
 + <​source src="​sounds/​400to1200Hzramp30s.ogg">​ 
 +  <source src="​sounds/​400to1200Hzramp30s.mp3">​ 
 +</​audio>​ 
 +</​html>​**Ramp from 400 to 1200Hz**\\ 
 + 
 + 
 +===== 18.P.014 ===== 
  
 ===== Standing Waves on a String - Both ends fixed ===== ===== Standing Waves on a String - Both ends fixed =====
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 Recall that for waves on a string $v=\sqrt{\frac{F_{T}}{\mu}}$ so if you take a string and stretch it further you need to take in to account both changes in $l$ and $v$. Recall that for waves on a string $v=\sqrt{\frac{F_{T}}{\mu}}$ so if you take a string and stretch it further you need to take in to account both changes in $l$ and $v$.
 +
 +===== 18.P.016 =====
 +
  
 ===== Making Sound - String Instruments ===== ===== Making Sound - String Instruments =====
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 +===== 18.P.043 =====
 +
 +===== 18.P.047 =====
  
 ===== A flaming tube ===== ===== A flaming tube =====
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 It's worth noting that as the speed of sound is $v=\sqrt{\frac{B}{\rho}}$ and the density $\rho$ can be approximated as the propane pressure the standing wave frequencies depend on the propane pressure and will not be the same as the frequencies when the tube is just full of air at external pressure. It's worth noting that as the speed of sound is $v=\sqrt{\frac{B}{\rho}}$ and the density $\rho$ can be approximated as the propane pressure the standing wave frequencies depend on the propane pressure and will not be the same as the frequencies when the tube is just full of air at external pressure.
 +
 +
 +===== Beats=====
 +
 +
 +[[http://​www.phys.unsw.edu.au/​jw/​beats.html|Beats]] occur when two waves with frequencies close to one another interfere.
 +
 +If the two waves are described by
 +
 +$D_{1}=A\sin2\pi f_{1}t$
 +
 +and
 +
 +$D_{2}=A\sin2\pi f_{2}t$
 +
 +$D=D_{1}+D_{2}$
 +
 +Using $\sin\theta_{1}+\sin\theta_{2}=2\sin\frac{1}{2}(\theta_{1}+\theta_{2})\cos\frac{1}{2}(\theta_{1}-\theta_{2})$
 +
 +$D=2A\cos2\pi(\frac{f_{1}-f_{2}}{2})t\sin2\pi(\frac{f_{1}+f_{2}}{2})t$
 +
 +A maximum in the amplitude is heard whenever $\cos2\pi(\frac{f_{1}-f_{2}}{2})t$ is equal to 1 or -1. Which gives a beat frequency of $|f_{1}-f_{2}|$.
 +
 +===== 18.P.057 =====
 +
 +
 +===== Can you hear the beat? =====
 +
 +Typically when two tones are seperated by less than about 30-40Hz we hear beating, if the separation is more than that they tend to sound like to different tones. (You can try a similar experiment at a higher frequency at [[http://​www.phys.unsw.edu.au/​jw/​beats.html|Beats from Physclips]]).
 +
 +
 + <​html>​
 +<audio controls>​
 + <​source src="​sounds/​beats-01.ogg">​
 +  <source src="​sounds/​beats-01.mp3">​
 +</​audio>​
 +</​html>​
 +**190Hz**\\
 +<​html>​
 +<audio controls>​
 + <​source src="​sounds/​beats-02.ogg">​
 +  <source src="​sounds/​beats-02.mp3">​
 +</​audio>​
 +</​html>​
 +**191Hz**\\
 +<​html>​
 +<audio controls>​
 + <​source src="​sounds/​beats-03.ogg">​
 +  <source src="​sounds/​beats-03.mp3">​
 +</​audio>​
 +</​html>​
 +**192Hz**\\
 +<​html>​
 +<audio controls>​
 + <​source src="​sounds/​beats-04.ogg">​
 +  <source src="​sounds/​beats-04.mp3">​
 +</​audio>​
 +</​html>​
 +**194Hz**\\
 +<​html>​
 +<audio controls>​
 + <​source src="​sounds/​beats-05.ogg">​
 +  <source src="​sounds/​beats-05.mp3">​
 +</​audio>​
 +</​html>​
 +**198Hz**\\
 +<​html>​
 +<audio controls>​
 + <​source src="​sounds/​beats-06.ogg">​
 +  <source src="​sounds/​beats-06.mp3">​
 +</​audio>​
 +</​html>​
 +**206Hz**\\
 +<​html>​
 +<audio controls>​
 + <​source src="​sounds/​beats-07.ogg">​
 +  <source src="​sounds/​beats-07.mp3">​
 +</​audio>​
 +</​html>​
 +**222Hz**\\
 +<​html>​
 +<audio controls>​
 + <​source src="​sounds/​beats-08.ogg">​
 +  <source src="​sounds/​beats-08.mp3">​
 +</​audio>​
 +</​html>​
 +**254Hz**\\ ​ {{visualbeats.png}}
  
  
phy131studiof15/lectures/chapter18.txt · Last modified: 2015/11/09 08:51 by mdawber
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