# Differences

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 phy131studiof15:lectures:chapter18 [2015/07/22 11:32]mdawber created phy131studiof15:lectures:chapter18 [2015/11/09 08:51] (current)mdawber [Interference between two speakers] 2015/11/09 08:51 mdawber [Interference between two speakers] 2015/11/09 08:51 mdawber [Interference between two speakers] 2015/11/09 08:42 mdawber [Reflection and transmission] 2015/11/09 08:41 mdawber [Reflection and transmission] 2015/11/05 16:48 mdawber [Beats] 2015/11/05 16:46 mdawber [Open and closed pipes] 2015/11/05 14:09 mdawber [Standing Waves on a String - Both ends fixed] 2015/11/05 14:07 mdawber [Interference between two speakers] 2015/07/22 11:37 mdawber [A flaming tube] 2015/07/22 11:35 mdawber [Reflection and transmission] 2015/07/22 11:34 mdawber [Principle of Superposition] 2015/07/22 11:32 mdawber created Next revision Previous revision 2015/11/09 08:51 mdawber [Interference between two speakers] 2015/11/09 08:51 mdawber [Interference between two speakers] 2015/11/09 08:42 mdawber [Reflection and transmission] 2015/11/09 08:41 mdawber [Reflection and transmission] 2015/11/05 16:48 mdawber [Beats] 2015/11/05 16:46 mdawber [Open and closed pipes] 2015/11/05 14:09 mdawber [Standing Waves on a String - Both ends fixed] 2015/11/05 14:07 mdawber [Interference between two speakers] 2015/07/22 11:37 mdawber [A flaming tube] 2015/07/22 11:35 mdawber [Reflection and transmission] 2015/07/22 11:34 mdawber [Principle of Superposition] 2015/07/22 11:32 mdawber created Line 7: Line 7: When more than one wave passes through a region in space at the same time the displacement is the sum of the separate displacements. This may be a vector sum! When more than one wave passes through a region in space at the same time the displacement is the sum of the separate displacements. This may be a vector sum! + + ===== Reflection and transmission ===== + + 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 [[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 ​ . + + + + =====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>​ + 