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Standing waves, aluminium, speed of sound

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LFS
#1
Mar31-11, 12:05 PM
P: 7
1. I am a mathematician trying to understand a physics experiment where an aluminum rod is held in the middle and struck with a mallet. The rod is then placed next to a microphone, the sound recorded and the frequency determined using Audacity. Working backwards, one determines the speed of sound in aluminum.

2. I made this applet using GeoGebra: http://geogebrawiki.wikispaces.com/Speed+of+Sound
Most important question: Do I have it right?

3. Questions - Believe me, I have spent over 10 days researching these questions so any answers here would be GREATLY appreciated.
(a) Is there only the one standing wave with this amplitude or do we also get the upside down one (it seemed like it would cancel the one I graphed so I took it off, but I looked at other pictures and they have both).
(b) Are there other standing waves of different amplitudes? What does amplitude mean here?
(c) Is the sound wave 2D or does it wrap itself around the rod (i.e. the rod is 3D, is the wave only 2D and if so on what plane)?
(d) Clearly the length of the wave gets longer as more nodes appear. However the time from one end of the rod to the other is the same (speed of sound). Does the fact that a point on the wave travels faster have any physical significance? (As I understand frequency, it is the number of times a point peaks per second. But does the actual "linear speed" of the point mean anything other than relating frequency and amplitude?)

Thank-you!
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ehild
#2
Mar31-11, 08:24 PM
HW Helper
Thanks
P: 10,344
A wave means a function which depends both on space and time through the argument (r-vt) In case of a one-dimensional body (rod) along the axis x it is f(vt-x), v is defined as the speed of the wave. Physicist usually consider the wave as the sum of their Fourier components

f=Asin(ωt-kx+φ).

ω is the angular frequency, ω=2π f ( f is the frequency), φ is just a phase constant, and k=2π/λ (λ is the wavelength). A is called the amplitude of the wave. It is the highest value of the oscillating quantity which is a displacement of some kind. This displacement can be along the propagation direction (longitudinal wave) or normal to it (transversal wave). The sound in air is a longitudinal wave (change of pressure or velocity of air particles) but the vibrations of the rod can produce both transversal and longitudinal waves. From ωt - kx= constant you get the velocity of propagation as v=x/t=ω/k = λ f, so λ is the distance the wave travels in one period of time.

If case of different media like a rod in air, the motion of the rod is described by the sum of two waves, travelling in opposite directions. If the length of the rod matches the wavelength and both waves have the same amplitude, you get standing waves of form B sin(ωt) sin( kx+φ). The standing wave pattern depends on the boundary conditions. If one end of the rod is free, the phase of the reflected wave is the same as the "travelling" wave, which results in a maximum displacement (antinode) at that end. If the end is fixed, the wave reflects with pi phase change, and you get a node (zero displacement) at the end.

In your case, both ends are free, so there are maxima at the ends and one of more nodes in between. The distance between to subsequent maxima or two nodes is half the wavelength, λ/2. Remember the wavelength is related to the frequency and speed of propagation as λ=v/f.

ehild


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