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_{PH203KUZMASPRING2014}
Video 2.1: Ruben's tube, standing wave with explanation. | Video 2.2: Standing waves on a 2D surface. |
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From: Walker, James S. “Ch. 14 Waves and Sound.” Physics. Upper Saddle River, NJ: Pearson/Prentice Hall, 2004
Table 14.1: Speed of sound in various materials | ||
Material | Temperature | Speed of sound (m/s) |
---|---|---|
Aluminum | 6420 | |
Granite | 6000 | |
Steel | 5960 | |
Pyrex glass | 5640 | |
Copper | 5010 | |
Plastic | 2680 | |
Fresh water | 20 $^\circ$C | 1482 |
0 $^\circ$C | 1402 | |
Hydrogen | 0 $^\circ$C | 1284 |
Helium | 0 $^\circ$C | 965 |
Air | 20 $^\circ$C | 343 |
0 $^\circ$C | 331 |
Addition of trig functions | Doppler equation approximation for slow observer/source speeds $u\ll v$ | |||
---|---|---|---|---|
$\sin\alpha+\sin\beta=$ $2\sin\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)$ | $\left(1+\frac{u}{v}\right)^2$ $\approx 1+2\frac{u}{v}$ | $\left(1-\frac{u}{v}\right)^2$ $\approx 1-2\frac{u}{v}$ | ||
$\cos\alpha+\cos\beta=$ $2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)$ | $\left(1+\frac{u}{v}\right)\left(1-\frac{u}{v}\right)$ $\,=\, 1-\left(\frac{u}{v}\right)^2$ $\,\approx\, 1$ | $\left(1+\frac{u_1}{v}\right)\left(1-\frac{u_2}{v}\right)$ $\approx 1+\frac{u_1-u_2}{v}$ | ||
$\cos\alpha+\sin\beta=$ $-2\sin\left(\frac{\alpha+\beta}{2}+\frac{\pi}{4}\right)\sin\left(\frac{\alpha-\beta}{2}-\frac{\pi}{4}\right)$ | $\left(\frac{1}{1+\frac{u}{v}}\right)$ $\approx 1-\frac{u}{v}$ | $\left(\frac{1}{1-\frac{u}{v}}\right)$ $\approx 1+\frac{u}{v}$ | ||
$\cos\alpha-\cos\beta=$ $-2\sin\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)$ | $\left(\frac{1+\frac{u}{v}}{1-\frac{u}{v}}\right)$ $\approx 1+2\frac{u}{v}$ | $\left(\frac{1-\frac{u}{v}}{1+\frac{u}{v}}\right)$ $\approx 1-2\frac{u}{v}$ | $\left(\frac{1+\frac{u_1}{v}}{1-\frac{u_2}{v}}\right)$ $\approx 1+\frac{u_1+u_2}{v}$ |
Please update if an equation is not included
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