The Basics of Waves

Direction button: change wave direction
Longitudinal button: show/hide longitudinal wave

Combining Waves

• The black curve represents the sum of the blue and the red waves. When the two wavelengths are almost but not exactly the same, the total wave display a pattern called "beats".
• If you set the wavelengths and amplitude of both waves to be the same and reverse the direction of propagation of the blue wave, you will see a "standing wave".
• We fixed the wave speed to be $v=1m/s$. Since $v=\omega/k$, in the equations below you will see $\omega$ and $k$ have the same numerical values as a result.
• Reminder: $k=2\pi/k$.

Waves with Different Boundary Conditions

Since the wave speed $v=f\lambda$ is fixed by the string, as $\lambda$ changes, the frequency must change as well. This explains why a tube (e.g. a flute) produces different sound depending on the boundary condition and the length. A shorter tube produces higher frequency.

• Waves on a string depends on the boundary conditions.
• The end of a string could be either movable (open) or fixed (shown as a blue dot above).
• Both ends fixed: $L = n \frac{\lambda}{2}$
• One end open, one end fixed: $L = (n-1) \frac{\lambda}{2} + \frac{\lambda}{4}$
• Both ends open: $L = n \frac{\lambda}{2}$

The Doppler Effect

Adjust the velocity of the source and see how it affects the wavefronts.
Gradually change the source velocity from below the speed of sound to above. Could you see any qualitative differences between the two cases?