Wave Anatomy
x y A λ 0 λ 2λ Crest (+A) Trough (−A) y = A sin(kx − ωt)SHM Position Summary
| Position | x | v | a | KE | PE |
|---|---|---|---|---|---|
| Mean | 0 | Aω (max) | 0 | Max = ½mω^{2} | 0 |
| x = A/√2 | A/√2 | Aω/√2 | −Aω^{2}/√2 | E/2 | E/2 |
| Extreme | ±A | 0 | ∓Aω^{2} (max) | 0 | Max = ½mω^{2} |
Pipe Harmonics Comparison
| Harmonic | Open Pipe (Hz) | Closed Pipe (Hz) |
|---|---|---|
| 1st (fundamental) | v/(2L) | v/(4L) |
| 1st overtone | 2v/(2L) = v/L | 3v/(4L) |
| 2nd overtone | 3v/(2L) | 5v/(4L) |
| Pattern | All n | Odd n only |
Doppler Cases Digest
| Scenario | Formula | Result |
|---|---|---|
| Source approaches | f' = fv/(v−v_S) | f' > f |
| Source recedes | f' = fv/(v+v_S) | f' < f |
| Observer approaches | f' = f(v+v_O)/v | f' > f |
| Observer recedes | f' = f(v−v_O)/v | f' < f |