Waves

$$v = f\lambda \qquad f = \frac{1}{T}$$ $v$ = speed, $f$ = frequency, $\lambda$ = wavelength, $T$ = period.

Sound

$$\text{Speed of sound} = \frac{2d}{t} \ \text{(echo)} \qquad \text{Speed of sound in air} \approx 330\text{ - }340\ \text{m/s}$$

• Closed pipe (one end closed): $\lambda = 4L$ (fundamental). Open pipe: $\lambda = 2L$.

Reflection and refraction

Angle of incidence $=$ angle of reflection. $$n = \frac{\sin i}{\sin r} \qquad n = \frac{\text{real depth}}{\text{apparent depth}} \qquad n = \frac{1}{\sin C}\ (C = \text{critical angle})$$ $$n = \frac{c_{\text{air}}}{v_{\text{medium}}} = \frac{\lambda_{\text{air}}}{\lambda_{\text{medium}}}$$

Mirrors and lenses

$$\frac{1}{f} = \frac{1}{u} + \frac{1}{v} \qquad \text{Magnification } m = \frac{v}{u} = \frac{\text{image height}}{\text{object height}}$$

• Mirrors: $f = \dfrac{R}{2}$ ($R$ = radius of curvature). Lens power $P = \dfrac{1}{f}$ (dioptres, $f$ in metres).
• Real-is-positive convention for a convex lens: $u$ and $v$ positive for real objects and images.
• Two thin lenses in contact: $P = P_1 + P_2$.

Optical instruments

• Magnifying glass (image at the near point): $M = 1 + \dfrac{D}{f}$; (image at infinity) $M = \dfrac{D}{f}$, $D = 25$ cm.
• Astronomical telescope in normal adjustment: $M = \dfrac{f_o}{f_e}$.