Waves: Standing Waves, Beats & Doppler Effect

A progressive (travelling) wave transfers energy without transferring matter.
The general equation for a transverse wave travelling in the +x direction is $y = A\sin(kx - \omega t)$, where $A$ is amplitude, $k = 2\pi/\lambda$ is the wave number, and $\omega = 2\pi f$ is angular frequency.
Wave speed $v = f\lambda = \omega/k$.
For a string of linear mass density $\mu$ under tension $T$: $v = \sqrt{T/\mu}$.

Standing Waves form when two identical progressive waves travel in opposite directions.
Superposing $y_1 = A\sin(kx - \omega t)$ and $y_2 = A\sin(kx + \omega t)$ gives $y = 2A\sin(kx)\cos(\omega t)$. This is NOT a travelling wave — every point oscillates with amplitude $2A|\sin(kx)|$.
Points where $\sin(kx) = 0$ (i.e., $x = n\lambda/2$) are nodes (zero amplitude, maximum pressure variation).
Points where $|\sin(kx)| = 1$ (i.e., $x = (2n+1)\lambda/4$) are antinodes (maximum amplitude, zero pressure variation).
Distance between consecutive nodes = $\lambda/2$.

Vibrations of a Stretched String (Fixed at Both Ends): Boundary condition: nodes at both ends.
Allowed wavelengths: $\lambda_n = 2L/n$ for $n = 1, 2, 3, ...$.
Frequencies: $f_n = nv/(2L) = (n/2L)\sqrt{T/\mu}$.
The first harmonic ($n = 1$) is the fundamental with $f_1 = v/(2L)$. All harmonics (even and odd) are present. The sound produced is rich in overtones.

Vibrations of Air Columns:

• Closed pipe (one end closed, one open): node at closed end, antinode at open end. $\lambda_n = 4L/(2n-1)$, $f_n = (2n-1)v/(4L)$. Only odd harmonics: $f_1, 3f_1, 5f_1, ...$
• Open pipe (both ends open): antinodes at both ends. $\lambda_n = 2L/n$, $f_n = nv/(4L) \times 2 = nv/(2L)$. All harmonics present. Fundamental frequency of an open pipe = twice that of a closed pipe of the same length.

End Correction: Real pipes have an antinode slightly outside the open end. End correction $e \approx 0.6r$ where $r$ is the pipe radius.
Effective length: $L_{\text{eff}} = L + e$ (closed pipe) or $L_{\text{eff}} = L + 2e$ (open pipe).

Beats: When two waves of slightly different frequencies $f_1$ and $f_2$ ($f_1 \approx f_2$) superpose, the resultant amplitude varies periodically.
Beat frequency $f_{\text{beat}} = |f_1 - f_2|$. Maximum loudness occurs $|f_1 - f_2|$ times per second. Beats are audible only when $|f_1 - f_2| \leq 7$ Hz approximately.

Doppler Effect: The apparent frequency changes when source, observer, or medium moves. General formula (medium: air, speed of sound $v$): $f' = f\left(\frac{v \pm v_o}{v \mp v_s}\right)$ Convention: upper signs when source and observer approach each other; lower signs when they recede. If observer moves toward source: $f' = f(v+v_o)/v$ (higher). If source moves toward observer: $f' = fv/(v-v_s)$ (higher). Both approaching: $f' = f(v+v_o)/(v-v_s)$.

The key problem-solving concept is recognizing the boundary conditions (nodes vs. antinodes at each end) to determine which harmonics are allowed, and applying the Doppler formula with correct sign conventions.