Traveling waves and standing waves
Waves pass through each other and simply add — and when the additions happen between fixed ends, the result is a family of standing harmonics.
Wave anatomy
A traveling wave is a shape moving through a medium: y(x, t) = A sin(kx - omega t). The wavelength lambda sets the spatial repeat (k = 2 pi/lambda), the period T sets the temporal repeat (omega = 2 pi/T), and the wave speed ties them together: v = f lambda. On a given string the medium fixes the speed, so shaking it faster shrinks the wavelength rather than raising the speed.
v = f\lambdaTwo kinds, superposed
In a transverse wave (string, light) the medium moves across the propagation direction; in a longitudinal wave (sound) it moves along it. Waves pass through each other and add point by point: crest on crest doubles the displacement (constructive), crest on trough cancels it (destructive). Two identical waves traveling opposite directions superpose into a standing wave whose nodes never move at all.
y = y_1 + y_2Worked example
A string fixed at both ends is 2.0 m long and carries transverse waves at 8.0 m/s. The fixed ends force nodes, so only shapes fitting whole half-wavelengths survive: the fundamental has lambda = 2L = 4.0 m and f1 = v/lambda = 2.0 Hz; the second harmonic has lambda = L = 2.0 m and f2 = 4.0 Hz; the third has lambda = 2L/3 = 1.33 m and f3 = 6.0 Hz. Each harmonic adds one more loop to the pattern.
f_n = \frac{nv}{2L} = \frac{n(8.0)}{2(2.0)} = 2n\ \text{Hz}Flashcard
Wave speed ties frequency to wavelength
Check your understanding
A wave of frequency 50 Hz travels at 340 m/s (roughly the speed of room-temperature sound). What is its wavelength?